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Fluctuations of the oscillator coherent states and the position-momentum relation of the model

Physical scope

The preparation spreads of position and momentum in the coherent states of the one-dimensional oscillator model, the exact value hbar/2 of their product, the one-dimensional position-momentum relation Delta Q Delta P >= hbar/2 for every unit preparation vector in the common domain of the two model operators, and the saturation of that relation by every coherent preparation. The statement is a conditional consequence of the postulates P1, P2 and P5, of the adopted oscillator model, of the cited uncertainty and moment results and of the cited mathematics; it concerns the two separately assigned outcome laws in one preparation and asserts no joint outcome law, no apparatus resolution, no disturbance and no empirical result.

Statement

Adopt the state and composition postulate P1, the observable and probability postulate P2 and the kinematics and massive-particle model postulate P5, read through the one-dimensional oscillator model of The quadratic-potential oscillator model of the framework clauses (a), (b), (c), and the coherent-state family of The coherent states of the oscillator model; assume the Axiom of Countable Choice and the Axiom of Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ ), The Axiom of Choice). Let $Q$ and $P$ be the model's position and momentum observables with their declared units ( $m$ and $k g m s^{- 1}$ ), let $ℓ = \sqrt{ℏ / (m ω)}$ , let $Δ_{ρ} A$ be the preparation spread of an assigned sharp observable in the reading of Preparation variance and spread of an assigned sharp observable, and write $⟨ A ⟩_{z} : = ⟨ A Ψ_{z}, Ψ_{z} ⟩$ for the model's coherent preparation of label $z$ . Then:

(a) Coherent spreads. For every $z \in C$ the vector $Ψ_{z}$ lies in $D (Q) \cap D (P)$ , the preparation spreads of $Q$ and $P$ in the pure preparation of $Ψ_{z}$ exist, and $Δ_{Ψ_{z}} Q = \frac{ℓ}{\sqrt{2}}, Δ_{Ψ_{z}} P = \frac{ℏ}{ℓ \sqrt{2}}, Δ_{Ψ_{z}} Q Δ_{Ψ_{z}} P = \frac{ℏ}{2};$

(b) Position-momentum relation of the model. Every unit vector $f \in D (Q) \cap D (P)$ satisfies $Δ_{f} Q Δ_{f} P \geq \frac{ℏ}{2},$ read in the product unit $k g m^{2} s^{- 1}$ ; consequently the value $ℏ / 2$ attained in (a) is the minimum over the common domain, and it is attained by every coherent preparation;

(c) Saturation. For every $z \in C$ , with the centered vectors $x_{z} : = (Q - ⟨ Q ⟩_{z} I) Ψ_{z}, y_{z} : = (P - ⟨ P ⟩_{z} I) Ψ_{z},$ one has $⟨ x_{z}, y_{z} ⟩ = - i ℏ / 2$ ; in particular $Re ⟨ x_{z}, y_{z} ⟩ = 0$ and $x_{z}, y_{z}$ are linearly dependent, so $∥ x_{z} ∥ ∥ y_{z} ∥ = ∣ ⟨ x_{z}, y_{z} ⟩ ∣ = ℏ / 2$ ;

(d) Status. Clauses (a)-(c) are conditional consequences of P1, P2, P5, the adopted oscillator model, the cited preparation-moment and uncertainty results and the cited mathematics. They concern the spreads of the two separately assigned outcome laws of $Q$ and $P$ in one preparation; they introduce no joint outcome law of position and momentum, no measurement error, no disturbance of the preparation, no dynamics and no empirical value. Whether a laboratory preparation and apparatus realize the pair of observables and produce this spread is a tested identification under the preparation and apparatus postulate P7.

Facts & Assumptions

Given: the adopted postulates P1, P2, P5; the Axiom of Countable Choice and the Axiom of Choice; the one-dimensional oscillator model with parameters $m, ω, ℏ > 0$ and length $ℓ$ ; the coherent states $Ψ_{z}$ ; a unit vector $f \in D (Q) \cap D (P)$ ; a complex number $z$ .

