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Coherent evolution and variance

Statement

Assume the Axiom of Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )). Fix $m, ω, ℏ > 0$ and let $ℓ = \sqrt{ℏ / (m ω)}$ , $D$ , $ψ_{n} = D g_{n}$ , $H_{o s c} = \frac{ℏ ω}{2} D A D^{- 1}$ be as in Oscillator scaling and exact realization, with $(g_{n})$ the Hermite family and $a, a^{†}, N$ the closed ladder and number operators of Closed ladder domains; put $a_{ψ} : = D a D^{- 1}$ , $a_{ψ}^{†} : = D a^{†} D^{- 1}$ , $N_{ψ} : = D N D^{- 1}$ , acting on $L^{2} (R, d x)$ with the transported domains, and for $z \in C$ let $ψ_{z}^{(y)} \in L^{2} (R, d y)$ be the coherent vector of Coherent-state convergence and define $ψ_{z} : = D ψ_{z}^{(y)} = e^{- ∣ z ∣^{2} / 2} \sum_{n} z^{n} ψ_{n} / \sqrt{n!} \in L^{2} (R, d x)$ . Finally set

$Q : = M_{x}, D (Q) = {f \in L^{2} (R, d x) : x f \in L^{2}}, P : = F_{2}^{*} M_{2 π ℏ ξ} F_{2}, D (P) = {f : ξ F_{2} f \in L^{2}},$

the self-adjoint position and momentum operators (Real L2 multipliers and unitary transport, Plancherel theorem; the conjugation is discharged in the proof).

Then:

(a) $H_{o s c} = ℏ ω (N_{ψ} + \frac{1}{2} I)$ as operators, with common domain $D (H_{o s c}) = D (N_{ψ})$ ;

(b) the formula

$U (t) f : = \sum_{n} e^{- i ω (n + 1 / 2) t} ⟨ f, ψ_{n} ⟩ ψ_{n} (t \in R, f \in L^{2} (R, d x))$

defines a strongly continuous one-parameter unitary group $U$ on $L^{2} (R, d x)$ : each $U (t)$ is a surjective linear isometry, $U (0) = I$ and $U (t + s) = U (t) U (s)$ , and the generator of $U$ is $- \frac{i}{ℏ} H_{o s c}$ , i.e. the limit $\lim_{t \to 0} \frac{1}{t} (U (t) f - f)$ exists exactly for $f \in D (H_{o s c})$ and equals $- \frac{i}{ℏ} H_{o s c} f$ ;

(d) on the orthonormal basis one has

$Q ψ_{n} = \frac{ℓ}{\sqrt{2}} (\sqrt{n + 1} ψ_{n + 1} + \sqrt{n} ψ_{n - 1}), P ψ_{n} = \frac{ℏ}{i ℓ \sqrt{2}} (\sqrt{n} ψ_{n - 1} - \sqrt{n + 1} ψ_{n + 1}),$

with the convention $ψ_{- 1} : = 0$ ; consequently, for every $z \in C$ ,

$Q ψ_{z} = \frac{ℓ}{\sqrt{2}} (a_{ψ} + a_{ψ}^{†}) ψ_{z}, P ψ_{z} = \frac{ℏ}{i ℓ \sqrt{2}} (a_{ψ} - a_{ψ}^{†}) ψ_{z};$

(e) with means $⟨ Q ⟩_{z} : = ⟨ Q ψ_{z}, ψ_{z} ⟩$ , $⟨ P ⟩_{z} : = ⟨ P ψ_{z}, ψ_{z} ⟩$ and variances $Var (Q)_{z} : = ∥ (Q - ⟨ Q ⟩_{z} I) ψ_{z} ∥^{2}$ , $Var (P)_{z} : = ∥ (P - ⟨ P ⟩_{z} I) ψ_{z} ∥^{2}$ ,

$⟨ Q ⟩_{z} = \sqrt{2} ℓ Re z, ⟨ P ⟩_{z} = \frac{\sqrt{2} ℏ}{ℓ} Im z, Var (Q)_{z} = \frac{ℓ^{2}}{2}, Var (P)_{z} = \frac{ℏ^{2}}{2 ℓ^{2}},$

so $Var (Q)_{z} Var (P)_{z} = ℏ^{2} / 4$ for every $z$ (the value produced by these exact computations; no independent uncertainty-bound statement is asserted here);

