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Qm Math Magnetic And Detection

1 · Prerequisites

2 · Summary

The magnetic and detection layer of the non-relativistic quantum mechanics mathematical prerequisites, all of it mathematics: the position and momentum operators on Euclidean $L^{2}$ with their maximal domains, the bounded-coefficient magnetic Schrodinger operator and its Kato-Rellich self-adjointness, gauge covariance and the gauge-transported propagator, the smooth local continuity equation, the coherent and incoherent two-path bin laws, covariance-refined and maximal-domain preparation uncertainty, Galilean boost covariance, conserved spectral probability laws, graph-domain Ehrenfest identities with the conditional timescale bound, and the conditional Stern-Gerlach force. Every item carries domain: mathematics and depends only on mathematical suppliers, published or from the in-run pages qm-math-composites, qm-math-oscillator, qm-math-orbital-and-spin and qm-math-intervals-and-contacts; nothing here asserts a physical interpretation, a measurement model or an empirical claim, and the companion page qm-math-magnetic-and-detection-examples stays mathematics as well.

Position and momentum operators on Euclidean L2 fixes the notation: $Q_{j} f = x_{j} f$ on the classes with $x_{j} f \in L^{2}$ , and $P_{j} f = - i ℏ D_{j} f$ on the classes whose $j$ -th weak derivative is an $L^{2}$ class. Self-adjointness, Fourier representation and translation identities of position and momentum proves that both are self-adjoint, that $D (P_{j}) = {f : ξ_{j} F_{2} f \in L^{2}}$ with $F_{2} (P_{j} f) = 2 π ℏ ξ_{j} F_{2} f$ , and that translations $τ_{a} f (x) = f (x - a)$ are unitary, preserve $W^{1, 2}$ and $W^{2, 2}$ , and satisfy $τ_{a} P_{j} τ_{a}^{- 1} = P_{j}$ , $τ_{a} Q_{j} τ_{a}^{- 1} = Q_{j} - a_{j}$ . The analytic workhorse is Weak product rule for a bounded coefficient with bounded weak derivative: a bounded coefficient with bounded weak first derivatives multiplies $W^{1, 2}$ into $W^{1, 2}$ with the Leibniz derivative, proved by density of compactly supported smooth classes.

Bounded-coefficient magnetic Schrodinger operator fixes the admissible magnetic data $(A, Φ)$ and the operator $K_{A, Φ} u = \frac{1}{2 m} \sum_{j} P_{A, j} (P_{A, j} u) + q Φ u$ on $W^{2, 2}$ , and Self-adjointness of the bounded-coefficient magnetic Schrodinger operator proves that $P_{A, j} = P_{j} - q A_{j}$ is self-adjoint on $D (P_{j})$ , that $K_{A, Φ}$ is symmetric and self-adjoint on $W^{2, 2}$ , that the expansion $K_{A, Φ} = \frac{1}{2 m} P^{2} - \frac{q}{2 m} \sum_{j} (P_{j} A_{j} + A_{j} P_{j}) + \frac{q^{2}}{2 m} ∣ A ∣^{2} + q Φ$ holds, and that $K_{A, Φ} - K_{0}$ has $K_{0}$ -bound below one, giving a lower bound. Gauge multiplication preserves H2 and transports the magnetic momentum then proves that the gauge multipliers $G (t) = e^{i \frac{q}{ℏ} χ (\cdot, t)}$ are unitary, preserve $W^{1, 2}$ and $W^{2, 2}$ , satisfy the covariance identity $(P_{j} - q (A_{j} + D_{j} χ)) G = G (P_{j} - q A_{j})$ , and are strongly continuous and differentiable in $t$ ; Gauge transported propagator for the bounded-coefficient magnetic operator turns this into the transported operator identity $H^{'} (t) = G (t) K_{A, Φ} G (t)^{- 1} - q \partial_{t} χ$ , the explicit propagator $G (t) e^{- i (t - s) K_{A, Φ} / ℏ} G (s)^{- 1}$ , and the uniqueness of the $H^{'}$ -dynamics in the stated class.

Smooth local continuity equation and its integrated forms records the pointwise continuity equation $\partial_{t} ∣ ψ ∣^{2} + div j = 0$ for $C^{2}$ solutions of the Schrodinger equation, its integrated form on bounded $C^{1}$ regions, global norm conservation under domination whenever the two times lie in one interval contained in the time domain, and the unitary-dynamics norm conservation as an independent statement. On the detection side, Coherent and incoherent two-path bin laws and the marker overlap compares the coherent law $∥ a + b ∥^{- 2} \int_{B} ∣ a + b ∣^{2}$ with the weighted incoherent law, records the signed difference identity, and imports the marker-overlap law with its contraction $∣ ⟨ e_{1}, e_{2} ⟩ ∣ \leq 1$ and the conditional independent-trials frequencies. Covariance-refined preparation uncertainty without operator products gives $(Δ A)^{2} (Δ B)^{2} \geq {Cov}_{u} (A, B)^{2} + (Im ⟨ x, y ⟩)^{2}$ with its equality case and the commutator pairing, and Position-momentum uncertainty on the maximal domains of the line proves $Δ Q Δ P \geq ℏ / 2$ on the full maximal domain of the line, with the explicit identities $Re m_{1} = 0$ , $Re m_{2} = - \frac{1}{2}$ and $Im ⟨ x_{Q}, y_{P} ⟩ = - \frac{ℏ}{2}$ .

Galilean boost covariance of the Schrodinger model with bounded potential builds the boost family $B_{u} (t) ψ (x) = e^{i (\frac{m}{ℏ} u \cdot x - \frac{m}{2 ℏ} ∣ u ∣^{2} t)} ψ (x - u t)$ , proves that it carries $W^{2, 2}$ -solutions of the bounded-potential model to solutions with the moving potential $V (\cdot - u t)$ and that it intertwines the free propagator, $B_{u} (t) e^{- i (t - s) K_{0} / ℏ} = e^{- i (t - s) K_{0} / ℏ} B_{u} (s)$ . Conserved spectral probability laws for commuting observables and stationary eigenrays gives the commutation calculus of a self-adjoint generator with bounded observables and their projection-valued measures, the invariance of spectral laws and existing moments, and the stationarity of eigenrays. Graph-domain Ehrenfest identities and the conditional timescale bound proves the graph-domain Ehrenfest identities $f^{'} = \frac{i}{ℏ} ⟨ (K A - A K) ψ, ψ ⟩ = - \frac{2}{ℏ} Im ⟨ A ψ, K ψ ⟩$ under explicit product-domain hypotheses, the bounded time-dependent version, and the conditional timescale bound $Δ K τ_{A} \geq \frac{ℏ}{2}$ . Finally Conditional Stern-Gerlach force for a spin-coupled bounded field model treats the two-component spin model $K = \frac{1}{2 m} \sum_{j} P_{j}^{2} - γ_{s} S_{z} B_{z} (Q_{x})$ , proves its self-adjointness and the conservation of the spin-sector probabilities, computes the force identity $\frac{d}{d t} ⟨ P_{x} u (t), u (t) ⟩ = γ_{s} ⟨ S_{z} B_{z}^{'} (Q_{x}) u (t), u (t) ⟩$ for $u_{0} \in D (K^{2})$ , records the constant-gradient value $\frac{γ_{s} ℏ}{2} G$ at the preparation time, and verifies $div B = 0$ , $curl B = (0, - B_{z}^{'}, 0)$ for the model field. All items are draft mathematics under construction; physical pages consume them as stated mathematical premises and add their own postulates and interpretation.

3 · Logical flowchart

4 · Definitions, postulates, theorems and proofs

DefinitionDefinition: AI-adaptedProof: Not applicableOpen item page →

Position and momentum operators on Euclidean L2

Definition

Assume the Axiom of Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )). Fix $ℏ > 0$ and an integer $d \geq 1$ , and work in the complex Hilbert space $L^{2} (R^{d}; C)$ with inner product $⟨ f, g ⟩ = \int_{R^{d}} f \overline{g}$ linear in the first argument (Complex Lp classes and Euclidean test-function conventions, The space $L^{p} (μ)$ as the quotient by null functions, $L^{2}$ with the integral pairing is a Hilbert space).

Weak derivatives. For $f \in L^{2} (R^{d}; C)$ and $j \in {1, \dots, d}$ , say that the $j$ -th weak derivative of $f$ exists in $L^{2}$ when there is a class $g \in L^{2} (R^{d}; C)$ with $\int_{R^{d}} f D_{j} φ = (- 1) \int_{R^{d}} g φ for every φ \in C_{c}^{\infty} (R^{d}),$ where $D_{j} φ$ is the classical partial derivative and the pairing is bilinear without conjugation (Weak derivative of a locally integrable function). Such a $g$ is unique as an $L^{2}$ class (Uniqueness of a weak derivative as an almost-everywhere class); write $D_{j} f : = g$ . The Sobolev space $W^{1, 2} (R^{d}; C)$ is the set of classes $f \in L^{2}$ for which $D_{1} f, \dots, D_{d} f$ all exist in $L^{2}$ , and $W^{2, 2} (R^{d}; C)$ is the set of $f \in W^{1, 2}$ with $D_{j} f \in W^{1, 2}$ for every $j$ , with $D_{j} D_{k} f$ written for the iterated derivative (Integer-order Sobolev spaces and their norms).

(a) Position operators. For $j \in {1, \dots, d}$ define $D (Q_{j}) : = {f \in L^{2} (R^{d}; C) : x \mapsto x_{j} f (x) \in L^{2} (R^{d}; C)}, (Q_{j} f) (x) : = x_{j} f (x) .$ Thus $Q_{j} f$ is the class of the pointwise product with the $j$ -th coordinate function.

(b) Momentum operators. For $j \in {1, \dots, d}$ define $D (P_{j}) : = {f \in L^{2} (R^{d}; C) : D_{j} f exists in L^{2}}, P_{j} f : = - i ℏ D_{j} f .$

Both $Q_{j}$ and $P_{j}$ are densely defined $C$ -linear operators in $L^{2} (R^{d}; C)$ : for $Q_{j}$ the compactly supported smooth classes lie in the domain because $x_{j}$ is bounded there, and for $P_{j}$ they lie in the domain because their classical derivatives are weak derivatives in $L^{2}$ (Integer-order Sobolev spaces and their norms).

For density, the classes of $C_{c}^{\infty} (R^{d}; C)$ are dense in $W^{0, 2} (R^{d}; C) = L^{2} (R^{d}; C)$ by Compactly supported smooth functions are dense in W^{k,p}(R^n) and Integer-order Sobolev spaces and their norms. Since these classes lie in each domain as above, both operator domains are dense.

For a class $f \in C_{c}^{\infty} (R^{d})$ one has $Q_{j} f = x_{j} f$ and $P_{j} f = - i ℏ \partial_{j} f$ pointwise. The self-adjointness of these operators and their Fourier representation are proved in the companion self-adjointness lemma on this page, not asserted here; for $d = 1$ the domain $D (P)$ of the single momentum operator coincides with $W^{1, 2} (R; C)$ .

LemmaStatement: AI-adaptedProof: AI-adaptedOpen item page →

Weak product rule for a bounded coefficient with bounded weak derivative

Statement

Assume the Axiom of Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )). Let $d \geq 1$ and let $a \in L^{\infty} (R^{d}; C)$ be a class having weak first derivatives $D_{j} a \in L^{\infty} (R^{d}; C)$ for every $j \in {1, \dots, d}$ , in the sense of Weak derivative of a locally integrable function. Let $u \in W^{1, 2} (R^{d}; C)$ (Integer-order Sobolev spaces and their norms). Then $a u \in W^{1, 2} (R^{d}; C)$ and, for every $j$ , $D_{j} (a u) = a D_{j} u + (D_{j} a) u as L^{2} classes,$ with pointwise products of chosen representatives. No boundedness of $u$ is assumed.

Facts & Assumptions

Given: a class $a \in L^{\infty}$ with weak derivatives $D_{j} a \in L^{\infty}$ , and a class $u \in W^{1, 2}$ .

[F1]

$u \in W^{1, 2}$ means $u \in L^{2}$ and each weak derivative $D_{j} u$ is an $L^{2}$ class satisfying $\int u D_{j} φ = - \int (D_{j} u) φ$ for every $φ \in C_{c}^{\infty}$ ; the pairing is bilinear and the weak derivative class is unique (Integer-order Sobolev spaces and their norms, Weak derivative of a locally integrable function, Uniqueness of a weak derivative as an almost-everywhere class).

[F2]

The same identity $\int a D_{j} ψ = - \int (D_{j} a) ψ$ holds for every $ψ \in C_{c}^{\infty}$ because $D_{j} a$ is the weak derivative of $a$ ; both sides are finite since $a$ and $D_{j} a$ are bounded and $ψ$ is compactly supported.

[F3]

$C_{c}^{\infty} (R^{d}; C)$ is dense in $W^{1, 2} (R^{d}; C)$ in the $W^{1, 2}$ norm (Compactly supported smooth functions are dense in W^{k,p}(R^n)).

[F4]

Classical partial derivatives of complex smooth functions are weak derivatives (Classical derivatives agree with weak derivatives). The product rule of Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ is stated for real functions of one real variable; its use for complex partial derivatives is reduced to that case in step 1.1.

Proof

1.1givenF1F2F4algebra

[Compactly supported smooth case] Let $v, φ \in C_{c}^{\infty} (R^{d}; C)$ , and fix $j$ . Write $v = v_{1} + i v_{2}$ , $φ = φ_{1} + i φ_{2}$ with real smooth components. Holding all coordinates except $x_{j}$ fixed gives smooth real functions of one variable. Applying [F4]'s real product and sum rules to $v_{1} φ_{1} - v_{2} φ_{2} + i (v_{1} φ_{2} + v_{2} φ_{1})$ gives $D_{j} (v φ) = (D_{j} v) φ + v D_{j} φ$ . Iterating these rules proves smoothness of $v φ$ , whose support lies in the intersection of the two compact supports. Its classical derivative is its weak derivative by [F4], so [F2] with $ψ : = v φ$ gives $\int a ((D_{j} v) φ + v D_{j} φ) = - \int (D_{j} a) v φ$ . Rearranging, $\int a v D_{j} φ = - \int (a D_{j} v + (D_{j} a) v) φ$ . Boundedness of $a, D_{j} a$ and square integrability of $v, D_{j} v$ imply $a v, a D_{j} v + (D_{j} a) v \in L^{2}$ . Since $j$ and the test $φ$ were arbitrary, [F1] gives $a v \in W^{1, 2}$ with these weak derivatives.

1.2givenF1F3algebra

[Approximation] By [F3] choose $v_{n} \in C_{c}^{\infty} (R^{d})$ with $∥ v_{n} - u ∥_{W^{1, 2}} \to 0$ ; then $v_{n} \to u$ and $D_{j} v_{n} \to D_{j} u$ in $L^{2}$ . Put $M : = \max {∥ a ∥_{\infty}, ∥ D_{j} a ∥_{\infty}} < \infty$ . Then $∥ a v_{n} - a u ∥_{2} \leq M ∥ v_{n} - u ∥_{2} \to 0$ , $∥ a D_{j} v_{n} - a D_{j} u ∥_{2} \leq M ∥ D_{j} v_{n} - D_{j} u ∥_{2} \to 0$ , and $∥ (D_{j} a) v_{n} - (D_{j} a) u ∥_{2} \leq M ∥ v_{n} - u ∥_{2} \to 0$ : all three convergences hold in $L^{2}$ .

2.1step 1.1step 1.2F1algebra∎

[Passage to the limit] Fix $φ \in C_{c}^{\infty} (R^{d})$ . By step 1.1, $\int a v_{n} D_{j} φ = - \int (a D_{j} v_{n} + (D_{j} a) v_{n}) φ$ for every $n$ . As $n \to \infty$ , the left side tends to $\int a u D_{j} φ$ and the right side to $- \int (a D_{j} u + (D_{j} a) u) φ$ , because $a v_{n} \to a u$ , $a D_{j} v_{n} \to a D_{j} u$ and $(D_{j} a) v_{n} \to (D_{j} a) u$ in $L^{2}$ by step 1.2 while $D_{j} φ$ and $φ$ are bounded with compact support. Hence $\int a u D_{j} φ = - \int (a D_{j} u + (D_{j} a) u) φ$ for every test function $φ$ . The class $a D_{j} u + (D_{j} a) u$ lies in $L^{2}$ by step 1.2's estimates, so by [F1] the class $a u$ has weak $j$ -derivative $a D_{j} u + (D_{j} a) u$ and $a u \in W^{1, 2}$ .

LemmaStatement: AI-adaptedProof: AI-adaptedOpen item page →

Self-adjointness, Fourier representation and translation identities of position and momentum

Statement

Assume the Axiom of Choice and the Axiom of Countable Choice (The Axiom of Choice, The Axiom of Countable Choice ( ${A C}_{ω}$ )). Fix $d \geq 1$ and $ℏ > 0$ , and let $Q_{j}, P_{j}$ be the position and momentum operators of Position and momentum operators on Euclidean L2 on $L^{2} (R^{d}; C)$ ; let $F_{2}$ be the unitary Plancherel transform with $\hat{f} (ξ) = \int_{R^{d}} f (x) e^{- 2 π i x \cdot ξ} d x$ on $L^{1}$ (Fourier transform on complex L1 classes, Plancherel theorem).

(a) For every $j$ , $Q_{j}$ is self-adjoint with domain $D (Q_{j}) = {f \in L^{2} : x_{j} f \in L^{2}}$ .

(b) For every $j$ , $P_{j}$ is self-adjoint with domain $D (P_{j}) = {f \in L^{2} (R^{d}; C) : ξ_{j} F_{2} f \in L^{2} (R^{d}; C)},$ and for every $f \in D (P_{j})$ , $F_{2} (P_{j} f) (ξ) = 2 π ℏ ξ_{j} F_{2} f (ξ)$ for almost every $ξ$ . In particular $P_{j} f = - i ℏ D_{j} f$ for every $f \in W^{1, 2} (R^{d}; C)$ , and for $d = 1$ the domain of the single momentum operator is exactly $D (P) = W^{1, 2} (R; C)$ .

(c) For every $a \in R^{d}$ the translation $(T_{a} f) (x) : = f (x - a)$ is a unitary operator, $T_{a} D (P_{j}) = D (P_{j})$ , $T_{a} P_{j} T_{a}^{- 1} = P_{j}$ , $T_{a} D (Q_{j}) = D (Q_{j})$ and $T_{a} Q_{j} T_{a}^{- 1} = Q_{j} - a_{j}$ for every $j$ ; moreover $D_{j} (T_{a} f) = T_{a} (D_{j} f)$ as $L^{2}$ classes for every $f \in W^{1, 2}$ , so $T_{a}$ preserves $W^{1, 2}$ and $W^{2, 2}$ .

Facts & Assumptions

Given: $d \geq 1$ , $ℏ > 0$ , and the operators $Q_{j}, P_{j}$ of Position and momentum operators on Euclidean L2.

[F1]

$F_{2}$ is unitary on $L^{2} (R^{d}; C)$ and $⟨ F_{2} f, F_{2} g ⟩ = ⟨ f, g ⟩$ for all $L^{2}$ classes (Plancherel theorem, Fourier transform on complex L1 classes).

[F2]

If $f, g \in L^{2}$ and $D_{j} f = g$ weakly, then $F_{2} g (ξ) = 2 π i ξ_{j} F_{2} f (ξ)$ for a.e. $ξ$ (Distributional derivatives are polynomial Fourier multipliers).

[F3]

$W^{1, 2} (R^{d}; C)$ consists of the classes $f$ with $⟨ ξ ⟩ F_{2} f \in L^{2}$ , where $⟨ ξ ⟩ : = (1 + ∣ ξ ∣^{2})^{1 / 2}$ (Integer-order W^{k,2} and H^k agree with equivalent norms).

[F4]

For a real measurable finite-valued $a : R^{d} \to R$ , the multiplication operator $M_{a} h : = a h$ on $D (M_{a}) = {h \in L^{2} : a h \in L^{2}}$ is self-adjoint (Real multiplication operators and free plus bounded-potential Schrödinger operators, clause (a)).

[F5]

Translation $τ_{h} f (x) = f (x - h)$ is an isometry of complex $L^{p}$ , $1 \leq p \leq \infty$ , depending norm-continuously on $h$ for $p < \infty$ (Complex translation, convolution, approximate identities, and mollification).

[F6]

Change of variables for the $C^{1}$ translation $y \mapsto y + a$ with Jacobian determinant one: $\int h (y) d y = \int h (x - a) d x$ for every nonnegative Lebesgue measurable $h$ , and hence for every integrable $h$ (A C^1 diffeomorphism satisfies the change-of-variables formula for nonnegative Lebesgue measurable functions).

Proof

1.1givenF4

[Position operators] The function $x \mapsto x_{j}$ is real, measurable and finite-valued, so [F4] applied to $a (x) = x_{j}$ gives that $Q_{j} = M_{a}$ is self-adjoint with $D (Q_{j}) = {f : x_{j} f \in L^{2}}$ . This is (a).

1.2givenalgebra

[Unitary conjugation of a self-adjoint operator] Let $U$ be unitary and let $S$ be self-adjoint. Put $D (A) : = U D (S)$ and $A u : = U S U^{*} u$ for $u \in D (A)$ . The domain is dense because $D (S)$ is dense and $U$ is a surjective isometry. Then $A$ is symmetric: for $u = U z$ , $v = U w$ with $z, w \in D (S)$ , $⟨ A u, v ⟩ = ⟨ U S z, U w ⟩ = ⟨ S z, w ⟩ = ⟨ z, S w ⟩ = ⟨ U z, U S w ⟩ = ⟨ u, A v ⟩$ by unitarity. If now $v \in D (A^{*})$ with $A^{*} v = w^{'}$ , then for every $z \in D (S)$ , writing $y : = U^{*} v$ , $⟨ S z, y ⟩ = ⟨ U S z, v ⟩ = ⟨ A U z, v ⟩ = ⟨ U z, w^{'} ⟩ = ⟨ z, U^{*} w^{'} ⟩$ ; hence $y \in D (S^{*}) = D (S)$ and $S y = U^{*} w^{'}$ . Therefore $v = U y$ with $y \in D (S)$ , that is $v \in D (A)$ , and $A v = U S y = U U^{*} w^{'} = w^{'}$ . So $D (A^{*}) = D (A)$ and $A^{*} = A$ .