[F1]

The model's observables. $(Q, m)$ and $(P, k g m s^{- 1})$ are assigned sharp observables of the model with $D (Q) = {f : x f \in L^{2}}$ , $Q f = x f$ , $D (P) = {f : D_{x} f \in L^{2}}$ , $P f = - i ℏ D_{x} f$ , where $D_{x}$ is the weak derivative, and $Q, P$ are the $d = 1$ , $r = 1$ instances of the componentwise operators of Componentwise position and momentum operators on the component-valued wave-function space; the flow of the isolated model is generated by the assigned energy observable $H_{1}$ (The quadratic-potential oscillator model of the framework clauses (b), (c); Kinematics and massive-particle model postulate P5: Newtonian frames and one-particle operators; Assigned sharp observables and the outcome law of the framework clause (a)).

[F2]

Coherent data. For every $z$ : $Ψ_{z}$ is a unit vector with $a_{ψ} Ψ_{z} = z Ψ_{z}$ and $Ψ_{z} \in D (N_{ψ}^{k})$ for every $k$ ; $⟨ Ψ_{z}, Ψ_{w} ⟩ = e^{- (∣ z ∣^{2} + ∣ w ∣^{2}) / 2 + z \overline{w}}$ ; and $Ψ_{0} = ψ_{0}$ (The coherent states of the oscillator model clauses (b), (c); Coherent-state convergence clauses (a), (b), (c), (d), (e)).

[F3]

Exact coherent quantities. With $Q$ and $P$ the position and momentum operators on $L^{2} (R, d x)$ and $a_{ψ} = D a D^{- 1}$ , $a_{ψ}^{†} = D a^{†} D^{- 1}$ the transported ladder operators, $Q Ψ_{z} = \frac{ℓ}{\sqrt{2}} (a_{ψ} + a_{ψ}^{†}) Ψ_{z}, P Ψ_{z} = \frac{ℏ}{i ℓ \sqrt{2}} (a_{ψ} - a_{ψ}^{†}) Ψ_{z},$ and the quadratic-form means and variances of the coherent vectors are $⟨ Q Ψ_{z}, Ψ_{z} ⟩ = \sqrt{2} ℓ Re z, ⟨ P Ψ_{z}, Ψ_{z} ⟩ = \frac{\sqrt{2} ℏ}{ℓ} Im z,$ $∥ (Q - ⟨ Q ⟩_{z} I) Ψ_{z} ∥^{2} = \frac{ℓ^{2}}{2}, ∥ (P - ⟨ P ⟩_{z} I) Ψ_{z} ∥^{2} = \frac{ℏ^{2}}{2 ℓ^{2}};$ and $∥ a_{ψ} Ψ_{z} ∥^{2} = ∣ z ∣^{2}$ , $∥ a_{ψ}^{†} Ψ_{z} ∥^{2} = ∣ z ∣^{2} + 1$ ; also the adjoint relation $(a_{ψ})^{*} = a_{ψ}^{†}$ holds, the transport of $a^{*} = a^{†}$ along the unitary $D$ (Coherent evolution and variance clauses (d), (e); Coherent-state convergence clause (e); Closed ladder domains clauses (b), (d), (e)).

[F4]

Means, variances and spreads. For an assigned sharp observable $(A, u_{A})$ and a unit vector $f \in D (A)$ the mean and second moment of the outcome law exist, $\int λ d μ_{f}^{A} = ⟨ A f, f ⟩$ , $\int λ^{2} d μ_{f}^{A} = ∥ A f ∥^{2}$ , and the preparation variance and spread are ${Var}_{f} (A) = ∥ (A - ⟨ A f, f ⟩ I) f ∥^{2}, Δ_{f} A = {Var}_{f} (A)^{1 / 2},$ read in the unit $u_{A}$ and its square (Mean, second moment and variance of an assigned observable in a preparation clauses (a), (b); Preparation variance and spread of an assigned sharp observable clauses (b), (c), (d)).