(f) after the evolution, with $ψ_{z} (t) : = U (t) ψ_{z}$ and $z (t) : = z e^{- i ω t}$ ,

$⟨ Q ⟩_{z (t)} = \sqrt{2} ℓ Re z (t), ⟨ P ⟩_{z (t)} = \frac{\sqrt{2} ℏ}{ℓ} Im z (t), Var (Q)_{z (t)} = \frac{ℓ^{2}}{2}, Var (P)_{z (t)} = \frac{ℏ^{2}}{2 ℓ^{2}} .$

The parameters $m, ω, ℏ$ enter as positive real numbers and no physical identification of $Q, P, H_{o s c}$ with measured quantities is made.

Facts & Assumptions

Given: the unitary $D$ , the complete orthonormal family $(ψ_{n})$ of eigenfunctions of $H_{o s c}$ , the operators $a_{ψ}, a_{ψ}^{†}, N_{ψ}$ transported from Closed ladder domains, and the coherent vectors $ψ_{z}$ .

[F1]

$(g_{n})$ is a complete orthonormal family of $L^{2} (R, d y)$ ; $D$ is a bijective isometry with inverse $D^{- 1} g (y) = ℓ^{1 / 2} g (ℓ y)$ ; $(ψ_{n})$ is a complete orthonormal family of $L^{2} (R, d x)$ ; $D (H_{o s c}) = {f : \sum_{n} ℏ^{2} ω^{2} (n + \frac{1}{2})^{2} ∣ c_{n} ∣^{2} < \infty}$ with $c_{n} = ⟨ f, ψ_{n} ⟩$ , $H_{o s c} ψ_{n} = ℏ ω (n + \frac{1}{2}) ψ_{n}$ , and $σ (H_{o s c}) = {ℏ ω (n + \frac{1}{2})}$ (Oscillator scaling and exact realization).

[F2]

On $D (N_{ψ})$ , $a_{ψ}^{†} a_{ψ} = N_{ψ}$ and $a_{ψ} a_{ψ}^{†} = N_{ψ} + I$ ; $a_{ψ}$ is closed; $a_{ψ} ψ_{z} = z ψ_{z}$ ; $ψ_{z} \in D (N_{ψ}^{k})$ for every $k$ , $⟨ ψ_{z}, ψ_{w} ⟩ = e^{- (∣ z ∣^{2} + ∣ w ∣^{2}) / 2 + z \overline{w}}$ , $∥ a_{ψ} ψ_{z} ∥^{2} = ∣ z ∣^{2}$ , $∥ a_{ψ}^{†} ψ_{z} ∥^{2} = ∣ z ∣^{2} + 1$ , $⟨ N_{ψ} ψ_{z}, ψ_{z} ⟩ = ∣ z ∣^{2}$ . These are the facts of Closed ladder domains(b),(d) and Coherent-state convergence, transported by the unitary $D$ .

[F3]

$Y = \frac{a + a^{†}}{\sqrt{2}}$ and $- i \partial_{y} = \frac{a - a^{†}}{i \sqrt{2}}$ on the polynomial-Gaussian space $P : = {p g_{0} : p \in C [y]}$ (denoted $P$ in the supplier), whence $\sqrt{2} y g_{n} = \sqrt{n} g_{n - 1} + \sqrt{n + 1} g_{n + 1}$ and $\sqrt{2} g_{n}^{'} = \sqrt{n} g_{n - 1} - \sqrt{n + 1} g_{n + 1}$ (Hermite recurrence(g), Hermite orthogonality(b)).