1.3givenF1F2F4

[The two momentum domains agree] Fix $j$ , define $b_{j} (ξ) : = 2 π ℏ ξ_{j}$ and let ${\tilde{P}}_{j} : = F_{2}^{*} M_{b_{j}} F_{2}$ on $D ({\tilde{P}}_{j}) : = {f \in L^{2} : b_{j} F_{2} f \in L^{2}}$ , where $M_{b_{j}}$ is maximal multiplication as in [F4]. Let $f \in D (P_{j})$ . Then $D_{j} f \in L^{2}$ and [F2] gives $F_{2} (D_{j} f) = 2 π i ξ_{j} F_{2} f$ , so $b_{j} F_{2} f = - i ℏ F_{2} (D_{j} f) \in L^{2}$ ; thus $f \in D ({\tilde{P}}_{j})$ and ${\tilde{P}}_{j} f = F_{2}^{*} (b_{j} F_{2} f) = - i ℏ F_{2}^{*} F_{2} (D_{j} f) = - i ℏ D_{j} f = P_{j} f$ .

1.4givenF5F6algebra

[Translations commute with weak derivatives] Let $a \in R^{d}$ and assume only $f \in L^{2}$ and $D_{j} f \in L^{2}$ for one fixed $j$ ; by [F5] $T_{a}$ is an isometry of $L^{2}$ . For every $φ \in C_{c}^{\infty}$ the function $φ (\cdot + a)$ is again a test function. Its partial difference quotient at $y$ is exactly that of $φ$ at $y + a$ , so $\partial_{j} (φ (\cdot + a)) = (D_{j} φ) (\cdot + a)$ . Two uses of [F6], applied to positive and negative parts of real and imaginary components of the integrable functions, give $\int (T_{a} f) D_{j} φ = \int f (y) D_{j} φ (y + a) d y = \int f (y) D_{j} (φ (\cdot + a)) (y) d y = - \int (D_{j} f) (y) φ (y + a) d y = - \int (T_{a} D_{j} f) (x) φ (x) d x$ . Thus $D_{j} (T_{a} f) = T_{a} (D_{j} f) \in L^{2}$ , without assuming any other weak derivative of $f$ . Applying this separately for every $j$ when $f \in W^{1, 2}$ , and then to each $D_{k} f$ when $f \in W^{2, 2}$ , proves preservation of both Sobolev spaces. The inverse $T_{- a}$ gives equality of these spaces under translation. Moreover $T_{a}$ is complex-linear and [F6] gives $⟨ T_{a} f, T_{a} g ⟩ = \int f (y) \overline{g (y)} d y = ⟨ f, g ⟩$ ; with its inverse this proves unitarity.

2.1givenstep 1.3F1F2algebra

[Converse inclusion] Let $f \in D ({\tilde{P}}_{j})$ and set $g : = F_{2}^{*} (2 π i ξ_{j} F_{2} f) \in L^{2}$ . We show that $g$ is the weak $j$ -derivative of $f$ . Let $φ \in C_{c}^{\infty} (R^{d})$ be complex-valued; it is enough to verify the bilinear identity for the real and imaginary parts, so assume first that $φ$ is real-valued. Applying [F2] to $φ$ (whose classical derivative is its weak derivative as specified in the defining supplier) and [F1], $\int f D_{j} φ = ⟨ f, D_{j} φ ⟩ = ⟨ F_{2} f, F_{2} (D_{j} φ) ⟩ = ⟨ F_{2} f, 2 π i ξ_{j} F_{2} φ ⟩ = \int F_{2} f \overline{2 π i ξ_{j} F_{2} φ} = - \int (2 π i ξ_{j} F_{2} f) \overline{F_{2} φ} = - ⟨ g, φ ⟩ = - \int g φ$ . The Fourier pairings remain sesquilinear; real-valuedness of $φ$ identifies only the spatial inner products with the bilinear test integrals. By linearity the identity holds for all complex $φ \in C_{c}^{\infty}$ . Hence $g$ is the $L^{2}$ weak $j$ -derivative of $f$ , that is $f \in D (P_{j})$ and $P_{j} f = - i ℏ g = F_{2}^{*} (2 π ℏ ξ_{j} F_{2} f) = {\tilde{P}}_{j} f$ .

2.2givenF1F4step 1.2step 1.3

[Momentum as a Fourier multiplier] The function $b_{j}$ of step 1.3 is real, measurable and finite-valued. By [F4], $M_{b_{j}}$ is self-adjoint with domain $D (M_{b_{j}}) = {h \in L^{2} : b_{j} h \in L^{2}}$ . The operator ${\tilde{P}}_{j}$ of step 1.3 has domain $F_{2}^{- 1} D (M_{b_{j}})$ , so step 1.2 with $U = F_{2}^{*}$ and $S = M_{b_{j}}$ proves its self-adjointness.

2.3givenF5step 1.4algebra

[Translation identities] Since $T_{a}^{- 1} = T_{- a}$ and $T_{a}$ is unitary by [F5], step 1.4 gives $T_{a} P_{j} T_{a}^{- 1} h = - i ℏ T_{a} D_{j} (T_{- a} h) = - i ℏ T_{a} T_{- a} D_{j} h = - i ℏ D_{j} h = P_{j} h$ for $h \in D (P_{j})$ , and $T_{a} D (P_{j}) = D (P_{j})$ because the same computation applies to $T_{- a}$ . For $Q_{j}$ , $Q_{j} T_{- a} h$ is the class $x \mapsto x_{j} h (x + a)$ and applying $T_{a}$ gives $x \mapsto (x - a)_{j} h (x)$ ; hence $T_{a} Q_{j} T_{a}^{- 1} = Q_{j} - a_{j}$ as class-valued maps, and $D (Q_{j} - a_{j}) = {h : (x_{j} - a_{j}) h \in L^{2}} = D (Q_{j})$ because $∣ (x_{j} - a_{j}) h ∣ \leq ∣ x_{j} h ∣ + ∣ a_{j} ∣ ∣ h ∣$ and conversely $∣ x_{j} h ∣ \leq ∣ (x_{j} - a_{j}) h ∣ + ∣ a_{j} ∣ ∣ h ∣$ .

3.1step 2.2step 1.3step 2.1F3algebra

[Conclusion of (b)] Steps 1.3 and 2.1 give $D (P_{j}) = D ({\tilde{P}}_{j}) = {f : ξ_{j} F_{2} f \in L^{2}}$ and $P_{j} = {\tilde{P}}_{j}$ , which is self-adjoint by step 2.2; and for $f \in D (P_{j})$ , $F_{2} (P_{j} f) = F_{2} ({\tilde{P}}_{j} f) = 2 π ℏ ξ_{j} F_{2} f$ a.e. If $d = 1$ then ${f : ξ F_{2} f \in L^{2}} = {f : ⟨ ξ ⟩ F_{2} f \in L^{2}} = W^{1, 2} (R; C)$ by [F3], because $ξ^{2} \leq ⟨ ξ ⟩^{2} = 1 + ξ^{2}$ and $⟨ ξ ⟩^{2} \leq 1 + 2 ξ^{2}$ for $∣ ξ ∣ \geq 1$ while both weights are bounded by constants on $∣ ξ ∣ \leq 1$ . This is (b).

4.1step 1.4step 2.3F5∎

[Conclusion of (c)] Steps 1.4 and 2.3 give all assertions of (c): unitarity of $T_{a}$ is [F5], the momentum and position conjugation identities and the preservation of $W^{1, 2}$ and $W^{2, 2}$ are step 1.4 and its iteration.

DefinitionDefinition: AI-adaptedProof: Not applicableOpen item page →

Bounded-coefficient magnetic Schrodinger operator

Definition

Assume the Axiom of Choice and the Axiom of Countable Choice (The Axiom of Choice, The Axiom of Countable Choice ( ${A C}_{ω}$ )). Fix $ℏ > 0$ , $m > 0$ and $q \in R$ , and work on $L^{2} (R^{3}; C)$ with the position and momentum operators $Q_{j}, P_{j}$ of Position and momentum operators on Euclidean L2.

Admissible magnetic data. Call the pair $(A, Φ)$ admissible when $A = (A_{1}, A_{2}, A_{3})$ is a triple of real-valued classes $A_{j} \in L^{\infty} (R^{3}; R)$ having weak first derivatives $D_{k} A_{j} \in L^{\infty} (R^{3}; R)$ for all $j, k \in {1, 2, 3}$ , and $Φ \in L^{\infty} (R^{3}; R)$ is a real-valued bounded class (Weak derivative of a locally integrable function, Integer-order Sobolev spaces and their norms). Write $∣ A ∣^{2} : = A_{1}^{2} + A_{2}^{2} + A_{3}^{2}$ and $div A : = D_{1} A_{1} + D_{2} A_{2} + D_{3} A_{3}$ for chosen representatives; both are bounded classes.

Magnetic momentum operators. For admissible $(A, Φ)$ and $j \in {1, 2, 3}$ define the $C$ -linear operator $P_{A, j} : = P_{j} - q A_{j}, D (P_{A, j}) : = D (P_{j}) = {f \in L^{2} : D_{j} f \in L^{2}},$ where $A_{j}$ acts as multiplication; this is well defined into $L^{2}$ because $A_{j} \in L^{\infty}$ and $f \in L^{2}$ . The self-adjointness of $P_{A, j}$ is proved later on this page, not asserted here.

The operator. Put $D (K_{A, Φ}) : = W^{2, 2} (R^{3}; C)$ and $K_{A, Φ} u : = \frac{1}{2 m} \sum_{j = 1}^{3} P_{A, j} (P_{A, j} u) + q Φ u (u \in D (K_{A, Φ})) .$ This is well defined: if $u \in W^{2, 2}$ and $(A, Φ)$ is admissible, then $P_{A, j} u = P_{j} u - q A_{j} u \in W^{1, 2} (R^{3}; C)$ by Weak product rule for a bounded coefficient with bounded weak derivative together with $P_{j} u \in W^{1, 2}$ , so $P_{A, j} u \in D (P_{A, j})$ and every summand lies in $L^{2}$ . Write $K_{0} : = K_{(0, 0)}$ for the free operator, so that $K_{0} u = \frac{1}{2 m} \sum_{j} P_{j} (P_{j} u)$ on $W^{2, 2}$ . The same formula shows $K_{A, Φ}$ is $C$ -linear. Its symmetry, self-adjointness, expansion in the magnetic momenta and relative boundedness with respect to $K_{0}$ are the content of the self-adjointness theorem on this page, not asserted here.

TheoremStatement: AI-adaptedProof: AI-adaptedOpen item page →

Self-adjointness of the bounded-coefficient magnetic Schrodinger operator

Statement

Assume the Axiom of Choice and the Axiom of Countable Choice (The Axiom of Choice, The Axiom of Countable Choice ( ${A C}_{ω}$ )). Fix $ℏ > 0$ , $m > 0$ , $q \in R$ and an admissible pair $(A, Φ)$ in the sense of Bounded-coefficient magnetic Schrodinger operator, and let $P_{A, j}$ , $K_{A, Φ}$ and $K_{0}$ be the operators defined there on $L^{2} (R^{3}; C)$ . Then:

(a) for every $j$ , $P_{A, j} = P_{j} - q A_{j}$ is self-adjoint with domain $D (P_{A, j}) = D (P_{j}) = {f : D_{j} f \in L^{2}}$ ; and for every $u \in W^{2, 2} (R^{3}; C)$ one has $P_{A, j} u \in W^{1, 2}$ and $P_{A, j} u = P_{j} u - q A_{j} u$ ;

(b) for every $u \in W^{2, 2} (R^{3}; C)$ the expansion holds: $K_{A, Φ} u = \frac{1}{2 m} P^{2} u - \frac{q}{2 m} \sum_{j = 1}^{3} (P_{j} A_{j} + A_{j} P_{j}) u + \frac{q^{2}}{2 m} ∣ A ∣^{2} u + q Φ u, P^{2} u : = \sum_{j = 1}^{3} P_{j} (P_{j} u) = - ℏ^{2} \sum_{j = 1}^{3} D_{j} D_{j} u,$ where $(P_{j} A_{j} + A_{j} P_{j}) u : = P_{j} (A_{j} u) + A_{j} P_{j} u$ , and each term lies in $L^{2}$ ;

(d) for every $a \in (0, 1)$ there is $b \in [0, \infty)$ with $∥ (K_{A, Φ} - K_{0}) u ∥ \leq a ∥ K_{0} u ∥ + b ∥ u ∥ (u \in W^{2, 2} (R^{3}; C)),$ so $K_{A, Φ} - K_{0}$ has $K_{0}$ -bound strictly below one (Relative boundedness with respect to an operator); consequently $K_{A, Φ}$ is bounded below.

The assertion is a statement about the displayed differential expression with its stated coefficient hypotheses; it identifies no physical field, particle or measured Hamiltonian.

Facts & Assumptions

Given: $ℏ > 0$ , $m > 0$ , $q \in R$ ; an admissible pair $(A, Φ)$ with $∥ A_{j} ∥_{\infty}, ∥ D_{k} A_{j} ∥_{\infty}, ∥ Φ ∥_{\infty} < \infty$ ; the operators $P_{j}$ , $Q_{j}$ , $P_{A, j}$ , $K_{A, Φ}$ , $K_{0}$ of the definition.

[F1]

$P_{j}$ is self-adjoint with $D (P_{j}) = {f : ξ_{j} F_{2} f \in L^{2}}$ , and $F_{2} (P_{j} f) = 2 π ℏ ξ_{j} F_{2} f$ for $f \in D (P_{j})$ ; also $D_{j} f \in L^{2}$ and $P_{j} f = - i ℏ D_{j} f$ for $f \in D_{j}$ in this domain (Self-adjointness, Fourier representation and translation identities of position and momentum).

[F2]

Multiplication by a real bounded measurable class is a bounded self-adjoint operator, and $q A_{j}$ , $Φ$ , $∣ A ∣^{2}$ are real bounded classes (Real multiplication operators and free plus bounded-potential Schrödinger operators, clause (a); Bounded-coefficient magnetic Schrodinger operator).

[F3]

Kato-Rellich: if $S$ is self-adjoint and an operator $B$ is symmetric with $D (S) \subseteq D (B)$ and $S$ -bound less than one, then $S + B$ with domain $D (S)$ is self-adjoint; moreover if $S \geq γ$ and $(a, b)$ is an admissible pair for $B$ with $a < 1$ , then $S + B \geq γ - \max {a ∣ γ ∣ + b, b / (1 - a)}$ (Kato-Rellich theorem, Relative boundedness with respect to an operator).

[F4]

For $u \in W^{1, 2}$ , $A_{j} u \in W^{1, 2}$ and $D_{k} (A_{j} u) = A_{j} D_{k} u + (D_{k} A_{j}) u$ ; hence $A_{j} u \in D (P_{j})$ with $P_{j} (A_{j} u) = A_{j} P_{j} u - i ℏ (D_{j} A_{j}) u$ (Weak product rule for a bounded coefficient with bounded weak derivative).

[F5]

For $u \in W^{2, 2}$ and admissible $A$ , $P_{A, j} u = P_{j} u - q A_{j} u \in W^{1, 2}$ : indeed $P_{j} u = - i ℏ D_{j} u \in W^{1, 2}$ because $D_{j} u \in W^{1, 2}$ and $A_{j} u \in W^{1, 2}$ by [F4] (Bounded-coefficient magnetic Schrodinger operator, Integer-order Sobolev spaces and their norms).

[F6]

$F_{2}$ is unitary and $W^{2, 2} (R^{3}; C) = {f : ∣ ξ ∣^{2} F_{2} f \in L^{2}}$ with equivalent norms (Plancherel theorem, Integer-order W^{k,2} and H^k agree with equivalent norms).

Proof

1.1givenF1F2F3

[Self-adjoint magnetic momenta] For each $j$ , $q A_{j}$ is a bounded symmetric operator by [F2] with $∥ q A_{j} f ∥ \leq ∣ q ∣ ∥ A_{j} ∥_{\infty} ∥ f ∥$ for every $f \in L^{2}$ , so its $P_{j}$ -bound is $0$ : the pair $(a, b) = (0, ∣ q ∣ ∥ A_{j} ∥_{\infty})$ is admissible in the sense of [F3] since $D (P_{j}) \subseteq L^{2} = D (q A_{j})$ . By [F3] with $S = P_{j}$ and $B = - q A_{j}$ (symmetric), $P_{A, j} = P_{j} - q A_{j}$ is self-adjoint with domain $D (P_{j})$ . This is the first assertion of (a).

1.2givenF1F4F5

[Membership of the shifted momentum] Let $u \in W^{2, 2}$ . Then $D_{j} u \in W^{1, 2}$ , so $P_{j} u = - i ℏ D_{j} u \in W^{1, 2}$ , and $A_{j} u \in W^{1, 2}$ by [F4]; hence $P_{A, j} u = P_{j} u - q A_{j} u \in W^{1, 2}$ and $P_{A, j} u = P_{j} u - q A_{j} u$ holds as classes. This completes (a).

1.3givenF1F6algebra

[Momentum bound by the free operator] For $u \in W^{2, 2}$ , $P_{j} u \in D (P_{j})$ and $F_{2} (P_{j} u) = 2 π ℏ ξ_{j} F_{2} u$ by [F1], while $K_{0} u = F_{2}^{*} (c ∣ ξ ∣^{2} F_{2} u)$ with $c : = (2 π ℏ)^{2} / (2 m)$ by [F1] and the definition of $K_{0}$ : iterating the multiplier identity gives $F_{2} (\frac{1}{2 m} \sum_{j} P_{j} (P_{j} u)) = \frac{(2 π ℏ)^{2}}{2 m} ∣ ξ ∣^{2} F_{2} u$ . Hence $∥ P_{j} u ∥ = ∥ 2 π ℏ ξ_{j} F_{2} u ∥$ and $∥ K_{0} u ∥ = ∥ c ∣ ξ ∣^{2} F_{2} u ∥$ by unitarity of $F_{2}$ [F6]. Given $ε > 0$ , the elementary inequality $2 π ℏ ∣ t ∣ \leq ε c t^{2} + C_{ε}$ (with $t = ∣ ξ ∣$ , $C_{ε} = (2 π ℏ)^{2} / (4 ε c)$ ) gives $∥ P_{j} u ∥ \leq ε ∥ K_{0} u ∥ + C_{ε} ∥ u ∥$ .

2.1givenstep 1.1step 1.2algebra

[Symmetry of the individual magnetic squares] For $u, v \in W^{2, 2}$ and each $j$ , step 1.2 gives $P_{A, j} u, P_{A, j} v \in W^{1, 2} \subseteq D (P_{j}) = D (P_{A, j})$ by step 1.1, and $P_{j} u, P_{j} v \in W^{1, 2} \subseteq D (P_{j})$ (Bounded-coefficient magnetic Schrodinger operator, Integer-order Sobolev spaces and their norms). Using self-adjointness of $P_{A, j}$ and of $P_{j}$ (step 1.1, [F1]) on their domains, $⟨ P_{A, j} (P_{A, j} u), v ⟩ = ⟨ P_{A, j} u, P_{A, j} v ⟩ = ⟨ u, P_{A, j} (P_{A, j} v) ⟩$ and $⟨ P_{j} (P_{j} u), v ⟩ = ⟨ P_{j} u, P_{j} v ⟩ = ⟨ u, P_{j} (P_{j} v) ⟩$ . Subtracting, the operator $B_{j} : = \frac{1}{2 m} (P_{A, j}^{2} - P_{j}^{2})$ with $B_{j} u : = \frac{1}{2 m} (P_{A, j} (P_{A, j} u) - P_{j} (P_{j} u))$ on $W^{2, 2}$ is symmetric.

2.2givenstep 1.2F4algebra

[Expansion] Let $u \in W^{2, 2}$ . By [F4] applied to $D_{k} u$ (which lies in $W^{1, 2}$ ), or directly to $u$ , $P_{j} (A_{j} u) = A_{j} P_{j} u - i ℏ (D_{j} A_{j}) u$ ; hence $P_{A, j} (P_{A, j} u) = (P_{j} - q A_{j}) (P_{j} u - q A_{j} u) = P_{j} (P_{j} u) - q P_{j} (A_{j} u) - q A_{j} P_{j} u + q^{2} A_{j}^{2} u = P^{2} {-term}_{j} - q (P_{j} A_{j} + A_{j} P_{j}) u + q^{2} A_{j}^{2} u$ , where $(P_{j} A_{j} + A_{j} P_{j}) u : = P_{j} (A_{j} u) + A_{j} P_{j} u$ . Summing over $j$ and dividing by $2 m$ , and using $P^{2} u = \sum_{j} P_{j} (P_{j} u)$ and $∣ A ∣^{2} = \sum_{j} A_{j}^{2}$ , gives the displayed expansion; every term lies in $L^{2}$ by steps 1.2 and [F2]. This is (b).

3.1givenstep 2.1F2

[Symmetry of $K_{A, Φ}$ ] The zero-order term $q Φ$ is bounded symmetric by [F2]. Hence $K_{A, Φ} = K_{0} + \sum_{j} B_{j} + q Φ$ is symmetric on the common domain $W^{2, 2}$ , because $K_{0} u = \frac{1}{2 m} \sum_{j} P_{j} (P_{j} u)$ by definition and each summand is symmetric by step 2.1 and [F2].