[F5]

Product domain and commutator in one dimension. For $d = 1$ , $r = 1$ the product domain is $D (Q P) \cap D (P Q) = {f : D_{x} f, x f, x D_{x} f \in L^{2}}$ and $(Q P - P Q) φ = i ℏ φ$ for every $φ$ in it (The position-momentum commutator identity on the product domain clauses (b), (c)).

[F6]

Robertson relation. For two assigned sharp observables $(A, u_{A})$ , $(B, u_{B})$ and a unit vector $ψ \in D (A) \cap D (B)$ : $Δ_{ψ} A Δ_{ψ} B \geq ∣ Im ⟨ A ψ, B ψ ⟩ ∣$ ; and if in addition $ψ \in D (A B) \cap D (B A)$ , then $Δ_{ψ} A Δ_{ψ} B \geq \frac{1}{2} ∣ ⟨ [A, B] ψ, ψ ⟩ ∣$ (The Robertson relation for preparation spreads of assigned sharp observables clauses (a), (b)).

[F7]

Approximation by product-domain vectors. For every nonzero $f \in D (Q) \cap D (P)$ there is a sequence of unit vectors $φ_{k} \in D (Q P) \cap D (P Q)$ with $φ_{k} \to f / ∥ f ∥$ , $Q φ_{k} \to Q f / ∥ f ∥$ , $P φ_{k} \to P f / ∥ f ∥$ , and with $⟨ Q φ_{k}, φ_{k} ⟩ \to ⟨ Q f, f ⟩ / ∥ f ∥^{2}$ , $∥ Q φ_{k} ∥ \to ∥ Q f ∥ / ∥ f ∥$ , and the analogous momentum limits (Approximation of Q-P graph-domain vectors by product-domain vectors).

[F8]

Inner-product calculus and Cauchy-Schwarz. The pairing is linear in the first argument and conjugate-linear in the second, $∥ u ∥^{2} = ⟨ u, u ⟩$ , $∣ ⟨ u, v ⟩ ∣ \leq ∥ u ∥ ∥ v ∥$ with equality if and only if $u, v$ are linearly dependent, and inner products and norms are continuous for norm convergence (Real and complex inner product spaces, with the inner product linear in the first argument, Hilbert space, Cauchy–Schwarz: $∣ ⟨ u, v ⟩ ∣ \leq ∥ u ∥ ∥ v ∥$ , with equality exactly for linearly dependent vectors).

Proof

Given: the data of the Statement and [F1]-[F8].

1.1givenF1F2F3

[The model's operators and the coherent formulas] By [F1] the model's $Q$ , $P$ are the $d = 1$ , $r = 1$ componentwise operators with the displayed dense domains, and both are assigned sharp observables; by [F2] each $Ψ_{z}$ is a unit vector with $a_{ψ} Ψ_{z} = z Ψ_{z}$ , $∥ a_{ψ} Ψ_{z} ∥^{2} = ∣ z ∣^{2}$ and $∥ a_{ψ}^{†} Ψ_{z} ∥^{2} = ∣ z ∣^{2} + 1$ . By [F3] the vectors $Q Ψ_{z} = \frac{ℓ}{\sqrt{2}} (a_{ψ} + a_{ψ}^{†}) Ψ_{z}$ and $P Ψ_{z} = \frac{ℏ}{i ℓ \sqrt{2}} (a_{ψ} - a_{ψ}^{†}) Ψ_{z}$ lie in $H_{1}$ , so $Ψ_{z} \in D (Q) \cap D (P)$ , and the displayed coherent means and central norms hold.

1.2givenF4

[Spreads as centered norms] Let $f \in D (Q) \cap D (P)$ be a unit vector. By [F4] applied to the assigned observables $(Q, m)$ and $(P, k g m s^{- 1})$ , the outcome-law means and variances exist and $Δ_{f} Q^{2} = ∥ (Q - ⟨ Q f, f ⟩ I) f ∥^{2} = ∥ Q f ∥^{2} - ⟨ Q f, f ⟩^{2}$ , and likewise $Δ_{f} P^{2} = ∥ P f ∥^{2} - ⟨ P f, f ⟩^{2}$ .