[F4]

$M_{x}$ is self-adjoint on $D (M_{x}) = {f : x f \in L^{2}}$ for the real function $x$ (Real L2 multipliers and unitary transport); $F_{2}$ is unitary (Plancherel theorem); if $T$ is self-adjoint on $K$ and $U : K \to H$ is a bijective linear isometry, then $U T U^{*}$ is self-adjoint on $U D (T)$ (proved in step 1.3); and $D (P) = W^{1, 2} (R)$ with $P f = - i ℏ f^{'}$ there (Integer-order W^{k,2} and H^k agree with equivalent norms, Distributional derivatives are polynomial Fourier multipliers).

[F5]

$e^{i x} = 1 + i x + o (x)$ as $x \to 0$ and $∣ e^{i x} - 1 - i x ∣ \leq 2 ∣ x ∣$ for $x \in R$ : the complex exponential is entire with derivative itself at every point (The complex exponential is entire and its complex derivative is itself, The complex exponential by its power series); the estimate follows from $e^{i x} - 1 = \int_{0}^{1} i x e^{i s x} d s$ , obtained by the chain rule and the fundamental theorem of calculus, applied to real and imaginary parts (The chain rule, in one line from Carathéodory: if $g$ is differentiable at $c$ and $f$ is differentiable at $g (c)$ , then $f \circ g$ is differentiable at $c$ with $(f \circ g)^{'} (c) = f^{'} (g (c)) g^{'} (c)$ , The second fundamental theorem: if $G$ is differentiable on $[a, b]$ with $G^{'} = f$ and $f$ is integrable, then $\int_{a}^{b} f = G (b) - G (a)$ ), so $∣ e^{i x} - 1 ∣ \leq ∣ x ∣$ ; and $∥ f ∥^{2} = \sum_{n} ∣ ⟨ f, ψ_{n} ⟩ ∣^{2}$ with the finite-subset net convergence of the expansion (Parseval equivalences for an orthonormal family).

[F6]

$∥ u v ∥_{1} \leq ∥ u ∥_{2} ∥ v ∥_{2}$ and $∣ ⟨ u, v ⟩ ∣ \leq ∥ u ∥ ∥ v ∥$ (Cauchy-Schwarz inequality for $L^{2}$ , The space $L^{p} (μ)$ as the quotient by null functions, Complex Lp classes and Euclidean test-function conventions).

Proof

1.1givenF1algebra

[(a) Hamiltonian as a number operator] Let $f \in D (H_{o s c})$ with $c_{n} = ⟨ f, ψ_{n} ⟩$ ; by [F1] this means $\sum_{n} (n + \frac{1}{2})^{2} ∣ c_{n} ∣^{2} < \infty$ , which is equivalent to $\sum_{n} n^{2} ∣ c_{n} ∣^{2} < \infty$ because $n^{2} \leq 4 (n + \frac{1}{2})^{2}$ and $(n + \frac{1}{2})^{2} \leq 2 n^{2} + \frac{1}{2}$ for all $n \geq 0$ ; hence $D (H_{o s c}) = D (N_{ψ})$ by [F1] and the transported domain formula, and $H_{o s c} f = \sum_{n} ℏ ω (n + \frac{1}{2}) c_{n} ψ_{n} = ℏ ω (\sum_{n} n c_{n} ψ_{n} + \frac{1}{2} f) = ℏ ω (N_{ψ} f + \frac{1}{2} f)$ .

1.2givenF5

[(b) unitarity of $U (t)$ ] For $f \in L^{2}$ the coefficients $c_{n} = ⟨ f, ψ_{n} ⟩$ are square summable, so the series in (b) converges and $∥ U (t) f ∥^{2} = \sum_{n} ∣ c_{n} ∣^{2} = ∥ f ∥^{2}$ by [F5]; $U (t)$ is linear and isometric. Its range contains every $ψ_{n} = U (t) (e^{i ω (n + 1 / 2) t} ψ_{n})$ , hence is dense; an isometry has closed range, so $U (t)$ is surjective: it is a bijective isometry. Also $U (t) ψ_{n} = e^{- i ω (n + 1 / 2) t} ψ_{n}$ shows $U (t) U (t)^{*}$ fixes every $ψ_{n}$ , hence equals the identity on the dense span; by continuity $U (t)^{*} = U (t)^{- 1}$ , and $U (t)^{- 1} = U (- t)$ .