3.2givenstep 2.2F2F4algebra

[The relative bound] Let $u \in W^{2, 2}$ . From (b), $∥ (K_{A, Φ} - K_{0}) u ∥ \leq \frac{∣ q ∣}{2 m} \sum_{j} ∥ (P_{j} A_{j} + A_{j} P_{j}) u ∥ + \frac{q^{2}}{2 m} ∥ ∣ A ∣^{2} u ∥ + ∣ q ∣ ∥ Φ u ∥$ . By [F4], $∥ (P_{j} A_{j} + A_{j} P_{j}) u ∥ \leq 2 ∥ A_{j} ∥_{\infty} ∥ P_{j} u ∥ + ℏ ∥ D_{j} A_{j} ∥_{\infty} ∥ u ∥$ ; and $∥ ∣ A ∣^{2} u ∥ \leq ∥ A ∥_{\infty}^{2} ∥ u ∥$ , $∥ Φ u ∥ \leq ∥ Φ ∥_{\infty} ∥ u ∥$ . So it suffices to bound $∥ P_{j} u ∥$ in terms of $∥ K_{0} u ∥$ and $∥ u ∥$ .

4.1step 3.2step 1.3algebra

[Conclusion of (d)] Fix $a \in (0, 1)$ and put $C : = \frac{∣ q ∣}{m} \sum_{j} ∥ A_{j} ∥_{\infty}$ and $B_{0} : = \frac{∣ q ∣ ℏ}{2 m} \sum_{j} ∥ D_{j} A_{j} ∥_{\infty} + \frac{q^{2}}{2 m} ∥ ∣ A ∣^{2} ∥_{\infty} + ∣ q ∣ ∥ Φ ∥_{\infty}$ . Steps 3.2 and 1.3 give $∥ (K_{A, Φ} - K_{0}) u ∥ \leq C ε ∥ K_{0} u ∥ + (C C_{ε} + B_{0}) ∥ u ∥$ . If $C > 0$ , take $ε = a / C$ and $b = C C_{ε} + B_{0}$ ; if $C = 0$ , take $b = B_{0}$ . In both cases the asserted inequality holds for this prescribed $a$ . The infimum of admissible relative constants is therefore zero, but an estimate with $a = 0$ is not asserted.

5.1step 3.1step 4.1F3∎

[Self-adjointness] By step 2.2 the operator $W : = K_{A, Φ} - K_{0}$ is defined on $D (K_{0}) = W^{2, 2}$ and is symmetric by step 3.1 (it is the sum of the symmetric $B_{j}$ and the bounded symmetric $q Φ$ ). By [F3] (Kato-Rellich) with $S = K_{0}$ and $B = W$ , whose $K_{0}$ -bound is below one by step 4.1, the operator $K_{0} + W = K_{A, Φ}$ is self-adjoint with domain $D (K_{0}) = W^{2, 2}$ . Moreover $K_{0} u = F_{2}^{*} (c ∣ ξ ∣^{2} F_{2} u)$ is positive (its quadratic form is $\int c ∣ ξ ∣^{2} ∣ F_{2} u ∣^{2} \geq 0$ ), so $K_{0} \geq 0$ and [F3] gives the lower bound $K_{A, Φ} \geq - \max {0, b / (1 - a)} = - b / (1 - a)$ for the admissible pair of step 4.1. This is (c) and the final clause of (d).

TheoremStatement: AI-adaptedProof: AI-adaptedOpen item page →

Gauge multiplication preserves H2 and transports the magnetic momentum

Statement

Assume the Axiom of Choice and the Axiom of Countable Choice (The Axiom of Choice, The Axiom of Countable Choice ( ${A C}_{ω}$ )). Fix $ℏ > 0$ , $m > 0$ , $q \in R$ , an open interval $I \subseteq R$ , admissible magnetic data $(A, Φ)$ on $R^{3}$ and a real function $χ : R^{3} \times I \to R$ such that:

for every $t \in I$ , $χ (\cdot, t)$ is $C^{2}$ on $R^{3}$ , and the functions $D_{j} χ (\cdot, t)$ , $D_{k} D_{j} χ (\cdot, t)$ are bounded on $R^{3}$ with bounds independent of $t \in I$ ;
for every $x \in R^{3}$ , $t \mapsto χ (x, t)$ is differentiable on $I$ , and $\partial_{t} χ$ is bounded on $R^{3} \times I$ .

Put $θ (x, t) : = \frac{q}{ℏ} χ (x, t)$ and let $G (t) : L^{2} (R^{3}; C) \to L^{2} (R^{3}; C)$ be multiplication by $e^{i θ (\cdot, t)}$ (The complex exponential by its power series). Then for every $t \in I$ :

(a) $G (t)$ is unitary and preserves $W^{1, 2}$ and $W^{2, 2}$ : $G (t) W^{k, 2} (R^{3}; C) = W^{k, 2} (R^{3}; C)$ for $k = 1, 2$ ;

(b) for every $u \in W^{1, 2}$ and every $j$ , $D_{j} (G (t) u) = G (t) D_{j} u + i (D_{j} θ (\cdot, t)) G (t) u as L^{2} classes;$

(c) for every $u \in W^{1, 2}$ the gauge covariance identity holds: $(P_{j} - q (A_{j} + D_{j} χ (\cdot, t))) G (t) u = G (t) (P_{j} - q A_{j}) u;$

(d) $t \mapsto G (t) h$ is continuous on $I$ for every $h \in L^{2}$ , and for every $h \in L^{2}$ and $t \in I$ the difference quotients converge in $L^{2}$ : $\lim_{s \to 0} \frac{1}{s} (G (t + s) h - G (t) h) = \frac{i q}{ℏ} (\partial_{t} χ (\cdot, t)) G (t) h .$

The function $χ$ is an arbitrary function with the stated regularity; no identification with a physical gauge choice is asserted.

Facts & Assumptions

Given: $ℏ, m > 0$ , $q \in R$ , admissible $(A, Φ)$ , a gauge function $χ$ as in the Statement, $θ = q χ / ℏ$ and the multipliers $G (t)$ .

[F1]

Euler's formula and trigonometric parity give $∣ e^{i θ (x, t)} ∣ = 1$ and $e^{- i θ (x, t)} = \overline{e^{i θ (x, t)}}$ for real $θ$ ( $\exp (x + i y) = e^{x} (\cos y + i \sin y)$ , $∣ \exp (x + i y) ∣ = e^{x}$ , and $e^{i π} + 1 = 0$ ). For a measurable multiplier $a$ with $∣ a ∣ = 1$ , $M_{a} h : = a h$ satisfies $∥ M_{a} h ∥_{2}^{2} = \int ∣ a ∣^{2} ∣ h ∣^{2} = ∥ h ∥_{2}^{2}$ and $M_{\overline{a}} M_{a} = M_{a} M_{\overline{a}} = I$ , so $M_{a}$ is unitary; pointwise modulus also gives its $L^{p}$ isometry for $1 \leq p \leq \infty$ . In particular $G (t)^{- 1} = M_{e^{- i θ (\cdot, t)}}$ and $∣ G (t) h ∣ = ∣ h ∣$ pointwise for chosen representatives.

[F2]

Classical partial derivatives of smooth functions are weak derivatives (Classical derivatives agree with weak derivatives).

[F3]

The complex exponential is entire with derivative $\exp^{'} (z) = \exp z$ (The complex exponential is entire and its complex derivative is itself). For an increment $h \in C$ , this derivative means $\exp (z + h) = \exp z + \exp (z) h + o (∣ h ∣)$ , so, identifying $C$ with $R^{2}$ , its real total derivative is multiplication by $\exp z$ . The chain and product rules applied to the real $C^{2}$ function $θ (\cdot, t)$ therefore give $D_{j} (e^{i θ (x, t)}) = i D_{j} θ e^{i θ}$ and $D_{k} D_{j} (e^{i θ}) = (i D_{k} D_{j} θ - (D_{k} θ) (D_{j} θ)) e^{i θ}$ pointwise (The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ , Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ ).

[F4]

Weak product rule: if $a \in L^{\infty}$ has weak first derivatives $D_{k} a \in L^{\infty}$ and $u \in W^{1, 2}$ , then $a u \in W^{1, 2}$ and $D_{k} (a u) = a D_{k} u + (D_{k} a) u$ (Weak product rule for a bounded coefficient with bounded weak derivative).

[F5]

$P_{j} f = - i ℏ D_{j} f$ for $f \in D (P_{j})$ , and $W^{1, 2} \subseteq D (P_{j})$ ; $W^{2, 2} = {f : D_{j} f \in W^{1, 2} \forall j}$ (Self-adjointness, Fourier representation and translation identities of position and momentum, Position and momentum operators on Euclidean L2, Integer-order Sobolev spaces and their norms).

[F6]

Dominated convergence for a sequence of measurable functions dominated by an $L^{2}$ class (Dominated convergence).

[F7]

For fixed $t \in I$ , $x \mapsto \partial_{t} χ (x, t)$ is measurable: choose $h_{n} \to 0$ with $t + h_{n} \in I$ ; each difference quotient $x \mapsto (χ (x, t + h_{n}) - χ (x, t)) / h_{n}$ is continuous and converges pointwise to $\partial_{t} χ (x, t)$ . The stated uniform bound therefore makes this derivative slice an $L^{\infty}$ class.

[F8]

If $g$ is continuous on the closed interval with endpoints $a, b$ , differentiable in its interior, and $∣ g^{'} ∣ \leq M$ there, then $∣ g (b) - g (a) ∣ \leq M ∣ b - a ∣$ : when $g (b) \neq g (a)$ , choose $ζ : = \overline{g (b) - g (a)} / ∣ g (b) - g (a) ∣$ and apply the real mean value theorem to $Re (ζ g)$ ; the zero-difference case is immediate (The mean value theorem, as the case $g (x) = x$ of Cauchy's: for $f$ continuous on $[a, b]$ with $a < b$ and differentiable on $(a, b)$ there is $c \in (a, b)$ with $f (b) - f (a) = f^{'} (c) (b - a)$ ).

Proof

1.1givenF1

[Unitarity] For each $t$ , $∣ e^{i θ (x, t)} ∣ = 1$ for every $x$ , so [F1] gives that $G (t)$ is unitary on $L^{2}$ . This is the first assertion of (a).

1.2givenF2F3F4

[First-order preservation] Let $u \in W^{1, 2}$ . By [F3] the multiplier $a : = e^{i θ (\cdot, t)}$ is bounded with classical derivative $D_{j} a = i (D_{j} θ) a$ , and this classical derivative is its weak derivative by [F2]; the coefficient $D_{j} θ$ is bounded by hypothesis. So [F4] applies: $G (t) u \in W^{1, 2}$ and $D_{j} (G (t) u) = G (t) D_{j} u + i (D_{j} θ) G (t) u$ , which is (b).

2.1givenstep 1.2F2F3F4

[Second-order preservation] Let $u \in W^{2, 2}$ , so $D_{j} u \in W^{1, 2}$ for every $j$ . By step 1.2, $D_{j} (G (t) u) = G (t) D_{j} u + i (D_{j} θ) G (t) u$ . Apply [F4] to $a = G (t)$ and the class $D_{j} u \in W^{1, 2}$ : $G (t) D_{j} u \in W^{1, 2}$ , with $D_{k} (G (t) D_{j} u) = G (t) D_{k} D_{j} u + i (D_{k} θ) G (t) D_{j} u$ . Apply [F4] to the bounded coefficient $i D_{j} θ$ (whose weak derivative $i D_{k} D_{j} θ$ is bounded by [F2], [F3]) and $G (t) u \in W^{1, 2}$ from step 1.2: $i (D_{j} θ) G (t) u \in W^{1, 2}$ . Adding, $D_{j} (G (t) u) \in W^{1, 2}$ with an explicit formula, so $G (t) W^{2, 2} \subseteq W^{2, 2}$ ; applying the same argument to $- χ$ (which satisfies the same hypotheses) shows $G (t)^{- 1} W^{2, 2} \subseteq W^{2, 2}$ , hence equality, and with step 1.2 this proves (a).

2.2givenstep 1.2F5algebra

[Gauge covariance] Let $u \in W^{1, 2}$ . By [F5], $P_{j} (G (t) u) = - i ℏ D_{j} (G (t) u)$ and $P_{j} u = - i ℏ D_{j} u$ ; substituting step 1.2, $(P_{j} - q (A_{j} + D_{j} χ)) G (t) u = - i ℏ (G (t) D_{j} u + i (D_{j} θ) G (t) u) - q (A_{j} + D_{j} χ) G (t) u = G (t) P_{j} u + ℏ (D_{j} θ) G (t) u - q A_{j} G (t) u - q (D_{j} χ) G (t) u$ . Since $D_{j} θ = \frac{q}{ℏ} D_{j} χ$ , the middle two terms cancel and the right side equals $G (t) (P_{j} u - q A_{j} u) = G (t) (P_{j} - q A_{j}) u$ , which is (c).

3.1givenF1F3F6F7F8algebra∎

[Time continuity and differentiability] Fix $h \in L^{2}$ and $t \in I$ . For every sequence $s_{n} \to 0$ with $t + s_{n} \in I$ , $G (t + s_{n}) h \to G (t) h$ pointwise and $∣ G (t + s_{n}) h - G (t) h ∣^{2} \leq 4 ∣ h ∣^{2} \in L^{1}$ ; [F6] gives convergence in $L^{2}$ , and the sequential criterion gives continuity in (d). For the difference quotients, pointwise differentiability and [F3] give the limit $\frac{i q}{ℏ} (\partial_{t} χ (\cdot, t)) G (t) h$ . Put $M : = \sup_{R^{3} \times I} ∣ \partial_{t} θ ∣ < \infty$ . For each fixed $x$ , the complex-valued curve $r \mapsto e^{i θ (x, r)}$ has derivative of modulus $∣ \partial_{r} θ (x, r) ∣ \leq M$ by [F1], [F3]; [F8] applied on the interval between $t$ and $t + s_{n}$ therefore bounds each difference quotient by $M ∣ h (x) ∣$ . Its pointwise limit has the same bound, so the squared difference from the limit is dominated by $4 M^{2} ∣ h ∣^{2} \in L^{1}$ ; [F6] yields convergence in $L^{2}$ to the asserted derivative. This is the second half of (d).

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Gauge transported propagator for the bounded-coefficient magnetic operator

Statement

Assume the Axiom of Choice and the Axiom of Countable Choice (The Axiom of Choice, The Axiom of Countable Choice ( ${A C}_{ω}$ )). Fix $ℏ > 0$ , $m > 0$ , $q \in R$ , admissible magnetic data $(A, Φ)$ on $R^{3}$ and a gauge function $χ : R^{3} \times I \to R$ as in Gauge multiplication preserves H2 and transports the magnetic momentum, with underlying open interval $I \subseteq R$ . Put $A_{j}^{'} (\cdot, t) : = A_{j} + D_{j} χ (\cdot, t)$ , $Φ^{'} (\cdot, t) : = Φ - \partial_{t} χ (\cdot, t)$ and, for $t \in I$ , $H^{'} (t) : = K_{A^{'} (t), Φ^{'} (t)}, G (t) : = multiplication by e^{i \frac{q}{ℏ} χ (\cdot, t)},$ so that $(A^{'} (t), Φ^{'} (t))$ is admissible for every $t$ and $H^{'} (t)$ is self-adjoint with domain $W^{2, 2} (R^{3}; C)$ by Self-adjointness of the bounded-coefficient magnetic Schrodinger operator. Let $K : = K_{A, Φ}$ and $U (τ) : = e^{- i τ K / ℏ}$ for $τ \in R$ , the strongly continuous unitary group generated by $- i K / ℏ$ (A self-adjoint operator generates a strongly continuous unitary group, Integral of a measurable function against a projection-valued measure).

(a) For every $t \in I$ one has the operator identity on $W^{2, 2}$ $H^{'} (t) = G (t) K G (t)^{- 1} - q \partial_{t} χ (\cdot, t),$ where the last term is multiplication by $- q \partial_{t} χ (\cdot, t)$ ; explicitly, $G (t) K G (t)^{- 1} = K_{A^{'} (t), Φ}$ is the magnetic operator with the transported vector potential and the original scalar potential $Φ$ .

(b) For every $u_{0} \in W^{2, 2}$ and $s \in I$ the curve $u (t) : = G (t) U (t - s) G (s)^{- 1} u_{0} (t \in I)$ lies in $W^{2, 2}$ for every $t$ , is differentiable into $L^{2}$ on $I$ , and satisfies $u (s) = u_{0}$ and $i ℏ u^{'} (t) = H^{'} (t) u (t)$ for every $t \in I$ . It is the unique such curve: if $v : I \to W^{2, 2}$ is continuous into $W^{2, 2} (R^{3}; C)$ , differentiable into $L^{2}$ , with $i ℏ v^{'} (t) = H^{'} (t) v (t)$ for all $t$ and $v (s) = u_{0}$ , then $v = u$ .

(c) The two-parameter family $U^{'} (t, s) : = G (t) U (t - s) G (s)^{- 1} (t, s \in I)$ consists of unitary operators mapping $W^{2, 2}$ onto $W^{2, 2}$ , satisfies $U^{'} (t, t) = I$ and the composition law $U^{'} (t, r) U^{'} (r, s) = U^{'} (t, s)$ for all $t, r, s \in I$ , and $u (t) = U^{'} (t, s) u_{0}$ is the solution of (b).

The theorem provides dynamics for the transformed coefficient class only; it asserts no general well-posedness for arbitrary time-dependent unbounded generators.

Facts & Assumptions

Given: $ℏ, m > 0$ , $q \in R$ , admissible $(A, Φ)$ , the gauge function $χ$ , the transported data $A^{'} (t), Φ^{'} (t)$ , the operators $K = K_{A, Φ}$ , $H^{'} (t) = K_{A^{'} (t), Φ^{'} (t)}$ and the multipliers $G (t)$ .

[F1]

$K$ is self-adjoint with domain $D (K) = W^{2, 2}$ , and $U (τ) = e^{- i τ K / ℏ}$ is a strongly continuous unitary group with $U (τ) D (K) = D (K)$ , $K U (τ) = U (τ) K$ on $D (K)$ , and $\lim_{τ \to 0} \frac{1}{τ} (U (τ) x - x) = - \frac{i}{ℏ} K x$ exactly for $x \in D (K)$ (A self-adjoint operator generates a strongly continuous unitary group, Integral of a measurable function against a projection-valued measure).

[F2]

For each $t$ , $G (t)$ is unitary, preserves $W^{1, 2}$ and $W^{2, 2}$ , and $(P_{j} - q A_{j}^{'} (\cdot, t)) G (t) u = G (t) (P_{j} - q A_{j}) u$ for $u \in W^{1, 2}$ ; moreover $t \mapsto G (t) h$ is continuous for $h \in L^{2}$ and differentiable with $\frac{d}{d t} G (t) h = \frac{i q}{ℏ} (\partial_{t} χ (\cdot, t)) G (t) h$ (Gauge multiplication preserves H2 and transports the magnetic momentum). The same assertions hold for $- χ$ in place of $χ$ (Gauge multiplication preserves H2 and transports the magnetic momentum).

[F3]

For each $t$ , $(A^{'} (t), Φ^{'} (t))$ is admissible: $D_{k} A_{j}^{'} = D_{k} A_{j} + D_{k} D_{j} χ (\cdot, t)$ and $Φ^{'} = Φ - \partial_{t} χ (\cdot, t)$ are bounded with the uniform bounds of the hypotheses; and $H^{'} (t) = K_{A^{'} (t), Φ^{'} (t)}$ is self-adjoint with domain $W^{2, 2}$ (Self-adjointness of the bounded-coefficient magnetic Schrodinger operator, Bounded-coefficient magnetic Schrodinger operator).

[F4]

A real function continuous on a closed interval and differentiable in its interior with zero derivative is constant (The mean value theorem, as the case $g (x) = x$ of Cauchy's: for $f$ continuous on $[a, b]$ with $a < b$ and differentiable on $(a, b)$ there is $c \in (a, b)$ with $f (b) - f (a) = f^{'} (c) (b - a)$ ). For an $L^{2}$ -differentiable curve $z$ , expansion of the inner product gives $(∥ z ∥^{2})^{'} = 2 Re ⟨ z^{'}, z ⟩$ . If $T$ is self-adjoint and $z \in D (T)$ , then $⟨ T z, z ⟩$ is real, so $2 Re ⟨ - i T z / ℏ, z ⟩ = 0$ .

Proof

1.1givenF2F3algebra

[Transported magnetic square] Let $w \in W^{2, 2}$ and put $u : = G (t) w$ , so that $u \in W^{2, 2}$ by [F2] and $w = G (t)^{- 1} u$ . By Self-adjointness of the bounded-coefficient magnetic Schrodinger operator clause (a) and [F2], $P_{A, j} w \in W^{1, 2}$ and $P_{A^{'}, j} (t) G (t) P_{A, j} w = G (t) P_{A, j} (P_{A, j} w)$ for the transported momentum $P_{A^{'}, j} (t) = P_{j} - q A_{j}^{'} (\cdot, t)$ ; here the covariance identity is applied twice, the first time to $w \in W^{1, 2}$ and the second to the class $P_{A, j} w \in W^{1, 2}$ . Summing over $j$ , $\frac{1}{2 m} \sum_{j} P_{A^{'}, j} (t)^{2} u = G (t) \frac{1}{2 m} \sum_{j} P_{A, j}^{2} w$ , and adding the scalar potential term, which is multiplication by $q Φ$ commuting with $G (t)$ , gives $K_{A^{'} (t), Φ} u = G (t) K_{A, Φ} w = G (t) K G (t)^{- 1} u$ . Since $u \in W^{2, 2}$ was arbitrary, $G (t) K G (t)^{- 1} = K_{A^{'} (t), Φ}$ on $W^{2, 2}$ .

2.1givenstep 1.1F3algebra

[Conclusion of (a)] By the definition of $Φ^{'} (t)$ and the linearity of $K$ in the scalar potential, $H^{'} (t) = K_{A^{'} (t), Φ^{'} (t)} = K_{A^{'} (t), Φ} - q \partial_{t} χ (\cdot, t)$ as operators on $W^{2, 2}$ ; combining with step 1.1 gives (a).