1.3givenF5F6

[Bound on the product domain] Let $φ \in D (Q P) \cap D (P Q)$ be a unit vector. By [F5], $(Q P - P Q) φ = i ℏ φ$ , so $⟨ [Q, P] φ, φ ⟩ = ⟨ i ℏ φ, φ ⟩ = i ℏ ⟨ φ, φ ⟩ = i ℏ$ by linearity of the pairing in its first argument. By the Robertson form of [F6] applied to the assigned sharp observables $(Q, m)$ and $(P, k g m s^{- 1})$ , $Δ_{φ} Q Δ_{φ} P \geq \frac{1}{2} ∣ i ℏ ∣ = ℏ / 2$ .

1.4givenF7

[Approximating a common-domain vector] Let $f \in D (Q) \cap D (P)$ be a unit vector. By [F7] there is a sequence of unit vectors $φ_{k} \in D (Q P) \cap D (P Q)$ with $φ_{k} \to f$ , $Q φ_{k} \to Q f$ , $P φ_{k} \to P f$ , $⟨ Q φ_{k}, φ_{k} ⟩ \to ⟨ Q f, f ⟩$ , $∥ Q φ_{k} ∥ \to ∥ Q f ∥$ and the analogous momentum limits.

2.1step 1.1step 1.2F3

[Clause (a)] By step 1.1 and step 1.2 applied to the unit vector $Ψ_{z} \in D (Q) \cap D (P)$ , the preparation variances are the central squared norms of [F3]: ${Var}_{Ψ_{z}} (Q) = ∥ (Q - ⟨ Q ⟩_{z} I) Ψ_{z} ∥^{2} = ℓ^{2} / 2$ and ${Var}_{Ψ_{z}} (P) = ℏ^{2} / (2 ℓ^{2})$ ; taking square roots gives $Δ_{Ψ_{z}} Q = ℓ / \sqrt{2}$ and $Δ_{Ψ_{z}} P = ℏ / (ℓ \sqrt{2})$ , whose product is $ℏ / 2$ in the unit $m \cdot k g m s^{- 1} = k g m^{2} s^{- 1}$ .

2.2step 1.2step 1.3step 1.4F8

[Clause (b), relation] Let $f \in D (Q) \cap D (P)$ be a unit vector and let $(φ_{k})$ be the approximating sequence of step 1.4. For every $k$ , step 1.3 gives $Δ_{φ_{k}} Q Δ_{φ_{k}} P \geq ℏ / 2$ ; by step 1.2, $Δ_{φ_{k}} Q^{2} = ∥ Q φ_{k} ∥^{2} - ⟨ Q φ_{k}, φ_{k} ⟩^{2}$ , and the limits of step 1.4 together with the continuity of the inner product [F8] give $Δ_{φ_{k}} Q^{2} \to ∥ Q f ∥^{2} - ⟨ Q f, f ⟩^{2} = Δ_{f} Q^{2}$ , and likewise $Δ_{φ_{k}} P^{2} \to Δ_{f} P^{2}$ . Taking square roots and passing to the limit in the inequality gives $Δ_{f} Q Δ_{f} P \geq ℏ / 2$ .