1.3givenF3F4

[(d) the position and momentum operators] The conjugation claim of [F4]: if $T = T^{*}$ on $K$ and $U : K \to H$ is a bijective isometry, let $T^{'} : = U T U^{*}$ on $D (T^{'}) : = U D (T)$ . For $x, y \in D (T^{'})$ , writing $x = U ξ$ , $y = U η$ with $ξ, η \in D (T)$ , $⟨ T^{'} x, y ⟩ = ⟨ T ξ, η ⟩ = ⟨ ξ, T η ⟩ = ⟨ x, T^{'} y ⟩$ , so $T^{'}$ is symmetric. If $y \in D ((T^{'})^{*})$ then for all $ξ \in D (T)$ , $⟨ T ξ, U^{*} y ⟩ = ⟨ T^{'} U ξ, y ⟩ = ⟨ U ξ, (T^{'})^{*} y ⟩ = ⟨ ξ, U^{*} (T^{'})^{*} y ⟩$ , so $U^{*} y \in D (T^{*}) = D (T)$ and $T^{*} U^{*} y = U^{*} (T^{'})^{*} y$ ; hence $y \in U D (T) = D (T^{'})$ and $(T^{'})^{*} y = U T U^{*} y = T^{'} y$ : $D ((T^{'})^{*}) \subseteq D (T^{'})$ and $T^{'}$ is self-adjoint. Applying this with $U = F_{2}^{*}$ and the real multiplier $T = M_{2 π ℏ ξ}$ (self-adjoint by [F4]) makes $P$ self-adjoint with the stated domain, and [F4] identifies $D (P) = W^{1, 2}$ and $P f = - i ℏ f^{'}$ there. The position operator $Q = M_{x}$ is self-adjoint by [F4]. For the basis vectors, $Q ψ_{n} (x) = x ℓ^{- 1 / 2} g_{n} (x / ℓ) = ℓ D (y g_{n}) (x) = \frac{ℓ}{\sqrt{2}} D (\sqrt{n} g_{n - 1} + \sqrt{n + 1} g_{n + 1}) (x)$ by [F3], which is the first displayed formula with $ψ_{- 1} = 0$ ; and $P ψ_{n} = - i ℏ ψ_{n}^{'}$ with $ψ_{n}^{'} = ℓ^{- 1} D (g_{n}^{'}) = \frac{ℓ^{- 1}}{\sqrt{2}} D (\sqrt{n} g_{n - 1} - \sqrt{n + 1} g_{n + 1})$ , giving the second formula.

2.1step 1.2F5

[(b) group and strong continuity] $U (t + s) f = \sum_{n} e^{- i ω (n + 1 / 2) (t + s)} c_{n} ψ_{n} = U (t) U (s) f$ first on every finite linear combination of the basis vectors by scalar multiplication of their phases, and then on all $f$ by density and continuity of the bounded operators $U (t + s)$ and $U (t) U (s)$ from step 1.2; $U (0) = I$ . For continuity at $s$ : $∥ U (t) f - U (s) f ∥^{2} = \sum_{n} ∣ e^{- i ω (n + 1 / 2) t} - e^{- i ω (n + 1 / 2) s} ∣^{2} ∣ c_{n} ∣^{2}$ , and for $t \to s$ the summands converge to $0$ pointwise in $n$ while being bounded by $4 ∣ c_{n} ∣^{2}$ , so the sum tends to $0$ (the tail $\sum_{n > N} 4 ∣ c_{n} ∣^{2}$ is arbitrarily small and finitely many summands converge).

2.2step 1.2F1

[(c) evolution of coherent vectors] By the definition of $ψ_{z}$ in the $(ψ_{n})$ -basis, $U (t) ψ_{z} = e^{- ∣ z ∣^{2} / 2} \sum_{n} \frac{z^{n}}{\sqrt{n!}} e^{- i ω (n + 1 / 2) t} ψ_{n} = e^{- i ω t / 2} \cdot e^{- ∣ z ∣^{2} / 2} \sum_{n} \frac{(z e^{- i ω t})^{n}}{\sqrt{n!}} ψ_{n} = e^{- i ω t / 2} ψ_{z e^{- i ω t}}$ , the series converging by step 1.2 and the definition of $ψ_{w}$ .