3.1givenstep 2.1F1F2algebra

[The curve solves the equation] Let $u_{0} \in W^{2, 2}$ and $s \in I$ , and put $w_{0} : = G (s)^{- 1} u_{0}$ , $w (t) : = U (t - s) w_{0}$ and $u (t) : = G (t) w (t)$ . By [F1]-[F2], both $w (t)$ and $u (t)$ lie in $W^{2, 2}$ at each time, and $w^{'} = - i K w / ℏ$ in $L^{2}$ . Decompose the difference quotient of $u$ as $G (t + h) (w (t + h) - w (t)) / h + (G (t + h) - G (t)) w (t) / h$ . The first term tends to $G (t) w^{'} (t)$ by strong $L^{2}$ continuity and unitarity; the second tends to $(i q / ℏ) (\partial_{t} χ) G (t) w (t)$ by [F2] on the fixed vector $w (t)$ . Thus $i ℏ u^{'} = - q (\partial_{t} χ) u + G (t) K w = H^{'} (t) u$ by step 2.1, and $u (s) = G (s) w_{0} = u_{0}$ . No time continuity in the $W^{2, 2}$ norm is needed. This proves existence in (b).

4.1givenstep 3.1F3F4algebra

[Uniqueness] Let $v$ be as in (b), and put $z (t) : = v (t) - u (t)$ . At every $t$ , $z (t) \in W^{2, 2} = D (H^{'} (t))$ , and $z$ is differentiable into $L^{2}$ with $i ℏ z^{'} = H^{'} (t) z$ and $z (s) = 0$ . By self-adjointness from [F3] and the norm derivative identity of [F4], $(∥ z (t) ∥^{2})^{'} = 2 Re ⟨ - i H^{'} (t) z (t) / ℏ, z (t) ⟩ = 0$ . Apply [F4] on the closed interval joining $s$ and $t$ , which lies in the open interval $I$ : the norm squared is constant and equals zero. Hence $v (t) = u (t)$ for every $t$ , proving uniqueness without any assertion of $W^{2, 2}$ continuity for $G (t)^{- 1} v (t)$ .

5.1step 3.1step 4.1F1F2algebra∎

[Propagator properties] Each $U^{'} (t, s) = G (t) U (t - s) G (s)^{- 1}$ is unitary as a product of unitaries and maps $W^{2, 2}$ onto $W^{2, 2}$ by [F1] and [F2]. The identity $U^{'} (t, t) = G (t) G (t)^{- 1} = I$ is immediate; the composition law follows from $G (r)^{- 1} G (r) = I$ : $U^{'} (t, r) U^{'} (r, s) = G (t) U (t - r) G (r)^{- 1} G (r) U (r - s) G (s)^{- 1} = G (t) U (t - r + r - s) G (s)^{- 1} = U^{'} (t, s)$ . By step 3.1, $u (t) = G (t) U (t - s) G (s)^{- 1} u_{0} = U^{'} (t, s) u_{0}$ solves the equation, and by step 4.1 it is the unique solution. This is (c).

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Smooth local continuity equation and its integrated forms

Statement

Assume the Axiom of Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )). Fix $ℏ > 0$ , $m > 0$ , let $Ω \subseteq R^{3}$ and $I \subseteq R$ be nonempty open sets, let $V : Ω \to R$ be continuous and let $ψ : Ω \times I \to C$ be such that $ψ$ , $\partial_{t} ψ$ and all spatial partial derivatives $\partial_{j} ψ$ , $\partial_{k} \partial_{j} ψ$ ( $j, k \in {1, 2, 3}$ ) exist and are continuous on $Ω \times I$ , and $i ℏ \partial_{t} ψ (x, t) = - \frac{ℏ^{2}}{2 m} Δ ψ (x, t) + V (x) ψ (x, t) for every (x, t) \in Ω \times I,$ where $\nabla ψ : = (\partial_{1} ψ, \partial_{2} ψ, \partial_{3} ψ)$ is the spatial gradient and $Δ ψ : = \sum_{j = 1}^{3} \partial_{j} \partial_{j} ψ$ is the classical Laplacian on $R^{3}$ (The Laplacian of a $C^{2}$ function and of a $C^{2}$ vector field, Directional derivatives and partial derivatives of a map $U \subseteq R^{m} \to R^{n}$ ). Define the current $j (x, t) : = \frac{ℏ}{m} Im (\overline{ψ (x, t)} \nabla ψ (x, t)) \in R^{3} .$ Then:

(a) pointwise continuity equation. For every $(x, t) \in Ω \times I$ , $\partial_{t} ∣ ψ ∣^{2} (x, t) + div j (x, t) = 0,$ where the divergence is the classical one of the $C^{1}$ field $x \mapsto j (x, t)$ (Divergence and curl are linear and satisfy the scalar product rules);

(b) integrated balance. If $D \subseteq Ω$ is a bounded open set with $C^{1}$ boundary and $\overline{D} \subseteq Ω$ (Divergence on a bounded C1 Euclidean domain), then $t \mapsto \int_{D} ∣ ψ (x, t) ∣^{2} d x$ is differentiable on $I$ with $\frac{d}{d t} \int_{D} ∣ ψ (x, t) ∣^{2} d x = - \int_{\partial D} j (x, t) \cdot n (x) d S,$ $n$ the outward unit normal;

(c) global conservation under domination. If $Ω = R^{3}$ , if there is $g \in L^{1} (R^{3})$ with $∣ ψ (x, t) ∣^{2} \leq g (x)$ for almost every $x$ and every $t \in I$ , and if $\lim_{R \to \infty} \sup_{t \in I} \int_{{∣ x ∣ = R}} ∣ j (x, t) ∣ d S = 0,$ then, for all $s, t \in I$ such that the closed interval joining $s$ and $t$ is contained in $I$ , $∥ ψ (\cdot, s) ∥_{L^{2} (R^{3})}^{2} = ∥ ψ (\cdot, t) ∥_{L^{2} (R^{3})}^{2};$

(d) norm conservation from unitary dynamics. Assume additionally the Axiom of Choice (The Axiom of Choice). If $K$ is a self-adjoint operator on a complex Hilbert space $H$ and $ψ (t) : = e^{- i t K / ℏ} ψ_{0}$ for some $ψ_{0} \in H$ , then $∥ ψ (t) ∥ = ∥ ψ_{0} ∥$ for every $t \in R$ (A self-adjoint operator generates a strongly continuous unitary group, Integral of a measurable function against a projection-valued measure).

Assertion (d) is independent of the smooth local model of (a)-(c); no interpretation of $∣ ψ ∣^{2}$ as a measured counting rate is asserted.

Facts & Assumptions

Given: $ℏ, m > 0$ , open $Ω \subseteq R^{3}$ , nonempty open time set $I$ , continuous real $V$ , a $C^{2}$ solution $ψ$ of the displayed equation with continuous $\partial_{t} ψ$ , and the current $j$ .

[F1]

The real one-variable product and sum rules are supplied by Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ . Writing $ψ = u + i v$ with real $u, v$ and restricting to a time or spatial coordinate line gives $\partial_{t} ∣ ψ ∣^{2} = \partial_{t} (u^{2} + v^{2}) = 2 u \partial_{t} u + 2 v \partial_{t} v = 2 Re (\overline{ψ} \partial_{t} ψ)$ and likewise for $\partial_{j} ∣ ψ ∣^{2}$ . Also $j = \frac{ℏ}{m} (u \nabla v - v \nabla u)$ ; the real scalar-vector product rule of Divergence and curl are linear and satisfy the scalar product rules gives $div j = \frac{ℏ}{m} (u Δ v - v Δ u) = \frac{ℏ}{m} Im (\overline{ψ} Δ ψ)$ , since the two gradient inner products cancel.

[F2]

The conjugated equation: $\partial_{t} ψ = \frac{i}{ℏ} (\frac{ℏ^{2}}{2 m} Δ ψ - V ψ)$ , since $V$ is real-valued.

[F3]

Divergence theorem: for a bounded $C^{1}$ domain $D$ and $F \in C^{1} (\overline{D}; R^{3})$ , $\int_{D} div F d x = \int_{\partial D} F \cdot n d S$ (Divergence on a bounded C1 Euclidean domain).

[F4]

Differentiation under the integral sign on an open interval $J \subseteq R$ : if $f : D \times J \to C$ has integrable spatial sections, differentiable time sections outside one measurable null set, measurable spatial sections of its time derivative, and $∣ \partial_{t} f (x, t) ∣ \leq G (x)$ outside one measurable null set for every $t \in J$ , with one nonnegative $G \in L^{1} (D)$ , then $t \mapsto \int_{D} f (x, t) d x$ is differentiable on $J$ with derivative $\int_{D} \partial_{t} f (x, t) d x$ (Differentiation under the integral sign).

[F5]

Dominated convergence, and the fact that an integral over $B_{R}$ of an $L^{1}$ class converges to the integral over $R^{3}$ as $R \to \infty$ (Dominated convergence).

[F6]

A self-adjoint $K$ with spectral PVM $E$ has $e^{- i t K / ℏ} = \int e^{- i t λ / ℏ} d E (λ)$ , a strongly continuous unitary group: each $e^{- i t K / ℏ}$ is unitary, hence norm preserving (A self-adjoint operator generates a strongly continuous unitary group, Integral of a measurable function against a projection-valued measure).

[F7]

If a real function $F$ is continuous on $[s, t]$ , differentiable on $(s, t)$ , and $∣ F^{'} (τ) ∣ \leq M$ there, the mean value theorem gives $∣ F (t) - F (s) ∣ \leq M (t - s)$ (The mean value theorem, as the case $g (x) = x$ of Cauchy's: for $f$ continuous on $[a, b]$ with $a < b$ and differentiable on $(a, b)$ there is $c \in (a, b)$ with $f (b) - f (a) = f^{'} (c) (b - a)$ ).

Proof

1.1givenF1F2algebra

[The pointwise identity] At a point $(x, t) \in Ω \times I$ , [F2] gives $\partial_{t} ψ = \frac{i}{ℏ} (\frac{ℏ^{2}}{2 m} Δ ψ - V ψ)$ and hence, by [F1], $\partial_{t} ∣ ψ ∣^{2} = 2 Re (\overline{ψ} \partial_{t} ψ) = 2 Re [\frac{i}{ℏ} (\frac{ℏ^{2}}{2 m} \overline{ψ} Δ ψ - V ∣ ψ ∣^{2})] = \frac{ℏ}{m} Re (i \overline{ψ} Δ ψ) = - \frac{ℏ}{m} Im (\overline{ψ} Δ ψ)$ , because the term $- \frac{i}{ℏ} V ∣ ψ ∣^{2}$ is purely imaginary and $V$ is real. Comparing with [F1]'s expression for $div j$ gives $\partial_{t} ∣ ψ ∣^{2} + div j = 0$ , which is (a).

2.1givenstep 1.1F3F4algebra

[The integrated balance] Let $D$ be as in (b). If $D = \emptyset$ , then $\partial D = \emptyset$ , both integrals are zero and (b) holds. Suppose $D \neq \emptyset$ and fix $t_{0} \in I$ . Openness of $I$ gives $ε > 0$ with $[t_{0} - ε, t_{0} + ε] \subseteq I$ ; put $J : = (t_{0} - ε, t_{0} + ε)$ . By step 1.1, $\partial_{t} ∣ ψ ∣^{2} = - div j$ , continuous on the nonempty compact set $\overline{D} \times [t_{0} - ε, t_{0} + ε]$ , so $M : = \max_{\overline{D} \times [t_{0} - ε, t_{0} + ε]} ∣ div j ∣ < \infty$ . Each section $∣ ψ (\cdot, t) ∣^{2}$ is bounded on $\overline{D}$ and integrable on the bounded set $D$ ; its time derivative exists everywhere and has continuous, hence measurable, spatial sections. The fixed majorant $G (x) : = M$ on $D$ is integrable and bounds the derivative for every $t \in J$ , with no exceptional set. Thus [F4] applies to $D \times J$ , giving differentiability at $t_{0}$ and derivative $- \int_{D} div j (\cdot, t_{0})$ . By [F3] applied to the $C^{1}$ field $j (\cdot, t_{0})$ , this equals $- \int_{\partial D} j (\cdot, t_{0}) \cdot n d S$ . Since $t_{0}$ was arbitrary, (b) holds on the possibly disconnected open set $I$ .

3.1givenstep 2.1F5F7algebra

[Global conservation] Assume (c), and let $s, t \in I$ with $s < t$ and $[s, t] \subseteq I$ . For $R > 0$ with $K_{R} : = {x : ∣ x ∣ < R}$ we have by step 2.1 applied to $D = K_{R}$ (for $R$ so large that $\overline{K_{R}} \subseteq R^{3}$ , which always holds) $\frac{d}{d τ} \int_{K_{R}} ∣ ψ ∣^{2} = - \int_{\partial K_{R}} j \cdot n d S$ , so [F7] gives $∣ \int_{K_{R}} ∣ ψ (x, t) ∣^{2} d x - \int_{K_{R}} ∣ ψ (x, s) ∣^{2} d x ∣ \leq (t - s) \sup_{τ \in [s, t]} \int_{\partial K_{R}} ∣ j ∣ d S$ , and the right side tends to $0$ as $R \to \infty$ by the hypothesis on the flux. By [F5], $\int_{K_{R}} ∣ ψ (\cdot, τ) ∣^{2} \to ∥ ψ (\cdot, τ) ∥^{2}$ for $τ = t$ and $τ = s$ , using $∣ ψ ∣^{2} \leq g \in L^{1}$ . Letting $R \to \infty$ in the inequality gives $∣ ∥ ψ (\cdot, t) ∥^{2} - ∥ ψ (\cdot, s) ∥^{2} ∣ = 0$ . Thus the norms agree for every pair with the closed interval between them contained in $I$ , proving (c).

4.1givenF6algebra∎

[Unitary dynamics] By [F6], $ψ (t) = e^{- i t K / ℏ} ψ_{0}$ with $e^{- i t K / ℏ}$ unitary, so $∥ ψ (t) ∥ = ∥ ψ_{0} ∥$ for every $t$ . This is (d) and completes the proof.

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Coherent and incoherent two-path bin laws and the marker overlap

Statement

Assume the Axiom of Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )). Let $(D, A, d x)$ be a $σ$ -finite measure space, let $L^{2} (D)$ be the complex Hilbert space of The space $L^{p} (μ)$ as the quotient by null functions and Complex Lp classes and Euclidean test-function conventions, and let $a, b \in L^{2} (D)$ .

(a) Coherent law. If $a + b \neq 0$ as a class, i.e. $∥ a + b ∥ > 0$ , then $p_{c o h} (B) : = \frac{\int_{B} ∣ a + b ∣^{2} d x}{∥ a + b ∥^{2}}, B \in A,$ defines a probability measure on $(D, A)$ (Measures on sigma-algebras, Probability measures and probability spaces), and for every $B$ $\int_{B} ∣ a + b ∣^{2} d x = \int_{B} ∣ a ∣^{2} d x + \int_{B} ∣ b ∣^{2} d x + 2 Re \int_{B} a \overline{b} d x .$

(b) Incoherent law. If $a \neq 0$ , $b \neq 0$ and $w \in [0, 1]$ , then $p_{m i x} (B) : = w \frac{\int_{B} ∣ a ∣^{2} d x}{∥ a ∥^{2}} + (1 - w) \frac{\int_{B} ∣ b ∣^{2} d x}{∥ b ∥^{2}}$ defines a probability measure on $(D, A)$ .

(c) Signed difference. Let $a, b$ be unit vectors with $Re ⟨ a, b ⟩ \neq - 1$ , and put $N (B) : = \int_{B} (∣ a ∣^{2} + ∣ b ∣^{2}) d x, c (B) : = Re \int_{B} a \overline{b} d x, p_{c o h} (B) : = \frac{N (B) + 2 c (B)}{2 + 2 c (D)}, p_{m i x} (B) : = \frac{N (B)}{2} .$ Then for every $B$ , $N (B) - 2 c (B) = \int_{B} ∣ a - b ∣^{2} d x \geq 0$ , and $p_{c o h} (B) - p_{m i x} (B) = \frac{2 c (B) - c (D) N (B)}{2 (1 + c (D))} .$ Equality $p_{c o h} (B) = p_{m i x} (B)$ for every measurable bin holds exactly when $2 Re (a \overline{b}) = c (D) (∣ a ∣^{2} + ∣ b ∣^{2})$ almost everywhere. In particular, if $a ⊥ b$ then $c (D) = 0$ and the signed difference is $c (B)$ , so orthogonality alone does not imply equality on each bin. If $b = a$ then equality holds on every bin. Coherence of the preparation changes the bin law according to the amplitudes and their actual normalization.

(d) Marker overlap. Let $K$ be a complex Hilbert space (Hilbert space) with unit vectors $e_{1}, e_{2}$ , let $α, β \in C$ with $∣ α ∣^{2} + ∣ β ∣^{2} = 1$ , let $a, b$ be unit vectors and put $Ψ : = α a \otimes e_{1} + β b \otimes e_{2}$ in the completed tensor product $L^{2} (D) \hat{\otimes} K$ . The reduced weight $ρ : = {Tr}_{K} T_{Ψ, Ψ}$ is defined even when $Ψ = 0$ , with $tr ρ = ∥ Ψ ∥^{2}$ . If $Ψ \neq 0$ , $B \in A$ has finite measure and $P_{B}$ is multiplication by $1_{B}$ , then the normalized bin law of the reduced weight $ρ$ is $\frac{tr (ρ P_{B})}{tr ρ} = \frac{1}{∥ Ψ ∥^{2}} \int_{B} (∣ α ∣^{2} ∣ a ∣^{2} + ∣ β ∣^{2} ∣ b ∣^{2} + 2 Re (α \overline{β} ⟨ e_{1}, e_{2} ⟩ a (x) \overline{b (x)})) d x,$ and $∣ ⟨ e_{1}, e_{2} ⟩ ∣ \leq 1$ : the marker overlap bounds the complex cross coefficient in modulus (its phase may rotate the real interference contribution); that contribution vanishes identically when $e_{1} ⊥ e_{2}$ , leaving the incoherent weight. If $Ψ = 0$ , no normalized law is defined.

(e) Finite sampling. If a detector model produces, for a fixed bin $B$ , a positive integer $N$ of independent trials each with success probability $p = p_{c o h} (B)$ or $p = p_{m i x} (B)$ , and $S_{N}$ is the number of successes, then $S_{N}$ is binomial $(N, p)$ and $E [S_{N} / N] = p$ , $Var (S_{N} / N) = p (1 - p) / N$ (The CHSH bound and the finite-sampling law for independent trials, clause (c)).

All assertions are identities and inequalities for the specified preparation vectors; no detector law, no measured counting rate and no empirical claim is asserted, and (e) is conditional on the stated independent-trials model.

Facts & Assumptions

Given: a $σ$ -finite measure space $(D, A, d x)$ , classes $a, b \in L^{2} (D)$ ; in (d) a Hilbert space $K$ with unit vectors $e_{1}, e_{2}$ and coefficients $α, β$ .

[F1]

For measurable $B$ the set function $B \mapsto \int_{B} g d x$ is countably additive for $g \in L^{1}$ : for pairwise disjoint $B_{n}$ with union $B$ , the partial sums of $\int ∣ g ∣ 1_{B_{n}}$ increase to $\int ∣ g ∣ 1_{B} \leq \int ∣ g ∣$ , and monotone convergence applied to the positive and negative parts gives $\int_{B} g = \sum_{n} \int_{B_{n}} g$ (Monotone convergence for the integral).

[F2]

For $a, b \in L^{2} (D)$ and measurable $B$ , $\int_{B} ∣ a + b ∣^{2} = \int_{B} ∣ a ∣^{2} + \int_{B} ∣ b ∣^{2} + 2 Re \int_{B} a \overline{b}$ , and $∣ \int_{B} a \overline{b} ∣ \leq ∥ a ∥_{L^{2} (B)} ∥ b ∥_{L^{2} (B)} \leq ∥ a ∥ ∥ b ∥$ (Cauchy-Schwarz inequality for $L^{2}$ , Cauchy–Schwarz: $∣ ⟨ u, v ⟩ ∣ \leq ∥ u ∥ ∥ v ∥$ , with equality exactly for linearly dependent vectors, $L^{2}$ with the integral pairing is a Hilbert space).

[F3]

Marker overlap contraction of Marker overlap contraction of the two-path cross term: with $Ψ = α a \otimes e_{1} + β b \otimes e_{2}$ , $ρ = {Tr}_{K} T_{Ψ, Ψ}$ is a positive trace-class weight with $tr ρ = ∥ Ψ ∥^{2}$ and, for every measurable $B$ of finite measure, $tr (ρ P_{B}) = \int_{B} (∣ α ∣^{2} ∣ a ∣^{2} + ∣ β ∣^{2} ∣ b ∣^{2} + 2 Re (α \overline{β} ⟨ e_{1}, e_{2} ⟩ a \overline{b})) d x$ ; also $∣ ⟨ e_{1}, e_{2} ⟩ ∣ \leq 1$ .

[F4]

Binomial frequencies: if $S_{N}$ is binomial $(N, p)$ with $N \geq 1$ , $p \in [0, 1]$ , then $E [S_{N} / N] = p$ and $Var (S_{N} / N) = p (1 - p) / N$ (The CHSH bound and the finite-sampling law for independent trials, clause (c)).

Proof

1.1givenF1F2algebra

[Coherent law] Assume $∥ a + b ∥ > 0$ ; then $∥ a + b ∥^{2} = \int_{D} ∣ a + b ∣^{2} > 0$ is finite, and $p_{c o h} (D) = 1$ . The set function $B \mapsto \int_{B} ∣ a + b ∣^{2}$ is countably additive by [F1] applied to $g = ∣ a + b ∣^{2} \in L^{1}$ , and nonnegative; dividing by the finite positive number $∥ a + b ∥^{2}$ preserves countable additivity and total mass one, so $p_{c o h}$ is a probability measure. The expansion is the pointwise identity $∣ a + b ∣^{2} = ∣ a ∣^{2} + ∣ b ∣^{2} + 2 Re (a \overline{b})$ integrated over $B$ , legitimate for $L^{1}$ classes. This is (a).

1.2givenF1algebra

[Incoherent law] If $a \neq 0$ and $b \neq 0$ , the two normalized set functions $B \mapsto ∥ a ∥^{- 2} \int_{B} ∣ a ∣^{2}$ and $B \mapsto ∥ b ∥^{- 2} \int_{B} ∣ b ∣^{2}$ are probability measures by [F1] applied to $∣ a ∣^{2}$ and $∣ b ∣^{2}$ , so any convex combination with $w \in [0, 1]$ is a probability measure. This is (b).