3.1step 1.1step 2.1F3F8

[Clause (c), saturation] Fix $z$ and put $u : = (a_{ψ} + a_{ψ}^{†}) Ψ_{z}$ , $v : = (a_{ψ} - a_{ψ}^{†}) Ψ_{z}$ , both in $H_{1}$ by step 1.1. Using the conjugate-linearity in the second argument, the coherent data $a_{ψ} Ψ_{z} = z Ψ_{z}$ and the adjoint relation $(a_{ψ})^{*} = a_{ψ}^{†}$ of [F2] and [F3], so that $⟨ a_{ψ} Ψ_{z}, Ψ_{z} ⟩ = z$ , $⟨ Ψ_{z}, a_{ψ}^{†} Ψ_{z} ⟩ = ⟨ a_{ψ} Ψ_{z}, Ψ_{z} ⟩ = z$ and $⟨ a_{ψ}^{†} Ψ_{z}, Ψ_{z} ⟩ = \overline{z}$ , together with $∥ Ψ_{z} ∥ = 1$ and $∥ a_{ψ}^{†} Ψ_{z} ∥^{2} = ∣ z ∣^{2} + 1$ , the four pairings are $⟨ a_{ψ} Ψ_{z}, a_{ψ} Ψ_{z} ⟩ = ∣ z ∣^{2}$ , $⟨ a_{ψ} Ψ_{z}, a_{ψ}^{†} Ψ_{z} ⟩ = z ⟨ Ψ_{z}, a_{ψ}^{†} Ψ_{z} ⟩ = z^{2}$ , $⟨ a_{ψ}^{†} Ψ_{z}, a_{ψ} Ψ_{z} ⟩ = \overline{z} ⟨ a_{ψ}^{†} Ψ_{z}, Ψ_{z} ⟩ = {\overline{z}}^{2}$ and $⟨ a_{ψ}^{†} Ψ_{z}, a_{ψ}^{†} Ψ_{z} ⟩ = ∣ z ∣^{2} + 1$ , so $⟨ u, v ⟩ = ∣ z ∣^{2} - z^{2} + {\overline{z}}^{2} - (∣ z ∣^{2} + 1) = {\overline{z}}^{2} - z^{2} - 1$ ; hence, using the first formula of [F3] and $\overline{ℏ / (i ℓ \sqrt{2})} = i ℏ / (ℓ \sqrt{2})$ , $⟨ Q Ψ_{z}, P Ψ_{z} ⟩ = \frac{ℓ}{\sqrt{2}} \cdot \frac{i ℏ}{ℓ \sqrt{2}} ⟨ u, v ⟩ = \frac{i ℏ}{2} ({\overline{z}}^{2} - z^{2} - 1) = 2 ℏ Re z Im z - \frac{i ℏ}{2}$ , because ${\overline{z}}^{2} - z^{2} = (\overline{z} - z) (\overline{z} + z) = - 4 i Re z Im z$ . By [F3], $⟨ Q ⟩_{z} ⟨ P ⟩_{z} = \sqrt{2} ℓ Re z \cdot \frac{\sqrt{2} ℏ}{ℓ} Im z = 2 ℏ Re z Im z$ , so $⟨ x_{z}, y_{z} ⟩ = ⟨ Q Ψ_{z}, P Ψ_{z} ⟩ - ⟨ Q ⟩_{z} ⟨ P ⟩_{z} = - \frac{i ℏ}{2}$ . Therefore $Re ⟨ x_{z}, y_{z} ⟩ = 0$ and $∣ ⟨ x_{z}, y_{z} ⟩ ∣ = ℏ / 2$ , while step 2.1 gives $∥ x_{z} ∥ ∥ y_{z} ∥ = Δ_{Ψ_{z}} Q Δ_{Ψ_{z}} P = ℏ / 2$ , so equality holds in the Cauchy-Schwarz inequality of [F8] and $x_{z}, y_{z}$ are linearly dependent.

3.2step 2.1step 2.2

[Clause (b), minimality] By step 2.2 every unit vector in $D (Q) \cap D (P)$ has spread product at least $ℏ / 2$ , and by step 2.1 every coherent preparation attains the value $ℏ / 2$ ; hence the minimum of the spread product over the unit vectors of the common domain is $ℏ / 2$ , attained by each $[Ψ_{z}]$ .

4.1step 2.1step 2.2step 3.1step 3.2∎

[Clause (d)] The spread values and the relation were obtained from the model's assigned observables, the preparation-moment theorem, the product-domain commutator, the cutoff approximation and the Robertson relation; the saturation computation used the same operators and the exact coherent-state quantities. No joint outcome law, apparatus resolution, disturbance, dynamics or empirical value entered, and the identification of a laboratory pair with $(Q, P)$ is a tested identification under P7. This is clause (d).

Fluctuations of the oscillator coherent states and the position-momentum relation of the model

Statement

Facts & Assumptions

Proof

Depends on

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