2.3step 1.3F2F6algebra

[(d),(e) action on coherent vectors and the moments] Since $ψ_{z} \in D (N_{ψ}) \subseteq D (a_{ψ}^{†}) = D (a_{ψ}) \cap D (a_{ψ}^{†})$ and $a_{ψ}, a_{ψ}^{†}$ are (transports of) the closed operators of [F2], the operators $\frac{ℓ}{\sqrt{2}} (a_{ψ} + a_{ψ}^{†})$ and $\frac{ℏ}{i ℓ \sqrt{2}} (a_{ψ} - a_{ψ}^{†})$ agree with $Q$ and $P$ on every $ψ_{n}$ by step 1.3, and $Q, P$ are closed; the partial sums $ψ_{z}^{(N)} = e^{- ∣ z ∣^{2} / 2} \sum_{n \leq N} \frac{z^{n}}{\sqrt{n!}} ψ_{n}$ converge to $ψ_{z}$ and their $Q$ -images converge: $∥ (a_{ψ} + a_{ψ}^{†}) (ψ_{z}^{(N)} - ψ_{z}) ∥^{2} \leq 2 (∥ a_{ψ} (\cdot) ∥^{2} + ∥ a_{ψ}^{†} (\cdot) ∥^{2}) \leq 4 \sum_{n > N} (n + 1) ∣ c_{n} ∣^{2} \to 0$ , and likewise for $a_{ψ} - a_{ψ}^{†}$ ; hence the displayed identities for $Q ψ_{z}$ and $P ψ_{z}$ hold, and $Q ψ_{z}, P ψ_{z} \in L^{2}$ . Therefore $⟨ Q ⟩_{z} = \frac{ℓ}{\sqrt{2}} (z + \overline{z}) = \sqrt{2} ℓ Re z$ and $⟨ P ⟩_{z} = \frac{ℏ}{i ℓ \sqrt{2}} (z - \overline{z}) = \frac{\sqrt{2} ℏ}{ℓ} Im z$ . For the variances, $Var (Q)_{z} = ∥ Q ψ_{z} ∥^{2} - ⟨ Q ⟩_{z}^{2}$ because the mean is real, and $∥ Q ψ_{z} ∥^{2} = \frac{ℓ^{2}}{2} ∥ (a_{ψ} + a_{ψ}^{†}) ψ_{z} ∥^{2} = \frac{ℓ^{2}}{2} (∥ a_{ψ} ψ_{z} ∥^{2} + ∥ a_{ψ}^{†} ψ_{z} ∥^{2} + 2 Re ⟨ a_{ψ} ψ_{z}, a_{ψ}^{†} ψ_{z} ⟩)$ . Now $⟨ a_{ψ} ψ_{z}, a_{ψ}^{†} ψ_{z} ⟩ = ⟨ z ψ_{z}, a_{ψ}^{†} ψ_{z} ⟩ = z \overline{⟨ a_{ψ}^{†} ψ_{z}, ψ_{z} ⟩} = z^{2}$ because $⟨ a_{ψ}^{†} ψ_{z}, ψ_{z} ⟩ = ⟨ ψ_{z}, a_{ψ} ψ_{z} ⟩ = \overline{z}$ ; and $∥ a_{ψ} ψ_{z} ∥^{2} = ∣ z ∣^{2}$ , $∥ a_{ψ}^{†} ψ_{z} ∥^{2} = ∣ z ∣^{2} + 1$ by [F2]. So $∥ Q ψ_{z} ∥^{2} = \frac{ℓ^{2}}{2} (2 ∣ z ∣^{2} + 1 + 2 Re z^{2}) = \frac{ℓ^{2}}{2} ((z + \overline{z})^{2} + 1)$ and $Var (Q)_{z} = \frac{ℓ^{2}}{2}$ . The same computation with a minus sign gives $∥ P ψ_{z} ∥^{2} = \frac{ℏ^{2}}{2 ℓ^{2}} (2 ∣ z ∣^{2} + 1 - 2 Re z^{2})$ and $Var (P)_{z} = \frac{ℏ^{2}}{2 ℓ^{2}}$ because $⟨ P ⟩_{z}^{2} = \frac{2 ℏ^{2}}{ℓ^{2}} (Im z)^{2} = \frac{ℏ^{2}}{2 ℓ^{2}} (2 ∣ z ∣^{2} - 2 Re z^{2})$ ; the product is $\frac{ℓ^{2}}{2} \cdot \frac{ℏ^{2}}{2 ℓ^{2}} = \frac{ℏ^{2}}{4}$ . No minimality among all states is asserted; the displayed values are those computed for the coherent vectors.