1.3givenF2algebra

[Signed difference] Let $a, b$ be unit vectors; then $N (D) = 2$ and, Cauchy–Schwarz gives $c (D) \geq - 1$ , and the hypothesis $c (D) \neq - 1$ therefore gives $∥ a + b ∥^{2} = 2 + 2 c (D) > 0$ . Expanding, $p_{c o h} (B) - p_{m i x} (B) = \frac{N (B) + 2 c (B)}{2 + 2 c (D)} - \frac{N (B)}{2} = \frac{2 N (B) + 4 c (B) - 2 N (B) - 2 c (D) N (B)}{4 (1 + c (D))} = \frac{2 c (B) - c (D) N (B)}{2 (1 + c (D))}$ , and $N (B) - 2 c (B) = \int_{B} (∣ a ∣^{2} + ∣ b ∣^{2} - 2 Re a \overline{b}) = \int_{B} ∣ a - b ∣^{2} \geq 0$ . The signed difference is the integral over $B$ of the integrable real function $[2 Re (a \overline{b}) - c (D) (∣ a ∣^{2} + ∣ b ∣^{2})] / [2 (1 + c (D))]$ . Its integral vanishes on every measurable bin exactly when this density is zero almost everywhere: otherwise one of its positive or negative level sets has a strictly signed integral. This proves the stated all-bin criterion. If $a ⊥ b$ , only $c (D) = 0$ follows, giving difference $c (B)$ . For an explicit counterexample, take $D = {1, 2}$ with $d x ({1}) = d x ({2}) = 1$ , $a = (1, 1) / \sqrt{2}$ and $b = (1, - 1) / \sqrt{2}$ . Their inner product is zero, but $c ({1}) = 1 / 2$ , so their coherent and balanced mixed laws differ on that bin. If $b = a$ (unit), then $N (B) = 2 \int_{B} ∣ a ∣^{2}$ and $c (B) = \int_{B} ∣ a ∣^{2}$ , so $2 c (B) - c (D) N (B) = 2 \int_{B} ∣ a ∣^{2} - 2 \int_{B} ∣ a ∣^{2} = 0$ ; the difference vanishes for every $B$ . This is (c).

1.4givenF3algebra

[Marker law] Under the hypotheses of (d), $tr ρ = ∥ Ψ ∥^{2}$ by [F3]; if $Ψ \neq 0$ the normalized law is $tr (ρ P_{B}) / tr ρ$ , and substituting [F3]'s formula gives the displayed identity. The bound $∣ ⟨ e_{1}, e_{2} ⟩ ∣ \leq 1$ bounds the complex cross coefficient by $∣ α β ∣$ , while its phase need not contract the real interference part; this is [F3]. When $Ψ = 0$ , the raw weight remains defined and has zero trace, so normalization is omitted; if $e_{1} ⊥ e_{2}$ the cross term $2 Re (α \overline{β} ⟨ e_{1}, e_{2} ⟩ a \overline{b})$ vanishes pointwise, leaving the incoherent weight $∣ α ∣^{2} ∣ a ∣^{2} + ∣ β ∣^{2} ∣ b ∣^{2}$ . This is (d).

2.1givenF4algebra∎

[Finite sampling] Under the stated independent-trials model, the number of successes $S_{N}$ is by definition binomial $(N, p)$ , and the mean and variance of $S_{N} / N$ are those of [F4]. This is (e), and it is conditional on that model.

TheoremStatement: AI-adaptedProof: AI-adaptedOpen item page →

Covariance-refined preparation uncertainty without operator products

Statement

Assume the Axiom of Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )). Let $H$ be a complex Hilbert space with inner product linear in the first argument (Real and complex inner product spaces, with the inner product linear in the first argument, Hilbert space), let $A, B$ be symmetric densely defined operators on $H$ (Symmetric, self-adjoint and essentially self-adjoint operators, Adjoint of a densely defined operator), and let $u \in D (A) \cap D (B)$ satisfy $∥ u ∥ = 1$ . Put $a : = ⟨ A u, u ⟩, b : = ⟨ B u, u ⟩, x : = (A - a) u, y : = (B - b) u, Δ A : = ∥ x ∥, Δ B : = ∥ y ∥,$ and define the covariance of the pair $(A, B)$ in the preparation $u$ by ${Cov}_{u} (A, B) : = Re ⟨ x, y ⟩$ . Then:

(a) $a, b \in R$ ; $x, y \in H$ ; and ${Cov}_{u} (A, B) \in R$ is symmetric in its two arguments: ${Cov}_{u} (A, B) = {Cov}_{u} (B, A)$ ;

(b) covariance-refined bound: $(Δ A)^{2} (Δ B)^{2} \geq [{Cov}_{u} (A, B)]^{2} + [Im ⟨ x, y ⟩]^{2} \geq [{Cov}_{u} (A, B)]^{2},$ so in particular $Δ A Δ B \geq ∣ {Cov}_{u} (A, B) ∣$ ;

(c) equality holds in (b)'s first inequality if and only if $x$ and $y$ are linearly dependent (in the convention that $x = 0$ or $y = 0$ counts as dependence);

(d) if in addition $u \in D (A B) \cap D (B A)$ — that is $B u \in D (A)$ and $A u \in D (B)$ — then the pairing $⟨ [A, B] u, u ⟩ : = ⟨ A B u, u ⟩ - ⟨ B A u, u ⟩$ is defined and $⟨ [A, B] u, u ⟩ = - 2 i Im ⟨ x, y ⟩,$ in particular it is purely imaginary, and consequently $(Δ A)^{2} (Δ B)^{2} \geq [{Cov}_{u} (A, B)]^{2} + \frac{1}{4} ∣ ⟨ [A, B] u, u ⟩ ∣^{2}, Δ A Δ B \geq \frac{1}{2} ∣ ⟨ [A, B] u, u ⟩ ∣ .$

Clauses (a)-(c) use no product $A B$ or $B A$ and no assumption on the domains of such products. The quantities are preparation quantities of the vector $u$ ; no measurement model, apparatus error or energy-time interpretation is asserted.

Facts & Assumptions

Given: symmetric densely defined $A, B$ on a complex Hilbert space $H$ , a unit vector $u \in D (A) \cap D (B)$ , the numbers $a, b$ , vectors $x, y$ and $Δ A, Δ B$ .

[F1]

$(T - λ) v \in H$ for $T$ symmetric, $v \in D (T)$ , $λ \in R$ ; and $⟨ T v, v ⟩ = \overline{⟨ T v, v ⟩}$ , so the mean of a symmetric operator in any vector of its domain is real (Symmetric, self-adjoint and essentially self-adjoint operators, Adjoint of a densely defined operator).

[F2]

Cauchy–Schwarz in a complex inner product space: $∣ ⟨ v, w ⟩ ∣ \leq ∥ v ∥ ∥ w ∥$ , with equality if and only if $v, w$ are linearly dependent; and $∣ ⟨ v, w ⟩ ∣^{2} = (Re ⟨ v, w ⟩)^{2} + (Im ⟨ v, w ⟩)^{2}$ (Cauchy–Schwarz: $∣ ⟨ u, v ⟩ ∣ \leq ∥ u ∥ ∥ v ∥$ , with equality exactly for linearly dependent vectors).

[F3]

The weak-form Robertson lemma of Weak-form Robertson uncertainty on a common domain states in clause (b) exactly the inequality $(Δ A)^{2} (Δ B)^{2} \geq [Re ⟨ x, y ⟩]^{2} + [Im ⟨ x, y ⟩]^{2}$ for the present data; it is recorded here for comparison, and is derived below from [F2] without operator products.

[F4]

If $v, w \in H$ then $⟨ w, v ⟩ = \overline{⟨ v, w ⟩}$ , hence $Re ⟨ w, v ⟩ = Re ⟨ v, w ⟩$ .

Proof

1.1givenF1F4

[Means, centered vectors and covariance] By [F1], $a$ and $b$ are real, so $x = (A - a) u$ and $y = (B - b) u$ are well-defined vectors of $H$ (the sums $(A - a) u = A u - a u$ lie in $H$ because $A u, B u \in H$ and $a, b$ are scalars). By [F4], ${Cov}_{u} (B, A) = Re ⟨ y, x ⟩ = Re ⟨ x, y ⟩ = {Cov}_{u} (A, B)$ , which is real by definition. This is (a).

1.2givenF2F3algebra

[The refined bound] Apply [F2] to $x, y$ : $Δ A Δ B = ∥ x ∥ ∥ y ∥ \geq ∣ ⟨ x, y ⟩ ∣$ , and [F2] again gives $∣ ⟨ x, y ⟩ ∣^{2} = (Re ⟨ x, y ⟩)^{2} + (Im ⟨ x, y ⟩)^{2} = {Cov}_{u} (A, B)^{2} + (Im ⟨ x, y ⟩)^{2}$ . Squaring the first inequality and substituting the second gives $(Δ A)^{2} (Δ B)^{2} \geq ∣ ⟨ x, y ⟩ ∣^{2} \geq {Cov}_{u} (A, B)^{2}$ , which is (b) and reproduces [F3] without any operator product.

2.1givenF2step 1.2algebra

[Equality case] Equality holds in $(Δ A)^{2} (Δ B)^{2} \geq [{Cov}_{u} (A, B)]^{2} + [Im ⟨ x, y ⟩]^{2}$ exactly when equality holds in $∥ x ∥ ∥ y ∥ \geq ∣ ⟨ x, y ⟩ ∣$ , since the second step $∣ ⟨ x, y ⟩ ∣^{2} \geq (Re ⟨ x, y ⟩)^{2}$ is an identity with $[Im ⟨ x, y ⟩]^{2}$ added, i.e. it is always an equality. By [F2] that holds exactly for linearly dependent $x, y$ . This is (c).

3.1givenstep 1.2F1F4algebra∎

[Commutator pairing] Assume $u \in D (A B) \cap D (B A)$ . Then $B u \in D (A)$ and $A u \in D (B)$ , so both $⟨ A B u, u ⟩ = ⟨ B u, A u ⟩$ (symmetry of $A$ applied to $B u \in D (A)$ and $u$ ) and $⟨ B A u, u ⟩ = ⟨ A u, B u ⟩$ (symmetry of $B$ ) are defined. Writing $z : = ⟨ x, y ⟩$ , the computation $z = ⟨ A u, B u ⟩ - a b - a b + a b = ⟨ A u, B u ⟩ - a b$ with real $a, b$ gives $\overline{z} = ⟨ B u, A u ⟩ - a b$ by [F4]. Hence $⟨ [A, B] u, u ⟩ = (\overline{z} + a b) - (z + a b) = \overline{z} - z = - 2 i Im z = - 2 i Im ⟨ x, y ⟩$ , which is purely imaginary. Substituting into (b) gives $(Δ A)^{2} (Δ B)^{2} \geq {Cov}_{u} (A, B)^{2} + \frac{1}{4} ∣ ⟨ [A, B] u, u ⟩ ∣^{2}$ and, taking square roots, $Δ A Δ B \geq \frac{1}{2} ∣ ⟨ [A, B] u, u ⟩ ∣$ . This is (d).

TheoremStatement: AI-adaptedProof: AI-adaptedOpen item page →

Position-momentum uncertainty on the maximal domains of the line

Statement

Assume the Axiom of Choice and the Axiom of Countable Choice (The Axiom of Choice, The Axiom of Countable Choice ( ${A C}_{ω}$ )). Fix $ℏ > 0$ and let $Q$ and $P$ be the position and momentum operators of Position and momentum operators on Euclidean L2 on $L^{2} (R; C)$ ; by Self-adjointness, Fourier representation and translation identities of position and momentum both are self-adjoint, $D (P) = W^{1, 2} (R; C)$ and $D (Q) = {f : x f \in L^{2}}$ . Let $f \in D (Q) \cap D (P)$ have $∥ f ∥ = 1$ , and put $a : = ⟨ Q f, f ⟩ = \int_{R} x ∣ f (x) ∣^{2} d x, b : = ⟨ P f, f ⟩,$ $x_{Q} : = (Q - a) f, y_{P} : = (P - b) f, Δ Q : = ∥ x_{Q} ∥, Δ P : = ∥ y_{P} ∥ .$ Then:

(a) $a, b \in R$ , and with $m_{1} : = \int_{R} f^{'} \overline{f} d x$ and $m_{2} : = \int_{R} x f^{'} \overline{f} d x$ (absolutely convergent complex integrals, $f^{'}$ the weak derivative) one has $Re m_{1} = 0, Re m_{2} = - \frac{1}{2}, b = ℏ Im m_{1};$

(b) $⟨ x_{Q}, y_{P} ⟩ = - \frac{i ℏ}{2} + ℏ Im m_{2} - a b$ ; consequently $Im ⟨ x_{Q}, y_{P} ⟩ = - \frac{ℏ}{2}$ and $Δ Q Δ P \geq \frac{ℏ}{2};$

(c) equality $Δ Q Δ P = \frac{ℏ}{2}$ holds if and only if $x_{Q}, y_{P}$ are linearly dependent and $Re ⟨ x_{Q}, y_{P} ⟩ = 0$ ;

(d) no assumption on the products $Q P$ or $P Q$ is used: the assertion holds on the full maximal domain of the pair, and the means $a, b$ , the centered vectors $x_{Q}, y_{P}$ and the variances are finite on it.

The statement concerns a preparation vector only; it asserts no apparatus error, jointly measured distribution or physical interpretation.

Facts & Assumptions

Given: a unit vector $f \in W^{1, 2} (R; C)$ with $x f \in L^{2}$ , its weak derivative $f^{'} \in L^{2}$ , the numbers $a, b$ , the vectors $x_{Q}, y_{P}$ and the variances.

[F1]

$f$ has a unique continuous locally absolutely continuous representative $f^{*}$ with $f^{*} (x) - f^{*} (y) = \int_{y}^{x} f^{'}$ ; moreover $∣ f^{*} (x) ∣ \leq C_{f} : = ∥ f ∥ + ∥ f^{'} ∥ < \infty$ for all $x$ and $f^{*} (x) \to 0$ as $∣ x ∣ \to \infty$ (One-dimensional $W^{1, p}$ functions have unique absolutely continuous representatives, Endpoint trace bound and decay of H1 functions clause (c)).

[F2]

There is $χ \in C_{c}^{\infty} (R)$ with $0 \leq χ \leq 1$ , $χ = 1$ on $[- 1, 1]$ , $χ = 0$ outside $[- 2, 2]$ ; for $R > 0$ , $χ_{R} (x) : = χ (x / R)$ satisfies $χ_{R} = 1$ on $[- R, R]$ , $χ_{R} = 0$ outside $[- 2 R, 2 R]$ , and $∣ x χ_{R}^{'} (x) ∣ \leq \sup_{t} ∣ t χ^{'} (t) ∣ = : C_{χ} < \infty$ (Explicit compactly supported smooth cutoffs).

[F3]

Complex conjugation preserves $W^{1, 2}$ and $D (\overline{f}) = \overline{D f}$ : this follows from the weak-derivative definition applied to the conjugate test functions, since $\int \overline{f} φ^{'} = \overline{\int f \overline{φ^{'}}} = - \overline{\int D f \overline{φ}} = - \int \overline{D f} φ$ for real-valued $φ$ and hence for all test functions by linearity.

[F4]

Smooth-factor Leibniz rule: if $η$ is smooth and $D^{β} η$ is bounded for $∣ β ∣ \leq 1$ and $u \in W^{1, 2}$ , then $η u \in W^{1, 2}$ with $D (η u) = (D η) u + η D u$ (Weak Leibniz rule with a smooth factor).

[F5]

Weak product rule for bounded Sobolev factors: if $u, v \in W^{1, 2} (R; C) \cap L^{\infty}$ , then $u v \in W^{1, 2}$ with $D (u v) = (D u) v + u (D v)$ (Weak product rule for bounded Sobolev functions).

[F6]

If $g \in W^{1, 1} (R; C)$ has compact support, then $\int_{R} D g = 0$ : the absolutely continuous representative of $g$ (for $p = 1$ ) vanishes outside a compact set, so its limiting difference and the difference formula give zero (One-dimensional $W^{1, p}$ functions have unique absolutely continuous representatives, applied with $p = 1$ ).

[F7]

Cauchy–Schwarz and its consequence: $∣ ⟨ v, w ⟩ ∣^{2} = (Re ⟨ v, w ⟩)^{2} + (Im ⟨ v, w ⟩)^{2}$ and $∥ v ∥ ∥ w ∥ \geq ∣ ⟨ v, w ⟩ ∣ \geq ∣ Im ⟨ v, w ⟩ ∣$ (Cauchy–Schwarz: $∣ ⟨ u, v ⟩ ∣ \leq ∥ u ∥ ∥ v ∥$ , with equality exactly for linearly dependent vectors).

[F8]

Dominated convergence: if $g_{R} \to g$ pointwise and $∣ g_{R} ∣ \leq G$ with $\int G < \infty$ , then $\int g_{R} \to \int g$ (Dominated convergence).

[F9]

Covariance-refined uncertainty of Covariance-refined preparation uncertainty without operator products: for symmetric densely defined operators $A, B$ and a unit vector $u \in D (A) \cap D (B)$ , with centered vectors $x = (A - a) u$ , $y = (B - b) u$ and $z : = ⟨ x, y ⟩$ , one has $Δ A Δ B \geq ∣ z ∣ \geq ∣ Im z ∣$ , with $∣ z ∣^{2} = (Re z)^{2} + (Im z)^{2}$ ; equality in $Δ A Δ B \geq ∣ z ∣$ holds exactly when $x$ and $y$ are linearly dependent (clauses (b) and (c) there, the equality clause being that of the Cauchy–Schwarz step).

Proof

1.1givenF7algebra

[Integrability of the two pairings] Since $f, x f, f^{'} \in L^{2} (R)$ , Cauchy–Schwarz gives $\int ∣ f^{'} \overline{f} ∣ \leq ∥ f^{'} ∥ ∥ f ∥ < \infty$ and $\int ∣ x f^{'} \overline{f} ∣ \leq ∥ x f ∥ ∥ f^{'} ∥ < \infty$ ; so $m_{1}, m_{2}$ are well-defined complex numbers.

1.2givenF2F3F4F5algebra

[Smooth cutoffs and products in W^{1,2}] For $R > 0$ put $η_{R} (x) : = x χ_{R} (x)$ and $u_{R} : = η_{R} f$ . By [F2], $η_{R}$ and $η_{R}^{'}$ are bounded (by $2 R$ and $1 + C_{χ}$ ), so [F4] gives $u_{R} \in W^{1, 2}$ with $D u_{R} = (χ_{R} + x χ_{R}^{'}) f + x χ_{R} f^{'}$ ; also $u_{R}$ is bounded and compactly supported. By [F3] and [F1], $\overline{f} \in W^{1, 2} \cap L^{\infty}$ with $D \overline{f} = \overline{f^{'}}$ . Hence [F5] applies to $u_{R}$ and $\overline{f}$ : $g_{R} : = u_{R} \overline{f} \in W^{1, 2} (R; C)$ , compactly supported, with $D g_{R} = (χ_{R} + x χ_{R}^{'}) ∣ f ∣^{2} + 2 x χ_{R} Re (f^{'} \overline{f}),$ using $f^{'} \overline{f} + \overline{f^{'}} f = 2 Re (f^{'} \overline{f})$ .

2.1givenstep 1.2F6algebra

[The integrated identity] By [F6] applied to the compactly supported $W^{1, 2}$ (hence $W^{1, 1}$ ) class $g_{R}$ , $0 = \int_{R} D g_{R} = \int_{R} χ_{R} ∣ f ∣^{2} + \int_{R} x χ_{R}^{'} ∣ f ∣^{2} + 2 Re \int_{R} x χ_{R} f^{'} \overline{f}$ .

3.1givenstep 1.1step 2.1F8algebra

[Limits as $R \to \infty$ ] As $R \to \infty$ : (i) $χ_{R} ∣ f ∣^{2} \to ∣ f ∣^{2}$ pointwise with $0 \leq χ_{R} ∣ f ∣^{2} \leq ∣ f ∣^{2} \in L^{1}$ , so $\int χ_{R} ∣ f ∣^{2} \to ∥ f ∥^{2} = 1$ by [F8]; (ii) $∣ x χ_{R}^{'} ∣$ is bounded by $C_{χ}$ and supported in ${∣ x ∣ \geq R}$ , so $∣ \int x χ_{R}^{'} ∣ f ∣^{2} ∣ \leq C_{χ} \int_{{∣ x ∣ \geq R}} ∣ f ∣^{2} \to 0$ by absolute continuity of the integral; (iii) $x χ_{R} f^{'} \overline{f} \to x f^{'} \overline{f}$ pointwise with $∣ x χ_{R} f^{'} \overline{f} ∣ \leq ∣ x f^{'} \overline{f} ∣ \in L^{1}$ by step 1.1, so $\int x χ_{R} f^{'} \overline{f} \to m_{2}$ by [F8]. Passing to the limit in step 2.1 gives $1 + 2 Re m_{2} = 0$ , that is $Re m_{2} = - \frac{1}{2}$ .

4.1givenstep 3.1F6F8algebra

[The second identity] Applying the same construction with $η_{R}$ replaced by $χ_{R}$ (so $g_{R}^{0} : = χ_{R} f \overline{f} \in W^{1, 2}$ , compactly supported, with $D g_{R}^{0} = χ_{R}^{'} ∣ f ∣^{2} + 2 χ_{R} Re (f^{'} \overline{f})$ ) and using [F6], $0 = \int χ_{R}^{'} ∣ f ∣^{2} + 2 Re \int χ_{R} f^{'} \overline{f}$ ; as $R \to \infty$ the first term tends to $0$ as in step 3.1(ii) and the second to $2 Re m_{1}$ by dominated convergence ( $∣ f^{'} \overline{f} ∣ \leq \frac{1}{2} (∣ f ∣^{2} + ∣ f^{'} ∣^{2}) \in L^{1}$ ). Hence $Re m_{1} = 0$ .

5.1givenstep 4.1F3algebra

[The means] $a$ is real because $Q$ is self-adjoint and $f \in D (Q)$ ; $b$ is real because $P$ is self-adjoint and $f \in D (P)$ . Moreover $b = ⟨ P f, f ⟩ = \int_{R} (- i ℏ f^{'}) \overline{f} d x = - i ℏ m_{1}$ , and $Re m_{1} = 0$ gives $- i ℏ m_{1} = ℏ Im m_{1}$ , real as required. This is (a).

6.1givenstep 3.1step 5.1algebra

[The pairing] Expand $⟨ x_{Q}, y_{P} ⟩ = ⟨ Q f, P f ⟩ - b ⟨ Q f, f ⟩ - a ⟨ f, P f ⟩ + a b ⟨ f, f ⟩$ . Here $⟨ Q f, f ⟩ = a$ and $⟨ f, P f ⟩ = \overline{⟨ P f, f ⟩} = b$ (both means real), so the last three terms sum to $- a b + a b - a b = - a b$ . And $⟨ Q f, P f ⟩ = \int x f \overline{(- i ℏ f^{'})} d x = i ℏ \int x f \overline{f^{'}} d x = i ℏ \overline{m_{2}} = i ℏ (- \frac{1}{2} - i Im m_{2}) = - \frac{i ℏ}{2} + ℏ Im m_{2}$ , using step 3.1. Hence $⟨ x_{Q}, y_{P} ⟩ = - \frac{i ℏ}{2} + ℏ Im m_{2} - a b$ , with imaginary part $- \frac{ℏ}{2}$ .

7.1givenstep 6.1F7F9algebra

[The uncertainty bound] By [F9] (or [F7] applied to $x_{Q}$ and $y_{P}$ ), $Δ Q Δ P = ∥ x_{Q} ∥ ∥ y_{P} ∥ \geq ∣ ⟨ x_{Q}, y_{P} ⟩ ∣ \geq ∣ Im ⟨ x_{Q}, y_{P} ⟩ ∣ = \frac{ℏ}{2}$ . This is (b).

8.1givenstep 6.1step 7.1F9algebra∎

[Equality and scope] By the equality clause of [F9], equality in $∥ x_{Q} ∥ ∥ y_{P} ∥ \geq ∣ ⟨ x_{Q}, y_{P} ⟩ ∣$ holds exactly when $x_{Q}, y_{P}$ are linearly dependent; given step 6.1, the further condition $∣ ⟨ x_{Q}, y_{P} ⟩ ∣ = \frac{ℏ}{2}$ is exactly $Re ⟨ x_{Q}, y_{P} ⟩ = 0$ by the identity $∣ ⟨ x_{Q}, y_{P} ⟩ ∣^{2} = (Re ⟨ x_{Q}, y_{P} ⟩)^{2} + (Im ⟨ x_{Q}, y_{P} ⟩)^{2}$ of [F9]. This is (c). No product $Q P$ or $P Q$ was used anywhere: the means, the centered vectors and the pairings were formed from $Q f$ and $P f$ only, so (d) holds.

TheoremStatement: AI-adaptedProof: AI-adaptedOpen item page →

Galilean boost covariance of the Schrodinger model with bounded potential

Statement

Assume the Axiom of Choice and the Axiom of Countable Choice (The Axiom of Choice, The Axiom of Countable Choice ( ${A C}_{ω}$ )). Fix $ℏ > 0$ , $m > 0$ , a real bounded measurable $V : R^{3} \to R$ , and let $K : = \frac{1}{2 m} \sum_{j} P_{j}^{2} + M_{V}$ with domain $W^{2, 2} (R^{3}; C)$ be the free-plus-bounded-potential operator of Real multiplication operators and free plus bounded-potential Schrödinger operators clause (c); write $V_{u} (x, t) : = V (x - u t)$ for $u \in R^{3}$ . For $u \in R^{3}$ and $t \in R$ define the boost operator $(B_{u} (t) ψ) (x) : = e^{i (\frac{m}{ℏ} u \cdot x - \frac{m ∣ u ∣^{2}}{2 ℏ} t)} ψ (x - u t), ψ \in L^{2} (R^{3}; C) .$ Then:

(a) for every $t$ , $B_{u} (t)$ is unitary with $B_{u} (t)^{- 1} = B_{u} (t)^{*}$ , and $B_{u} (t)$ maps $W^{1, 2}$ and $W^{2, 2}$ onto themselves;

(b) for every $u_{0} \in W^{2, 2}$ and every $s \in R$ the curve $v (t) : = B_{u} (t) e^{- i (t - s) K / ℏ} B_{u} (s)^{- 1} u_{0}$ lies in $W^{2, 2}$ , is differentiable into $L^{2}$ on $R$ , and satisfies $v (s) = u_{0}$ and $i ℏ v^{'} (t) = (\frac{1}{2 m} \sum_{j} P_{j}^{2} + M_{V_{u} (\cdot, t)}) v (t) (t \in R),$ where $M_{V_{u} (\cdot, t)}$ is multiplication by $x \mapsto V (x - u t)$ . It is the unique such curve: if $w : R \to W^{2, 2}$ is continuous into $W^{2, 2}$ , differentiable into $L^{2}$ and satisfies the same equation with $w (s) = u_{0}$ , then $w = v$ ;

(c) the two-parameter family $U_{u} (t, s) : = B_{u} (t) e^{- i (t - s) K / ℏ} B_{u} (s)^{- 1}$ consists of unitaries preserving $W^{2, 2}$ , satisfies $U_{u} (t, t) = I$ , $U_{u} (t, r) U_{u} (r, s) = U_{u} (t, s)$ , and $v (t) = U_{u} (t, s) v_{0}$ is the solution of (b) at time $s$ for every $v_{0} \in W^{2, 2}$ ;

(d) free propagator covariance. With $K_{0} : = \frac{1}{2 m} \sum_{j} P_{j}^{2}$ (that is $V = 0$ ) one has $B_{u} (t) e^{- i (t - s) K_{0} / ℏ} = e^{- i (t - s) K_{0} / ℏ} B_{u} (s)$ for all $t, s$ .

All assertions concern the displayed unitary family and the Schrodinger-type equation; no Lorentz structure, no photon limit and no worldline of an unmeasured particle is asserted.

Facts & Assumptions

Given: $ℏ, m > 0$ , $u \in R^{3}$ , a real bounded measurable $V$ , the boost $B_{u} (t) = M_{e^{i φ (\cdot, t)}} T_{u t}$ with phase $φ (x, t) : = \frac{m}{ℏ} u \cdot x - \frac{m ∣ u ∣^{2}}{2 ℏ} t$ , translation $T_{a} f (x) = f (x - a)$ , the operator $K = \frac{1}{2 m} \sum_{j} P_{j}^{2} + M_{V}$ , and $A : = \frac{1}{2 m} \sum_{j} P_{j}^{2} + M_{V_{u} (\cdot, t)}$ .

[F1]

$M_{V}$ is bounded self-adjoint; $K$ is self-adjoint with $D (K) = W^{2, 2}$ , and $e^{- i t K / ℏ}$ is a strongly continuous unitary group with $e^{- i t K / ℏ} D (K) = D (K)$ , commuting with $K$ on $D (K)$ , and derivative $- \frac{i}{ℏ} K x$ exactly on $D (K)$ (Real multiplication operators and free plus bounded-potential Schrödinger operators clause (c), A self-adjoint operator generates a strongly continuous unitary group, Integral of a measurable function against a projection-valued measure).

[F2]

Translations and unimodular multiplications: $T_{a}$ is unitary and preserves $W^{1, 2}$ , $W^{2, 2}$ , with $D_{j} (T_{a} f) = T_{a} (D_{j} f)$ (Self-adjointness, Fourier representation and translation identities of position and momentum clause (c)); the phase family $M_{e^{i φ (\cdot, t)}}$ is unitary, preserves $W^{1, 2}$ and $W^{2, 2}$ , and has $t$ -derivative $i (\partial_{t} φ) M_{e^{i φ (\cdot, t)}} = - \frac{i m ∣ u ∣^{2}}{2 ℏ} M_{e^{i φ (\cdot, t)}}$ on $L^{2}$ (Gauge multiplication preserves H2 and transports the magnetic momentum applied with $(q, ℏ, χ) = (1, 1, φ)$ , whose hypotheses hold because $φ$ is affine in $x$ and $\partial_{t} φ = - \frac{m ∣ u ∣^{2}}{2 ℏ}$ ).

[F3]

For a smooth scalar $f : R^{3} \to C$ , define $\nabla f : = (\partial_{1} f, \partial_{2} f, \partial_{3} f)$ and $Δ f : = \sum_{j = 1}^{3} \partial_{j}^{2} f$ . The complex exponential addition law $\exp (z + w) = \exp z \exp w$ holds ( $\exp (z + w) = \exp z \exp w$ , and the complex exponential extends the real exponential). For fixed $t$ , put $α_{j} : = \partial_{j} φ = m u_{j} / ℏ$ ; the gauge-domain theorem F2, clause (b), gives the first derivative identity as an $L^{2}$ class, which is classical for smooth $ψ$ ; differentiating it with the product rule Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ gives $\partial_{j} (e^{i φ} ψ) = e^{i φ} (\partial_{j} ψ + i α_{j} ψ)$ , and, since $α_{j}$ is constant in $x$ , $\partial_{j}^{2} (e^{i φ} ψ) = e^{i φ} (\partial_{j}^{2} ψ + 2 i α_{j} \partial_{j} ψ - α_{j}^{2} ψ)$ . Summing these coordinate identities yields the gradient and Laplacian formulas used below.

[F4]

Density extends the identities of [F3] to the Sobolev classes. First, for fixed $ψ \in W^{1, 2}$ the translation-difference-quotient identity $\frac{1}{h} (T_{u h} ψ - ψ) \to - u \cdot \nabla ψ$ in $L^{2}$ as $h \to 0$ : for smooth $ψ$ this is the classical chain rule, $\frac{1}{h} (T_{u h} ψ - ψ) (x) = - \frac{1}{h} \int_{0}^{h} (u \cdot \nabla ψ) (x - s u) d s$ , and the bound $∥ \frac{1}{h} (T_{u h} ν - ν) ∥ \leq ∥ u \cdot \nabla ν ∥$ for smooth $ν$ together with the density of $C_{c}^{\infty}$ in $W^{1, 2}$ extends the identity to all $ψ \in W^{1, 2}$ (the translation is isometric by [F2]). Second, for $ρ \in W^{1, 2}$ and $w : = B_{u} ρ$ , one has $B_{u} (\nabla ρ) = \nabla w - i \frac{m}{ℏ} u w$ as $L^{2}$ vector classes; and for $ρ \in W^{2, 2}$ also $B_{u} (Δ ρ) = Δ w - 2 i \frac{m}{ℏ} u \cdot \nabla w - \frac{m^{2} ∣ u ∣^{2}}{ℏ^{2}} w$ as multiplier classes. Both follow from the smooth case in [F3] by density of $C_{c}^{\infty}$ in $W^{1, 2}$ and $W^{2, 2}$ (Integer-order Sobolev spaces and their norms, Compactly supported smooth functions are dense in W^{k,p}(R^n)) and continuity of the maps involved in the relevant norms.

[F5]

$P_{j} = - i ℏ D_{j}$ on its domain, so $u \cdot P ψ = - i ℏ u \cdot \nabla ψ$ for $\nabla ψ \in L^{2}$ ; and $B_{u} (t)^{- 1} = T_{- u t} M_{e^{- i φ (\cdot, t)}}$ with the analogous derivative $i (\partial_{t} φ)$ term (Self-adjointness, Fourier representation and translation identities of position and momentum, Position and momentum operators on Euclidean L2, [F2]).

[F6]

A real function continuous on a closed interval and differentiable in its interior with zero derivative is constant, by The mean value theorem, as the case $g (x) = x$ of Cauchy's: for $f$ continuous on $[a, b]$ with $a < b$ and differentiable on $(a, b)$ there is $c \in (a, b)$ with $f (b) - f (a) = f^{'} (c) (b - a)$ ; restrictions of a differentiable curve to compact intervals satisfy these continuity hypotheses.

Proof

1.1givenF2

[Unitarity and domain preservation] By [F2], $B_{u} (t)$ is a product of two unitary operators preserving $W^{1, 2}$ and $W^{2, 2}$ , hence unitary on $L^{2}$ and preserving both Sobolev spaces; its inverse is $T_{- u t} M_{e^{- i φ}}$ , so $B_{u} (t)^{- 1} = B_{u} (t)^{*}$ and (a) holds.

1.2givenF2F3F4algebra

[Derivative identities] Put $λ : = \frac{m ∣ u ∣^{2}}{2 ℏ}$ . Direct computation from the definition gives $B_{u} (t + h) = B_{u} (t) e^{- i λ h} T_{u h}$ , so for fixed $ψ \in W^{1, 2}$ the difference quotient is $\frac{1}{h} (B_{u} (t + h) ψ - B_{u} (t) ψ) = B_{u} (t) [e^{- i λ h} \frac{T_{u h} ψ - ψ}{h} + \frac{e^{- i λ h} - 1}{h} ψ]$ . The bracket tends to $- u \cdot \nabla ψ - i λ ψ$ in $L^{2}$ : the first term by the translation-difference-quotient identity of [F4] (classical chain rule [F3] extended by density), the second by [F2] instantiated at the affine phase $φ^{'} (x, t) : = - λ t$ , which gives $\frac{d}{d t} e^{- i λ t} = - \frac{i m ∣ u ∣^{2}}{2 ℏ} e^{- i λ t}$ . Hence $\frac{d}{d t} [B_{u} (t) ψ] = - i λ B_{u} (t) ψ - B_{u} (t) (u \cdot \nabla ψ)$ . By the transported-gradient identity of [F4], $B_{u} (t) (u \cdot \nabla ψ) = u \cdot \nabla (B_{u} (t) ψ) - 2 i λ B_{u} (t) ψ$ , so the identity also reads $\frac{d}{d t} [B_{u} (t) ψ] = i λ B_{u} (t) ψ - u \cdot \nabla (B_{u} (t) ψ)$ ; replacing $u$ by $- u$ , which gives $B_{- u} (t) = B_{u} (t)^{- 1}$ directly from the definition, yields $\frac{d}{d t} [B_{u} (t)^{- 1} η] = i λ B_{u} (t)^{- 1} η + u \cdot \nabla (B_{u} (t)^{- 1} η)$ for fixed $η \in W^{1, 2}$ .

1.3givenF2algebra

[Transported potential] $B_{u} (t) M_{V} B_{u} (t)^{- 1} = M_{V (\cdot - u t)}$ : indeed $T_{u t} M_{V} T_{- u t} = M_{V (\cdot - u t)}$ by the change of variables $y = x - u t$ , and conjugation by a unimodular multiplication fixes multiplication by a real function.

1.4givenF4algebra

[Transported Laplacian] Let $ρ \in W^{2, 2}$ and $w : = B_{u} (t) ρ$ . Applying [F4] twice, $B_{u} (\nabla ρ) = \nabla w - i \frac{m}{ℏ} u w$ and $B_{u} (Δ ρ) = Δ w - 2 i \frac{m}{ℏ} u \cdot \nabla w - \frac{m^{2} ∣ u ∣^{2}}{ℏ^{2}} w$ ; consequently $B_{u} (t) (- \frac{ℏ^{2}}{2 m} Δ ρ) = - \frac{ℏ^{2}}{2 m} Δ w + i ℏ u \cdot \nabla w + \frac{m ∣ u ∣^{2}}{2} w$ .

2.1givenstep 1.2step 1.3step 1.4F1F2F4F5algebra

[Forward equation and initial data] Let $u_{0} \in W^{2, 2}$ and $s \in R$ , put $w_{0} : = B_{u} (s)^{- 1} u_{0}$ , $w (t) : = e^{- i (t - s) K / ℏ} w_{0}$ and $v (t) : = B_{u} (t) w (t)$ . By [F1] the group preserves $D (K)$ , commutes with $K$ there and is strongly continuous both on $L^{2}$ and on $D (K)$ with graph norm: apply strong continuity to $w_{0}$ and $K w_{0}$ . That graph norm is equivalent to the $W^{2, 2}$ norm, since the free operator has this equivalence and the bounded perturbation satisfies $∥ K_{0} f ∥ \leq ∥ K f ∥ + ∥ V ∥_{\infty} ∥ f ∥$ and $∥ K f ∥ \leq ∥ K_{0} f ∥ + ∥ V ∥_{\infty} ∥ f ∥$ . Hence $w$ is continuous into $W^{2, 2}$ and differentiable into $L^{2}$ with $w^{'} = - i K w / ℏ$ . The boosts are strongly continuous on $W^{2, 2}$ : translations are isometries in each weak-derivative norm and are continuous by smooth density, while derivatives of the affine-phase multiplier are constant multiples of that multiplier; its time dependence is a scalar phase. They are locally uniformly bounded on $W^{2, 2}$ . Thus $v$ is continuous into $W^{2, 2}$ . For differentiability, decompose its quotient into $B_{u} (t + h) (w (t + h) - w (t)) / h + (B_{u} (t + h) - B_{u} (t)) w (t) / h$ ; strong continuity on $L^{2}$ , unitarity and step 1.2 give $v^{'} = B_{u} (t) w^{'} + i λ v - u \cdot \nabla v$ , where $λ = m ∣ u ∣^{2} / (2 ℏ)$ . By steps 1.3-1.4, $B_{u} (t) K w = - ℏ^{2} Δ v / (2 m) + i ℏ u \cdot \nabla v + m ∣ u ∣^{2} v / 2 + M_{V_{u} (\cdot, t)} v$ . Therefore $i ℏ v^{'} = - m ∣ u ∣^{2} v / 2 - i ℏ u \cdot \nabla v + B_{u} (t) K w = (- ℏ^{2} Δ / (2 m) + M_{V_{u} (\cdot, t)}) v$ ; both extra terms cancel. At the actual initial time, $v (s) = B_{u} (s) w_{0} = u_{0}$ , with no assumption that $s = 0$ . This proves existence and the initial condition in (b).

3.1givenstep 1.2step 1.3step 1.4step 2.1F1F5algebraF6

[Uniqueness with the transformed initial datum] Let $w$ be another solution in the stated class, and put $ρ (t) : = B_{u} (t)^{- 1} w (t)$ . The continuity argument of step 2.1 applies also to the inverse boost. For its $L^{2}$ derivative, decompose the difference quotient just as in step 2.1, using the inverse derivative from step 1.2; this gives $ρ^{'} = i λ ρ + u \cdot \nabla ρ + B_{u} (t)^{- 1} w^{'}$ . The inverse transported identities of steps 1.3-1.4 give $B_{u} (t)^{- 1} (- ℏ^{2} Δ / (2 m) + M_{V_{u} (\cdot, t)}) B_{u} (t) ρ = K ρ - i ℏ u \cdot \nabla ρ + m ∣ u ∣^{2} ρ / 2$ . Substituting the equation for $w$ cancels the scalar and first-order terms, so $i ℏ ρ^{'} = K ρ$ and $ρ (s) = B_{u} (s)^{- 1} u_{0}$ . Set $G (t) : = e^{i (t - s) K / ℏ}$ and $F (t) : = G (t) ρ (t)$ . The quotient decomposition $G (t + h) (ρ (t + h) - ρ (t)) / h + (G (t + h) - G (t)) ρ (t) / h$ converges in $L^{2}$ : the first term by strong continuity and unitarity, the second by [F1] on the fixed vector $ρ (t) \in D (K)$ . Hence $F^{'} = G (t) (ρ^{'} + i K ρ / ℏ) = 0$ . A differentiable Hilbert-space curve with zero derivative is constant, by applying the real scalar mean-value theorem to every bounded real linear functional. Thus $F (t) = F (s) = B_{u} (s)^{- 1} u_{0}$ , and $ρ (t) = e^{- i (t - s) K / ℏ} B_{u} (s)^{- 1} u_{0}$ . Multiplying by $B_{u} (t)$ proves $w (t) = B_{u} (t) e^{- i (t - s) K / ℏ} B_{u} (s)^{- 1} u_{0} = v (t)$ . This proves the uniqueness part of (b) with its actual initial datum.

4.1givenstep 2.1step 3.1F1algebra

[Propagator properties] Each $U_{u} (t, s)$ is unitary and preserves $W^{2, 2}$ by (a) and [F1]; $U_{u} (t, t) = B_{u} (t) B_{u} (t)^{- 1} = I$ ; and the composition law follows from $B_{u} (r)^{- 1} B_{u} (r) = I$ and the group law of $e^{- i τ K / ℏ}$ : $U_{u} (t, r) U_{u} (r, s) = B_{u} (t) e^{- i (t - r) K / ℏ} e^{- i (r - s) K / ℏ} B_{u} (s)^{- 1} = U_{u} (t, s)$ . By step 2.1, $t \mapsto U_{u} (t, s) v_{0}$ solves the moving-potential equation with initial value $v_{0}$ , and step 3.1 makes it unique. This is (c).

5.1step 3.1step 4.1F1F4algebra∎

[Free covariance] Let $V = 0$ , so $V_{u} = 0$ , $K = K_{0}$ and the equation of (b) is the free equation. By step 4.1 with $V = 0$ , the propagator for the free equation is $B_{u} (t) e^{- i (t - s) K_{0} / ℏ} B_{u} (s)^{- 1}$ , and it equals $e^{- i (t - s) K_{0} / ℏ}$ because the free equation has at most one $W^{2, 2}$ -valued solution for each initial datum (step 3.1 with $V = 0$ ). Multiplying the identity $B_{u} (t) e^{- i (t - s) K_{0} / ℏ} B_{u} (s)^{- 1} = e^{- i (t - s) K_{0} / ℏ}$ on the right by $B_{u} (s)$ gives (d). This equality first holds on $W^{2, 2}$ and extends to all $L^{2}$ because both sides are bounded unitaries and $C_{c}^{\infty} \subset W^{2, 2}$ is dense in $L^{2}$ by [F4].

TheoremStatement: AI-adaptedProof: AI-adaptedOpen item page →

Conserved spectral probability laws for commuting observables and stationary eigenrays

Statement

Assume the Axiom of Choice and the Axiom of Countable Choice (The Axiom of Choice, The Axiom of Countable Choice ( ${A C}_{ω}$ )). Let $H$ be a nonzero complex Hilbert space (Hilbert space), let $K$ be a self-adjoint operator on $H$ with spectral projection-valued measure $E = E_{K}$ on the Borel subsets of $R$ (Spectral theorem for unbounded self-adjoint operators (PVM form), Projection valued measure), and put $U (t) : = e^{- i t K / ℏ} = \int_{R} e^{- i t λ / ℏ} d E (λ)$ for $ℏ > 0$ , the strongly continuous unitary group generated by $- i K / ℏ$ (A self-adjoint operator generates a strongly continuous unitary group, Integral of a measurable function against a projection-valued measure).

(a) If a bounded operator $B \in B (H)$ satisfies $B E (C) = E (C) B$ for every Borel set $C \subseteq R$ , then $B U (t) = U (t) B$ for every $t \in R$ .

(a') If a bounded self-adjoint operator $S$ satisfies $S K \subseteq K S$ — that is $S (K x) = K (S x)$ for every $x \in D (K)$ — then $S U (t) = U (t) S$ for every $t$ .

(b) Let $F$ be a projection-valued measure on a measurable space $(Y, Σ)$ with $F (S) E (C) = E (C) F (S)$ for all $S \in Σ$ and all Borel $C \subseteq R$ . Then:

(i) $F (S) U (t) = U (t) F (S)$ for all $t$ and $S$ ;

(ii) for every density operator $ρ$ on $H$ — a self-adjoint positive trace-class operator with $tr ρ = 1$ (Trace class operator, Density operators and scalar spectral probability clause (a)) — the numbers $tr (ρ F (S))$ are independent of $t$ : $tr (U (t) ρ U (t)^{*} F (S)) = tr (ρ F (S)) (t \in R, S \in Σ);$

(iii) for every unit vector $ψ \in H$ and every $S$ , $∥ F (S) U (t) ψ ∥^{2} = ∥ F (S) ψ ∥^{2}$ for all $t$ ;

(c) applying (b) with $F = E$ : for every density operator $ρ$ the energy law $tr (ρ E (C))$ is independent of $t$ , for every unit $ψ$ the law $∥ E (C) U (t) ψ ∥^{2} = ∥ E (C) ψ ∥^{2}$ is independent of $t$ , and if $ψ \in D (K)$ then both existing scalar moments are conserved: $\int_{R} λ^{k} d E_{K, U (t) ψ} (λ) = \int_{R} λ^{k} d E_{K, ψ} (λ), k = 1, 2,$ with the integrals finite at $t = 0$ and equal to $⟨ K ψ, ψ ⟩$ and $∥ K ψ ∥^{2}$ respectively; moments that are infinite at $t = 0$ remain outside the assertion;

(d) if $K ψ = λ ψ$ for some $λ \in R$ and unit $ψ \in D (K)$ , then $U (t) ψ = e^{- i t λ / ℏ} ψ$ for every $t$ , and every $S \mapsto ∥ F (S) ψ ∥^{2}$ of (b) agrees with $S \mapsto ∥ F (S) U (t) ψ ∥^{2}$ .

No equilibration, relaxation or convergence of the law of an arbitrary state is asserted, and no moment beyond those finite at $t = 0$ is claimed.

Facts & Assumptions

Given: a nonzero complex Hilbert space $H$ ; a self-adjoint $K$ with spectral PVM $E$ ; $U (t) = e^{- i t K / ℏ}$ ; in (b) a PVM $F$ commuting with $E$ and a density operator $ρ$ .

[F1]

For bounded Borel $h : R \to C$ the bounded integral $Φ_{E} (h) = \int h d E$ satisfies $∥ Φ_{E} (h) x ∥^{2} = \int ∣ h ∣^{2} d E_{x}$ , $Φ_{E} (h_{1} h_{2}) = Φ_{E} (h_{1}) Φ_{E} (h_{2})$ , and $Φ_{E} (h)$ is a norm limit of integrals of simple functions: there are simple Borel functions $s_{n}$ with $∥ h - s_{n} ∥_{\infty} \to 0$ and $∥ Φ_{E} (h) - \int s_{n} d E ∥ \to 0$ (Bounded borel pvm integral, Pvm integral is a star homomorphism).

[F2]

$U (t) = Φ_{E} (f_{t})$ with $f_{t} (λ) = e^{- i t λ / ℏ}$ , bounded with $∣ f_{t} ∣ = 1$ ; each $U (t)$ is unitary and $U (t) D (K) = D (K)$ (A self-adjoint operator generates a strongly continuous unitary group, Integral of a measurable function against a projection-valued measure).

[F3]

For a self-adjoint operator $T$ with PVM $E_{T}$ and $ψ \in D (T)$ : $∥ (T - μ) ψ ∥^{2} = \int (λ - μ)^{2} d E_{T, ψ} (λ)$ , $\int λ d E_{T, ψ} = ⟨ T ψ, ψ ⟩$ and $\int λ^{2} d E_{T, ψ} = ∥ T ψ ∥^{2}$ (The unbounded PVM integral is densely defined, closed and normal, Density operators and scalar spectral probability clause (c)).

[F4]

Trace cyclicity: $tr (B T) = tr (T B)$ for trace-class $T$ and bounded $B$ ; the trace-class operators form a two-sided operator ideal, so $U ρ U^{*}$ is trace class for a density operator $ρ$ (Nuclear bounded cyclicity for trace-class operators clause (b), Rank-one operator calculus).

[F5]

Measures on $R$ are determined by their values on Borel sets; a positive measure with $μ (R ∖ {λ}) = 0$ is the point mass at $λ$ (Scalar and complex measures from a pvm).

Proof

1.1givenF1F2algebra

[Commutation passes to bounded Borel functions] Assume $B$ commutes with every $E (C)$ , hence with every simple integral $\int s d E = \sum_{k} c_{k} E (B_{k})$ . Choose simple $s_{n} \to f_{t}$ uniformly by [F1]; then $\int s_{n} d E \to U (t)$ in operator norm, and $∥ B \int s_{n} d E - \int s_{n} d E B ∥ \leq ∥ B ∥ ∥ \int s_{n} d E - U (t) ∥ + ∥ \int s_{n} d E - U (t) ∥ ∥ B ∥ \to 0$ , while each commutator is zero; hence $B U (t) = U (t) B$ . This is (a).

1.2givenF1F2algebra

[Resolvent commutation and the group] Assume $S K \subseteq K S$ for a bounded self-adjoint $S$ . For $z \in C ∖ R$ the resolvent $R (z) = (K - z)^{- 1}$ is a bounded operator with range $D (K)$ , and $S (K - z) = (K - z) S$ on $D (K)$ implies $S R (z) = R (z) S$ . For $t \in R$ and $n \geq 1$ put $W_{n} (t) : = (1 + \frac{i t}{ℏ n} K)^{- 1}$ , which is defined and bounded because it equals $- \frac{i ℏ n}{t} (K - \frac{i ℏ n}{t})^{- 1}$ for $t \neq 0$ (and $W_{n} (0) = I$ ), so $S W_{n} (t) = W_{n} (t) S$ . By the spectral theorem, $(1 + \frac{i t λ}{ℏ n})^{- n} \to e^{- i t λ / ℏ}$ boundedly for every $λ$ , so $W_{n} (t)^{n} \to U (t)$ strongly by dominated convergence in the spectral measures [F1]; since every $W_{n} (t)^{n}$ commutes with $S$ , the strong limit $U (t)$ does, by $∥ (S U (t) - U (t) S) x ∥ \leq ∥ S ∥ ∥ (W_{n}^{n} - U (t)) x ∥ + ∥ (W_{n}^{n} - U (t)) S x ∥$ . This is (a').

2.1givenstep 1.1

[Commuting sharp observable] Under the hypotheses of (b), each $F (S)$ is bounded and commutes with every $E (C)$ , so step 1.1 with $B = F (S)$ gives $F (S) U (t) = U (t) F (S)$ for all $t, S$ . This is (b)(i).

3.1givenstep 2.1F4algebra

[Invariant law of a density operator] Let $ρ$ be a density operator. Then $U (t) ρ U (t)^{*}$ is trace class by [F4], and by trace cyclicity twice, $tr (U (t) ρ U (t)^{*} F (S)) = tr (ρ U (t)^{*} F (S) U (t))$ . By step 2.1, $U (t)^{*} F (S) U (t) = U (t)^{*} U (t) F (S) = F (S)$ ; substituting gives $tr (U (t) ρ U (t)^{*} F (S)) = tr (ρ F (S))$ , independent of $t$ . This is (b)(ii).

3.2givenstep 2.1F2algebra

[Invariant law of a pure state] For unit $ψ$ , $∥ F (S) U (t) ψ ∥^{2} = ∥ U (t) F (S) ψ ∥^{2}$ by step 2.1, and this equals $∥ F (S) ψ ∥^{2}$ because $U (t)$ is unitary. This is (b)(iii).

4.1givenstep 3.1step 3.2F5

[Energy law] Applying steps 3.1 and 3.2 with $F = E$ gives $tr (U (t) ρ U (t)^{*} E (C)) = tr (ρ E (C))$ and $∥ E (C) U (t) ψ ∥^{2} = ∥ E (C) ψ ∥^{2}$ for every Borel $C$ . Moreover $E_{K, U (t) ψ} (C) = ∥ E (C) U (t) ψ ∥^{2} = ∥ E (C) ψ ∥^{2} = E_{K, ψ} (C)$ for every $C$ , so the two scalar measures agree by [F5].

5.1givenstep 4.1F2F3

[Moment conservation] Let $ψ \in D (K)$ and $u_{0} : = ψ$ ; by [F2], $U (t) ψ \in D (K)$ for every $t$ . Since $E_{K, U (t) ψ} = E_{K, ψ}$ by step 4.1, the two integrals $\int λ^{k} d E_{K, U (t) ψ}$ and $\int λ^{k} d E_{K, ψ}$ coincide for $k = 1, 2$ (extended real values, both finite at $t = 0$ by [F3]); by [F3] the time-zero values are $⟨ K ψ, ψ ⟩$ and $∥ K ψ ∥^{2}$ . If a moment is infinite at $t = 0$ , no equality of finite quantities is claimed. This proves (c).

6.1givenstep 3.2F1F3F5algebra∎

[Stationary eigenrays] Let $K ψ = λ ψ$ with $∥ ψ ∥ = 1$ . By [F3], $∥ (K - λ) ψ ∥^{2} = \int (μ - λ)^{2} d E_{K, ψ} (μ) = 0$ , so the finite positive measure $E_{K, ψ}$ assigns mass zero to $R ∖ {λ}$ and is the point mass at $λ$ by [F5]. Then $∥ U (t) ψ - e^{- i t λ / ℏ} ψ ∥^{2} = \int ∣ e^{- i t μ / ℏ} - e^{- i t λ / ℏ} ∣^{2} d E_{K, ψ} (μ) = 0$ by [F1], so $U (t) ψ = e^{- i t λ / ℏ} ψ$ . The last assertion of (d) is step 3.2 with this $ψ$ .

TheoremStatement: AI-adaptedProof: AI-adaptedOpen item page →

Graph-domain Ehrenfest identities and the conditional timescale bound

Statement

Assume the Axiom of Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )). Let $H$ be a complex Hilbert space (Real and complex inner product spaces, with the inner product linear in the first argument, Hilbert space), let $K, A$ be self-adjoint operators on $H$ , let $I \subseteq R$ be a nonempty open interval, fix $ℏ > 0$ and let $ψ : I \to H$ satisfy, for every $t \in I$ :

$ψ (t) \in D (A)$ , and the curve is $C^{1}$ into $D (A)$ with the graph norm of $A$ (Unbounded linear operators: domain, graph and extension, Densely defined, closed and closable operators, and cores), meaning that both $t \mapsto ψ (t)$ and $t \mapsto A ψ (t)$ are $C^{1}$ into $H$ ;
$K ψ (t) \in D (A)$ , that is $ψ (t) \in D (A K)$ ; the additional condition $A ψ (t) \in D (K)$ (that is $ψ (t) \in D (K A)$ ) is imposed only in the clauses that state it;
$ψ^{'} (t) = - \frac{i}{ℏ} K ψ (t)$ in $H$ .

Then, writing $f (t) : = ⟨ A ψ (t), ψ (t) ⟩$ and $(K A - A K) ψ : = K (A ψ) - A (K ψ)$ for $ψ \in D (A K) \cap D (K A)$ :

(a) if hypotheses 1 and 3 and $K ψ (t) \in D (A)$ for all $t$ hold (that is, $ψ (t) \in D (A K)$ ), then $f$ is $C^{1}$ on $I$ and $f^{'} (t) = - \frac{2}{ℏ} Im ⟨ A ψ (t), K ψ (t) ⟩;$ if moreover $A ψ (t) \in D (K)$ for all $t$ (that is, $ψ (t) \in D (K A)$ ), then also $f^{'} (t) = \frac{i}{ℏ} ⟨ (K A - A K) ψ (t), ψ (t) ⟩$ ;

(b) if $(A_{t})_{t \in I}$ is a family of bounded self-adjoint operators that is $C^{1}$ in operator norm and $ψ$ satisfies 1 and 3 above (no product-domain hypothesis), then $g (t) : = ⟨ A_{t} ψ (t), ψ (t) ⟩$ is $C^{1}$ with $g^{'} (t) = ⟨ A_{t}^{'} ψ (t), ψ (t) ⟩ - \frac{2}{ℏ} Im ⟨ A_{t} ψ (t), K ψ (t) ⟩;$ if in addition $A_{t} ψ (t) \in D (K)$ and $K ψ (t) \in D (A_{t})$ for all $t$ , then this equals $⟨ A_{t}^{'} ψ (t), ψ (t) ⟩ + \frac{i}{ℏ} ⟨ [K, A_{t}] ψ (t), ψ (t) ⟩$ ;

(c) if moreover $∥ ψ (t) ∥ = 1$ for all $t$ and for a fixed self-adjoint $A$ as in (a) one has $f^{'} (t_{0}) \neq 0$ at some $t_{0} \in I$ , then with the preparation spreads $Δ K (t_{0}) : = ∥ (K - ⟨ K ψ (t_{0}), ψ (t_{0}) ⟩) ψ (t_{0}) ∥, Δ A (t_{0}) : = ∥ (A - f (t_{0})) ψ (t_{0}) ∥,$ and the timescale $τ_{A} (t_{0}) : = Δ A (t_{0}) / ∣ f^{'} (t_{0}) ∣$ , the bound $Δ K (t_{0}) τ_{A} (t_{0}) \geq \frac{ℏ}{2}$ holds;

(d) if $f^{'} (t_{0}) = 0$ then the definition of $τ_{A} (t_{0})$ supplies no timescale, and no universal time operator, apparatus error or energy-time observable is asserted by (a)-(c).

Facts & Assumptions

Given: self-adjoint $K, A$ on a complex Hilbert space $H$ , an open interval $I$ , a curve $ψ$ satisfying hypotheses 1-3 with $K ψ (t) \in D (A)$ (and $A ψ (t) \in D (K)$ where a clause states it), the function $f (t) = ⟨ A ψ (t), ψ (t) ⟩$ and the notation $(K A - A K) ψ = K (A ψ) - A (K ψ)$ .

[F1]

A self-adjoint operator is closed and symmetric; $T^{*} = T$ , $D (T) = D (T^{*})$ , and $⟨ T v, w ⟩ = ⟨ v, T w ⟩$ for $v, w \in D (T)$ (Symmetric, self-adjoint and essentially self-adjoint operators, Adjoint of a densely defined operator, Densely defined, closed and closable operators, and cores).

[F2]

Hypotheses 1 and 2: $t \mapsto ψ (t)$ and $t \mapsto A ψ (t)$ are $C^{1}$ into $H$ , and $K ψ (t) \in D (A)$ ; $A ψ (t) \in D (K)$ is assumed only for the commutator clauses. Because $A$ is closed and $t \mapsto ψ (t)$ is $C^{1}$ into $D (A)$ , the derivative of $t \mapsto A ψ (t)$ is $A ψ^{'} (t)$ : the graph-norm difference quotients converge to $ψ^{'} (t)$ , and applying the closed operator $A$ to the limit gives $(A ψ)^{'} (t) = A ψ^{'} (t)$ (Unbounded linear operators: domain, graph and extension, Densely defined, closed and closable operators, and cores).

[F3]

For symmetric $K, A$ and a unit $u \in D (K) \cap D (A)$ , clause (b) of Covariance-refined preparation uncertainty without operator products gives $Δ K Δ A \geq ∣ Im ⟨ (A - ⟨ A u, u ⟩) u, (K - ⟨ K u, u ⟩) u ⟩ ∣ = ∣ Im ⟨ A u, K u ⟩ ∣$ , since both means are real. No product-domain hypothesis is needed for this bound. The weak-form Robertson lemma of Weak-form Robertson uncertainty on a common domain: for symmetric $K, A$ , a unit $u \in D (K) \cap D (A)$ with $u \in D (K A) \cap D (A K)$ satisfies $Δ K Δ A \geq \frac{1}{2} ∣ ⟨ [K, A] u, u ⟩ ∣$ , where $[K, A] u = K A u - A K u$ (clause (c)); the covariance form of the same bound is Covariance-refined preparation uncertainty without operator products clause (d).

Proof

1.1givenF2algebra

[Differentiability of $f$ ] By [F2], $(A ψ)^{'} (t) = A ψ^{'} (t)$ and both factors $A ψ (t)$ and $ψ (t)$ are $C^{1}$ into $H$ ; the product rule for the inner product gives $f^{'} (t) = ⟨ (A ψ)^{'} (t), ψ (t) ⟩ + ⟨ A ψ (t), ψ^{'} (t) ⟩ = ⟨ A ψ^{'} (t), ψ (t) ⟩ + ⟨ A ψ (t), ψ^{'} (t) ⟩$ .

1.2givenF1algebra

[Bounded time-dependent observable] Let $A_{t}$ be a bounded self-adjoint family, $C^{1}$ in operator norm, and $g (t) = ⟨ A_{t} ψ (t), ψ (t) ⟩$ . By the product rule for the three $C^{1}$ factors, $g^{'} = ⟨ A_{t}^{'} ψ, ψ ⟩ + ⟨ A_{t} ψ^{'}, ψ ⟩ + ⟨ A_{t} ψ, ψ^{'} ⟩$ ; self-adjointness of $A_{t}$ combines the last two terms into $2 Re ⟨ A_{t} ψ^{'}, ψ ⟩ = 2 Re (- \frac{i}{ℏ} q)$ with $q : = ⟨ A_{t} K ψ, ψ ⟩$ , and for $q = x + i y$ one has $- \frac{i}{ℏ} q = \frac{y - i x}{ℏ}$ , so $2 Re (- \frac{i}{ℏ} q) = \frac{2 y}{ℏ} = \frac{2}{ℏ} Im q$ . Since $A_{t}$ is self-adjoint, $q = ⟨ A_{t} K ψ, ψ ⟩ = ⟨ K ψ, A_{t} ψ ⟩ = \overline{⟨ A_{t} ψ, K ψ ⟩}$ , hence $Im q = - Im ⟨ A_{t} ψ, K ψ ⟩$ and $g^{'} = ⟨ A_{t}^{'} ψ, ψ ⟩ - \frac{2}{ℏ} Im ⟨ A_{t} ψ, K ψ ⟩$ . If in addition $A_{t} ψ \in D (K)$ and $K ψ \in D (A_{t})$ , then $⟨ K A_{t} ψ, ψ ⟩ = ⟨ A_{t} ψ, K ψ ⟩$ and $⟨ A_{t} K ψ, ψ ⟩ = ⟨ K ψ, A_{t} ψ ⟩ = \overline{⟨ A_{t} ψ, K ψ ⟩}$ , so $⟨ [K, A_{t}] ψ, ψ ⟩ = 2 i Im ⟨ A_{t} ψ, K ψ ⟩$ and $- \frac{2}{ℏ} Im ⟨ A_{t} ψ, K ψ ⟩ = \frac{i}{ℏ} ⟨ [K, A_{t}] ψ, ψ ⟩$ , completing (b).

2.1givenstep 1.1F1algebra

[The commutator form] Since $ψ^{'} (t) \in D (A)$ by hypothesis 1 and $A$ is symmetric [F1], $⟨ A ψ^{'} (t), ψ (t) ⟩ = ⟨ ψ^{'} (t), A ψ (t) ⟩$ , so $f^{'} (t) = 2 Re ⟨ ψ^{'} (t), A ψ (t) ⟩$ . Substituting $ψ^{'} = - \frac{i}{ℏ} K ψ$ and writing $z : = ⟨ A ψ (t), K ψ (t) ⟩$ , we get $f^{'} = 2 Re (- \frac{i}{ℏ} ⟨ K ψ, A ψ ⟩) = 2 Re (- \frac{i}{ℏ} \overline{z})$ ; since $Re (- \frac{i}{ℏ} \overline{z}) = Re (- \frac{i}{ℏ} (Re z - i Im z)) = - \frac{1}{ℏ} Im z$ , this gives $f^{'} = - \frac{2}{ℏ} Im ⟨ A ψ, K ψ ⟩$ .

3.1givenstep 2.1F1algebra

[The commutator pairing] Assume in addition $A ψ (t) \in D (K)$ . Then the pairings $⟨ K A ψ, ψ ⟩ = ⟨ A ψ, K ψ ⟩$ (symmetry of $K$ with $A ψ (t) \in D (K)$ ) and $⟨ A K ψ, ψ ⟩ = ⟨ K ψ, A ψ ⟩ = \overline{⟨ A ψ, K ψ ⟩}$ (symmetry of $A$ with $K ψ (t) \in D (A)$ ) are defined, so $⟨ (K A - A K) ψ, ψ ⟩ = z - \overline{z} = 2 i Im z$ with $z$ as in step 2.1, and $\frac{i}{ℏ} (z - \overline{z}) = \frac{i}{ℏ} \cdot 2 i Im z = - \frac{2}{ℏ} Im z$ , which is the second displayed form of $f^{'}$ . This proves (a).

3.2givenstep 2.1F3algebra

[Timescale bound] Let $u : = ψ (t_{0})$ with $∥ u ∥ = 1$ and $f^{'} (t_{0}) \neq 0$ . By hypotheses 1-3, $u \in D (K) \cap D (A)$ and $K u \in D (A)$ , so step 2.1 gives $∣ f^{'} (t_{0}) ∣ = (2 / ℏ) ∣ Im ⟨ A u, K u ⟩ ∣$ . The weak bound in [F3] gives $Δ K (t_{0}) Δ A (t_{0}) \geq ∣ Im ⟨ A u, K u ⟩ ∣ = (ℏ / 2) ∣ f^{'} (t_{0}) ∣$ . Dividing by $∣ f^{'} (t_{0}) ∣ > 0$ proves (c), without assuming $A u \in D (K)$ .

4.1givenstep 3.2algebra∎

[Scope and vanishing derivative] If $f^{'} (t_{0}) = 0$ , the quotient $τ_{A} (t_{0}) = Δ A (t_{0}) / ∣ f^{'} (t_{0}) ∣$ is undefined, so no timescale is supplied; and nothing in the argument produces a time operator, a measurement error or an energy-time observable, since all quantities were formed from the specified curve and the two operators. This is (d).

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Conditional Stern-Gerlach force for a spin-coupled bounded field model

Statement

Assume the Axiom of Choice and the Axiom of Countable Choice (The Axiom of Choice, The Axiom of Countable Choice ( ${A C}_{ω}$ )). Fix $ℏ > 0$ , $m > 0$ , $γ_{s} \in R$ and a real function $B_{z} \in C^{1} (R)$ with $B_{z}$ and $B_{z}^{'}$ bounded. Let $H : = L^{2} (R^{3}; C) \oplus L^{2} (R^{3}; C), ⟨ (u_{+}, u_{-}), (v_{+}, v_{-}) ⟩ : = ⟨ u_{+}, v_{+} ⟩ + ⟨ u_{-}, v_{-} ⟩,$ let $P_{j}, Q_{j}$ act componentwise (Position and momentum operators on Euclidean L2, Self-adjointness, Fourier representation and translation identities of position and momentum), write $B_{z} (Q_{x})$ for componentwise multiplication by $x \mapsto B_{z} (x_{1})$ and $S_{z} (u_{+}, u_{-}) : = (\frac{ℏ}{2} u_{+}, - \frac{ℏ}{2} u_{-}), K (u_{+}, u_{-}) : = (\frac{1}{2 m} \sum_{j} P_{j}^{2} u_{+} - \frac{γ_{s} ℏ}{2} B_{z} (Q_{x}) u_{+}, \frac{1}{2 m} \sum_{j} P_{j}^{2} u_{-} + \frac{γ_{s} ℏ}{2} B_{z} (Q_{x}) u_{-}),$ with $D (K) : = {(u_{+}, u_{-}) : u_{+}, u_{-} \in W^{2, 2} (R^{3}; C)}$ , so that formally $K = \frac{1}{2 m} \sum_{j} P_{j}^{2} - γ_{s} S_{z} B_{z} (Q_{x})$ . Identify the spin factor with $V_{1 / 2} = C^{2}$ of Finite spin matrices by the basis $(e_{1 / 2}, e_{- 1 / 2})$ , so that on that factor $S_{z} = ℏ J_{z}$ with $J_{z} e_{m} = m e_{m}$ and eigenvalues $\pm ℏ / 2$ . Then:

(a) $K$ is self-adjoint with domain $D (K)$ , and $S_{z} K = K S_{z}$ on $D (K)$ ; consequently, for $u (t) : = e^{- i t K / ℏ} u_{0}$ and the sector projections $p_{\pm} : = \frac{1}{2} (I \pm \frac{2}{ℏ} S_{z})$ , $∥ p_{\pm} u (t) ∥^{2} = ∥ p_{\pm} u_{0} ∥^{2}, ⟨ S_{z} u (t), u (t) ⟩ = ⟨ S_{z} u_{0}, u_{0} ⟩ (t \in R, u_{0} \in H);$

(b) for every $u_{0} \in D (K^{2}) : = {u \in D (K) : K u \in D (K)}$ the function $t \mapsto ⟨ P_{x} u (t), u (t) ⟩$ is $C^{1}$ and $\frac{d}{d t} ⟨ P_{x} u (t), u (t) ⟩ = γ_{s} ⟨ S_{z} B_{z}^{'} (Q_{x}) u (t), u (t) ⟩,$ where $B_{z}^{'} (Q_{x})$ is componentwise multiplication by $x \mapsto B_{z}^{'} (x_{1})$ and $S_{z} B_{z}^{'} (Q_{x}) (u_{+}, u_{-}) = (\frac{ℏ}{2} B_{z}^{'} (Q_{x}) u_{+}, - \frac{ℏ}{2} B_{z}^{'} (Q_{x}) u_{-})$ ;

(c) if in addition $∥ u_{0} ∥ = 1$ , $p_{+} u_{0} = u_{0}$ , and $B_{z}^{'} (s) = G$ for every $s$ in a Borel set $N \subseteq R$ with $\int_{R ∖ N} η_{+} (x_{1}) d x_{1} = 0$ , where $η_{+} (x_{1}) : = \int_{R^{2}} ∣ u_{0}^{+} (x_{1}, x_{2}, x_{3}) ∣^{2} d x_{2} d x_{3}$ is the first-coordinate marginal density, then $\frac{d}{d t} ∣_{t = 0} ⟨ P_{x} u (t), u (t) ⟩ = \frac{γ_{s} ℏ}{2} G,$ an identity at the preparation time $t = 0$ ; for $t \neq 0$ the general formula of (b) applies and may differ;

(d) for the model field $B (x) : = (0, 0, B_{z} (x_{1}))$ one has $div B = 0$ and $curl B = (0, - B_{z}^{'} (x_{1}), 0)$ at every point.

The statements are conditional identities for the displayed model; no apparatus, measured trajectory, source-free Maxwell realisation of $B$ or inference about an individual particle is asserted.

Facts & Assumptions

Given: $ℏ, m > 0$ , $γ_{s} \in R$ , bounded $B_{z} \in C^{1} (R)$ with bounded $B_{z}^{'}$ ; the direct-sum Hilbert space $H$ ; the operators $S_{z}$ , $B_{z} (Q_{x})$ , $K$ on $H$ ; the componentwise operators $P_{j}, Q_{j}$ .

[F1]

On each component, $K_{\pm} : = \frac{1}{2 m} \sum_{j} P_{j}^{2} \mp \frac{γ_{s} ℏ}{2} M_{B_{z}}$ is self-adjoint with domain $W^{2, 2} (R^{3}; C)$ : this is Real multiplication operators and free plus bounded-potential Schrödinger operators clause (c) with the real bounded potential $\mp \frac{γ_{s} ℏ}{2} B_{z}$ ; $M_{B_{z}}$ is bounded self-adjoint (Real multiplication operators and free plus bounded-potential Schrödinger operators clause (a)).

[F2]

A direct sum $T = T_{+} \oplus T_{-}$ of self-adjoint operators is self-adjoint on $D (T_{+}) \oplus D (T_{-})$ : symmetry is componentwise, and surjectivity of $T \pm i$ follows from surjectivity of the components (Symmetric, self-adjoint and essentially self-adjoint operators).

[F3]

$T_{u (t), u (t)}$ denotes the rank-one operator $x \mapsto ⟨ x, u (t) ⟩ u (t)$ ; the PVM of the bounded self-adjoint $S_{z}$ is $F ({ℏ / 2}) = p_{+}$ , $F ({- ℏ / 2}) = p_{-}$ , $F (R ∖ {\pm ℏ / 2}) = 0$ , and $F (S) U (t) = U (t) F (S)$ whenever $S_{z} K \subseteq K S_{z}$ , by Conserved spectral probability laws for commuting observables and stationary eigenrays clauses (a') and (b)(iii). Also $p_{\pm} = \frac{1}{2} (I \pm \frac{2}{ℏ} S_{z})$ .

[F4]

Graph-domain Ehrenfest identities and the conditional timescale bound clause (a), one-sided form: if $ψ : I \to H$ is $C^{1}$ into $D (A)$ with the graph norm, $ψ^{'} (t) = - \frac{i}{ℏ} K ψ (t)$ and $K ψ (t) \in D (A)$ for all $t$ , then $t \mapsto ⟨ A ψ (t), ψ (t) ⟩$ is $C^{1}$ with derivative $- \frac{2}{ℏ} Im ⟨ A ψ (t), K ψ (t) ⟩$ .

[F5]

Plancherel and Fourier calculus: $F_{2}$ is unitary and $∥ g ∥_{2} = ∥ F_{2} g ∥_{2}$ (Plancherel theorem, Fourier transform on complex L1 classes). For $w \in W^{2, 2} (R^{3}; C)$ the class is bounded with $∣ w (x) ∣ \leq C ∥ w ∥_{W^{2, 2}}$ for a.e. $x$ , because $∣ w (x) ∣ = ∣ \int F_{2} w (ξ) e^{2 π i x \cdot ξ} d ξ ∣ \leq ∥ ⟨ ξ ⟩^{2} F_{2} w ∥_{2} ∥ ⟨ ξ ⟩^{- 2} ∥_{2}$ and $\int (1 + ∣ ξ ∣^{2})^{- 2} d ξ < \infty$ in three dimensions.

[F6]

Products of bounded Sobolev classes: for $w \in W^{2, 2} (R^{3}; C) \subseteq W^{1, 2} \cap L^{\infty}$ and its conjugate $\overline{w} \in W^{1, 2} \cap L^{\infty}$ , the product $∣ w ∣^{2} = w \overline{w}$ lies in $W^{1, 2}$ with $D_{j} ∣ w ∣^{2} = (D_{j} w) \overline{w} + w D_{j} \overline{w} = 2 Re ((D_{j} w) \overline{w})$ (Weak product rule for bounded Sobolev functions).

[F7]

Compactly supported integration by parts: if $u \in W^{1, 2} (R^{3})$ is compactly supported and $v \in W^{1, 2} (R^{3})$ , then $\int u D_{j} v = - \int v D_{j} u$ (Integration by parts for dual-exponent Sobolev functions, case $p = p^{'} = 2$ ). More generally a smooth compactly supported multiplier times a bounded Lipschitz function lies in $W^{1, 2}$ with the Leibniz derivative (Weak Leibniz rule with a smooth factor).

[F8]

Cutoffs and dominated convergence: the functions $χ_{R} (x) = χ (x / R)$ of Explicit compactly supported smooth cutoffs satisfy $0 \leq χ_{R} \leq 1$ , $χ_{R} = 1$ on $[- R, R]^{3}$ , $χ_{R} = 0$ outside $[- 2 R, 2 R]^{3}$ and $∣ D_{j} χ_{R} ∣ \leq C / R$ ; $L^{1}$ dominated convergence applies to integrals against $∣ w ∣^{2}$ and $∣ \partial_{j} ∣ w ∣^{2} ∣$ (Dominated convergence).

[F9]

For self-adjoint $K$ , the spectral unitary group preserves $D (K)$ , commutes with $K$ there and has derivative $- i K / ℏ$ for initial vectors in $D (K)$ . Tonelli on the sigma-finite product $R \times R^{2}$ defines the nonnegative marginal $η_{+}$ and gives $\int η_{+} = ∥ u_{0}^{+} ∥_{2}^{2}$ (A self-adjoint operator generates a strongly continuous unitary group, Tonelli's theorem for nonnegative measurable functions on a sigma-finite product).

Proof

1.1givenF1F2

[Self-adjointness and commutation] By [F1] each $K_{\pm}$ is self-adjoint on $W^{2, 2}$ , so [F2] makes $K = K_{+} \oplus K_{-}$ self-adjoint on $D (K) = W^{2, 2} \oplus W^{2, 2}$ . Also $S_{z}$ is bounded self-adjoint with eigenvalues $\pm ℏ / 2$ by the identification with $ℏ J_{z}$ of Finite spin matrices, and for $(u_{+}, u_{-}) \in D (K)$ , $S_{z} K (u_{+}, u_{-}) = (\frac{ℏ}{2} K_{+} u_{+}, - \frac{ℏ}{2} K_{-} u_{-}) = K S_{z} (u_{+}, u_{-})$ componentwise.

1.2givenF5

[The free part is real] For $w \in W^{2, 2} (R^{3}; C)$ put $K_{0} : = \frac{1}{2 m} \sum_{j} P_{j}^{2}$ and $c : = \frac{(2 π ℏ)^{2}}{2 m}$ . By [F5] and the multiplier identity of Self-adjointness, Fourier representation and translation identities of position and momentum, $⟨ P_{x} w, K_{0} w ⟩ = \int_{R^{3}} 2 π ℏ ξ_{1} c ∣ ξ ∣^{2} ∣ F_{2} w (ξ) ∣^{2} d ξ$ , an absolutely convergent integral of a real-valued function, hence a real number (Cauchy–Schwarz bounds it by $∥ P_{x} w ∥_{2} ∥ K_{0} w ∥_{2}$ ; no parity of $∣ F_{2} w ∣^{2}$ is assumed); so $Im ⟨ P_{x} w, K_{0} w ⟩ = 0$ .

1.3givenF6F7F8algebra

[The potential part] Let $w \in W^{2, 2}$ and let $χ_{R}$ be as in [F8]. By [F6], $∣ w ∣^{2} \in W^{1, 2}$ with $D_{x} ∣ w ∣^{2} = 2 Re ((D_{x} w) \overline{w})$ , and by [F7] the compactly supported class $χ_{R} B_{z} (x_{1}) \in W^{1, 2}$ satisfies $\int χ_{R} B_{z} D_{x} ∣ w ∣^{2} = - \int D_{x} (χ_{R} B_{z}) ∣ w ∣^{2} = - \int χ_{R} B_{z}^{'} ∣ w ∣^{2} - \int (D_{x} χ_{R}) B_{z} ∣ w ∣^{2}$ . Letting $R \to \infty$ , [F8] gives $\int B_{z} D_{x} ∣ w ∣^{2} = - \int B_{z}^{'} ∣ w ∣^{2}$ and $∣ \int (D_{x} χ_{R}) B_{z} ∣ w ∣^{2} ∣ \leq C ∥ B_{z} ∥_{\infty} R^{- 1} \int_{∣ x ∣ \geq R} ∣ w ∣^{2} \to 0$ . Since $Im ⟨ P_{x} w, M_{B_{z}} w ⟩ = Im \int (- i ℏ D_{x} w) B_{z} \overline{w} = - ℏ Re \int B_{z} D_{x} w \overline{w} = - \frac{ℏ}{2} \int B_{z} D_{x} ∣ w ∣^{2}$ , this gives $Im ⟨ P_{x} w, M_{B_{z}} w ⟩ = \frac{ℏ}{2} \int B_{z}^{'} ∣ w ∣^{2}$ .

2.1givenstep 1.1F3

[Conserved spin law] By step 1.1 the inclusion $S_{z} K \subseteq K S_{z}$ holds, so [F3] gives $S_{z} U (t) = U (t) S_{z}$ and hence $p_{\pm} U (t) = U (t) p_{\pm}$ because $p_{\pm}$ are polynomials in $S_{z}$ . For $u_{0} \in H$ , $∥ p_{\pm} u (t) ∥ = ∥ U (t) p_{\pm} u_{0} ∥ = ∥ p_{\pm} u_{0} ∥$ , and the mean identity follows from $⟨ S_{z} u, u ⟩ = \frac{ℏ}{2} (∥ p_{+} u ∥^{2} - ∥ p_{-} u ∥^{2})$ applied at times $t$ and $0$ . This is (a).

2.2givenstep 1.1F1F4F5F9algebra

[Graph differentiability for the full stated domain] For $w \in W^{2, 2}$ , the Fourier inequality $ξ_{1}^{2} \leq 1 + ∣ ξ ∣^{4}$ and Plancherel [F5] give $∥ P_{x} w ∥ \leq C (∥ K_{0} w ∥ + ∥ w ∥)$ . Since $K_{\pm} = K_{0} + V_{\pm}$ with bounded $V_{\pm}$ , it follows componentwise that $∥ P_{x} u ∥ \leq C^{'} (∥ K u ∥ + ∥ u ∥)$ on $D (K)$ , so $P_{x}$ is continuous from the $K$ graph norm into $H$ . If $u_{0} \in D (K^{2})$ , [F9] applied both to $u_{0}$ and to $K u_{0} \in D (K)$ makes $u (t) = U (t) u_{0}$ and $K u (t) = U (t) K u_{0}$ continuously differentiable in $H$ , with derivatives $- i K u (t) / ℏ$ and $- i K^{2} u (t) / ℏ$ . Thus $u (t)$ is $C^{1}$ in the $K$ graph norm, and applying the graph-continuous map $P_{x}$ yields $\frac{d}{d t} P_{x} u (t) = - i P_{x} K u (t) / ℏ$ . The domain inclusions $u (t), K u (t) \in D (K) \subseteq D (P_{x})$ therefore meet [F4], which gives $f (t) = ⟨ P_{x} u (t), u (t) ⟩ \in C^{1}$ and $f^{'} (t) = - 2 Im ⟨ P_{x} u (t), K u (t) ⟩ / ℏ$ without weakening $D (K^{2})$ .

3.1givenstep 2.2step 1.2step 1.3algebra

[Assembling the force] Write $u = (u_{+}, u_{-})$ , $K_{\pm} = K_{0} \mp \frac{γ_{s} ℏ}{2} M_{B_{z}}$ and $z (t) = \sum_{\pm} ⟨ P_{x} u_{\pm}, K_{\pm} u_{\pm} ⟩$ . By steps 1.2 and 1.3, $Im z (t) = \sum_{\pm} (\mp \frac{γ_{s} ℏ}{2}) \frac{ℏ}{2} \int B_{z}^{'} ∣ u_{\pm} ∣^{2} = - \frac{γ_{s} ℏ^{2}}{4} \int B_{z}^{'} ∣ u_{+} ∣^{2} + \frac{γ_{s} ℏ^{2}}{4} \int B_{z}^{'} ∣ u_{-} ∣^{2} = - \frac{γ_{s} ℏ}{2} ⟨ S_{z} B_{z}^{'} (Q_{x}) u, u ⟩$ , because $⟨ S_{z} B_{z}^{'} (Q_{x}) u, u ⟩ = \frac{ℏ}{2} \int B_{z}^{'} ∣ u_{+} ∣^{2} - \frac{ℏ}{2} \int B_{z}^{'} ∣ u_{-} ∣^{2}$ . Substituting into step 2.2, $f^{'} = - \frac{2}{ℏ} Im z = γ_{s} ⟨ S_{z} B_{z}^{'} (Q_{x}) u (t), u (t) ⟩$ , which is (b).

4.1givenstep 3.1F9algebra

[Constant-gradient preparation in three dimensions] Under (c), $u_{0} = (u_{0}^{+}, 0)$ and the marginal $η_{+} (x_{1})$ of [F9] satisfies $\int η_{+} = ∥ u_{0} ∥^{2} = 1$ and vanishes in integral off $N$ . Tonelli and boundedness of $B_{z}^{'}$ give $⟨ S_{z} B_{z}^{'} (Q_{x}) u_{0}, u_{0} ⟩ = \frac{ℏ}{2} \int_{R} B_{z}^{'} (x_{1}) η_{+} (x_{1}) d x_{1} = \frac{ℏ}{2} G \int_{N} η_{+} = \frac{ℏ}{2} G$ . By step 3.1 at preparation time $t = 0$ , $f^{'} (0) = γ_{s} ℏ G / 2$ . The general force formula retains the later-time dependence of the marginal and field gradient.

5.1givenalgebra∎

[Field identities] For $B (x) = (0, 0, B_{z} (x_{1}))$ : $div B = D_{1} B_{1} + D_{2} B_{2} + D_{3} B_{3} = 0$ because $B_{1} = B_{2} = 0$ and $B_{3}$ is independent of $x_{3}$ ; and $curl B = (D_{2} B_{3} - D_{3} B_{2}, D_{3} B_{1} - D_{1} B_{3}, D_{1} B_{2} - D_{2} B_{1}) = (0, - B_{z}^{'} (x_{1}), 0)$ (Divergence and curl are linear and satisfy the scalar product rules). This is (d).

5 · Examples, experiments and counterexamples

None yet.

Qm Math Magnetic And Detection

1 · Prerequisites

2 · Summary

3 · Logical flowchart

4 · Definitions, postulates, theorems and proofs

Position and momentum operators on Euclidean L2

Definition

Weak product rule for a bounded coefficient with bounded weak derivative

Statement

Facts & Assumptions

Proof

Self-adjointness, Fourier representation and translation identities of position and momentum

Statement

Facts & Assumptions

Proof

Bounded-coefficient magnetic Schrodinger operator

Definition

Self-adjointness of the bounded-coefficient magnetic Schrodinger operator

Statement

Facts & Assumptions

Proof

Gauge multiplication preserves H2 and transports the magnetic momentum

Statement

Facts & Assumptions

Proof

Gauge transported propagator for the bounded-coefficient magnetic operator

Statement

Facts & Assumptions

Proof

Smooth local continuity equation and its integrated forms

Statement

Facts & Assumptions

Proof

Coherent and incoherent two-path bin laws and the marker overlap

Statement

Facts & Assumptions

Proof

Covariance-refined preparation uncertainty without operator products

Statement

Facts & Assumptions

Proof

Position-momentum uncertainty on the maximal domains of the line

Statement

Facts & Assumptions

Proof

Galilean boost covariance of the Schrodinger model with bounded potential

Statement

Facts & Assumptions

Proof

Conserved spectral probability laws for commuting observables and stationary eigenrays

Statement

Facts & Assumptions

Proof

Graph-domain Ehrenfest identities and the conditional timescale bound

Statement

Facts & Assumptions

Proof

Conditional Stern-Gerlach force for a spin-coupled bounded field model

Statement

Facts & Assumptions

Proof

5 · Examples, experiments and counterexamples

Sources