3.1step 1.1step 1.2step 2.1F5

[(b) generator and its exact domain] Let $f \in D (H_{o s c})$ and $g_{t} : = \frac{1}{t} (U (t) f - f) + \frac{i}{ℏ} H_{o s c} f = \sum_{n} ε_{n} (t) c_{n} ψ_{n}$ with $ε_{n} (t) = \frac{e^{- i ω (n + 1 / 2) t} - 1}{t} + i ω (n + \frac{1}{2})$ . By [F5], $∣ ε_{n} (t) ∣ \leq 2 ω (n + \frac{1}{2})$ , so $\sum_{n} ∣ ε_{n} (t) ∣^{2} ∣ c_{n} ∣^{2} \leq 4 ω^{2} \sum_{n} (n + \frac{1}{2})^{2} ∣ c_{n} ∣^{2} < \infty$ by step 1.1, and $ε_{n} (t) \to 0$ as $t \to 0$ for each $n$ by $e^{i x} = 1 + i x + o (x)$ ; the same dominated-convergence argument as in step 2.1 gives $∥ g_{t} ∥ \to 0$ . Hence $t \mapsto U (t) f$ is differentiable at $0$ with derivative $- \frac{i}{ℏ} H_{o s c} f$ ; the general $t$ follows from the group law and $U (t) f \in D (H_{o s c})$ (the coefficients $e^{- i ω (n + 1 / 2) t} c_{n}$ have the same moduli as $c_{n}$ ). Conversely, if $\frac{1}{t} (U (t) f - f) \to g$ in $L^{2}$ , then for every $n$ , $⟨ \frac{1}{t} (U (t) f - f), ψ_{n} ⟩ = \frac{e^{- i ω (n + 1 / 2) t} - 1}{t} c_{n} \to - i ω (n + \frac{1}{2}) c_{n}$ while these pairings converge to $⟨ g, ψ_{n} ⟩$ ; hence $⟨ g, ψ_{n} ⟩ = - i ω (n + \frac{1}{2}) c_{n}$ for every $n$ and $\sum_{n} (n + \frac{1}{2})^{2} ∣ c_{n} ∣^{2} = ω^{- 2} \sum_{n} ∣ ⟨ g, ψ_{n} ⟩ ∣^{2} = ω^{- 2} ∥ g ∥^{2} < \infty$ by [F5], so $f \in D (H_{o s c})$ and $g = - \frac{i}{ℏ} H_{o s c} f$ .

4.1step 2.2step 2.3∎

[(f) evolved moments] By step 2.2 and the definitions, $⟨ Q ⟩_{z (t)} = ⟨ Q (e^{- i ω t / 2} ψ_{z (t)}), e^{- i ω t / 2} ψ_{z (t)} ⟩ = ⟨ Q ψ_{z (t)}, ψ_{z (t)} ⟩ = \sqrt{2} ℓ Re z (t)$ , and similarly $⟨ P ⟩_{z (t)} = \frac{\sqrt{2} ℏ}{ℓ} Im z (t)$ ; the variances in step 2.3 depend on $z$ only through the value of the displayed constant formulas, which are independent of $z$ , so they are unchanged: $Var (Q)_{z (t)} = ℓ^{2} / 2$ and $Var (P)_{z (t)} = ℏ^{2} / (2 ℓ^{2})$ .

Coherent evolution and variance

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources