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Evolution of the oscillator coherent states: classical mean motion and stationary fluctuations

Physical scope

The time evolution of the coherent states of the one-dimensional oscillator model: under the model's unitary flow the coherent preparation of label z passes to the coherent preparation of label z exp(-i omega t); the preparation means of position and momentum perform the classical harmonic motion through (x_0,p_0) with frequency omega, while the position and momentum spreads stay at l/sqrt(2) and hbar/(l sqrt(2)) and the saturated product hbar/2 is preserved; the excitation-number mean is |z|^2 in the assigned dimensionless unit 1; and the energy outcome law of the coherent preparation is the affine image of the Poisson law of parameter |z|^2 under n -> hbar omega (n+1/2) under the convention subtracting the potential-energy value V(0), with energy spread hbar omega |z|. The statement is a conditional consequence of P1, P2, P4 and P5, of the adopted oscillator model and of the cited coherent-state and excursion mathematics; it asserts no measurement record, no apparatus realisation and no empirical result.

Statement

Adopt the state and composition postulate P1, the observable and probability postulate P2, the autonomous-dynamics postulate P4 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 $U (t) = \exp (- i t H_{1} / ℏ)$ be the model's flow, let $Ψ_{z}$ be the coherent state of label $z \in C$ , put $z (t) : = z e^{- i ω t}$ , and define $x_{0} : = \sqrt{2} ℓ Re z$ , $p_{0} : = \frac{\sqrt{2} ℏ}{ℓ} Im z$ for every $z \in C$ . Then:

(a) Coherent evolution. For every $t \in R$ , $U (t)$ is a bijective linear isometry of $H_{1}$ and $U (t) Ψ_{z} = e^{- i ω t / 2} Ψ_{z (t)};$ consequently the preparation $[Ψ_{z}]$ evolves under the model's Schroedinger dynamics to the coherent preparation $[Ψ_{z (t)}]$ , and the labels of the evolved family rotate with angular frequency $ω$ ;

(b) Classical mean motion. The preparation means of position and momentum at time $t$ are $⟨ Q ⟩_{z (t)} = \sqrt{2} ℓ Re z (t), ⟨ P ⟩_{z (t)} = \frac{\sqrt{2} ℏ}{ℓ} Im z (t),$ equivalently, for every $t \in R$ , $x (t) = x_{0} \cos ω t + \frac{p_{0}}{m ω} \sin ω t$ and $p (t) = p_{0} \cos ω t - m ω x_{0} \sin ω t$ with $x (t) : = ⟨ Q ⟩_{z (t)}$ , $p (t) : = ⟨ P ⟩_{z (t)}$ ; these functions are differentiable with $\dot{x} (t) = \frac{p (t)}{m}$ and $\dot{p} (t) = - m ω^{2} x (t)$ for every $t$ , so the means follow the classical harmonic motion of frequency $ω$ through $(x_{0}, p_{0})$ ;

(c) Stationary fluctuations. For every $t \in R$ the evolved preparation has $Δ_{Ψ_{z (t)}} Q = \frac{ℓ}{\sqrt{2}}, Δ_{Ψ_{z (t)}} P = \frac{ℏ}{ℓ \sqrt{2}}, Δ_{Ψ_{z (t)}} Q Δ_{Ψ_{z (t)}} P = \frac{ℏ}{2};$ the position and momentum spreads and their product are constant along the evolution;

(d) Energy statistics. The mean and the outcome law of the assigned energy observable in the coherent preparation are $⟨ H_{1} ⟩_{z} = ℏ ω (∣ z ∣^{2} + \frac{1}{2}), μ_{Ψ_{z}}^{H_{1}} ({ℏ ω (n + \frac{1}{2})}) = e^{- ∣ z ∣^{2}} \frac{∣ z ∣^{2 n}}{n!} (n \in N),$ the pushforward under $n \mapsto ℏ ω (n + \frac{1}{2})$ of the Poisson law on $N$ with parameter $s : = ∣ z ∣^{2}$ , defined by the weights $p_{n} = e^{- s} s^{n} / n!$ , with $s^{0} = 1$ also at $s = 0$ ; consequently $Δ_{Ψ_{z}} H_{1} = ℏ ω ∣ z ∣$ ;

(e) Status. Clauses (a)-(d) are conditional consequences of P1, P2, P4, P5, the adopted oscillator model, the model's spectrum theorem, the coherent-state fluctuation theorem and the cited mathematics. They concern the exact evolution of the model's coherent preparations and the statistics of its assigned observables; no measurement record, no apparatus realization, no damping or driving and no empirical value is asserted, and the identification of the mean motion with the motion of a laboratory oscillator is a tested identification under P7.

Facts & Assumptions

Given: the adopted postulates P1, P2, P4, P5; the Axiom of Countable Choice and the Axiom of Choice; the one-dimensional oscillator model with parameters $m, ω, ℏ > 0$ , length $ℓ$ and energy operator $H_{1}$ ; the coherent states $Ψ_{z}$ ; a complex number $z$ and a real number $t$ .

[F1]

The model's flow. $H_{1}$ is the model's assigned energy observable, and the model's flow is the strongly continuous one-parameter unitary group $U (t) = \exp (- i t H_{1} / ℏ) = \int e^{- i t λ / ℏ} d E_{H_{1}} (λ)$ of the autonomous-dynamics reading of the model definition, whose generator is $- \frac{i}{ℏ} H_{1}$ with domain $D (H_{1})$ (The quadratic-potential oscillator model of the framework clause (c); Assigned energy observable, unitary flow and the two pictures of autonomous evolution clauses (a), (b), (c); Autonomous dynamics postulate P4: self-adjoint generators and unitary evolution).

[F2]

Explicit flow and coherent evolution. The family $U (t) f = \sum_{n} e^{- i ω (n + \frac{1}{2}) t} ⟨ f, ψ_{n} ⟩ ψ_{n}$ is the strongly continuous one-parameter unitary group with generator $- \frac{i}{ℏ} H_{1}$ , so it is the flow of [F1]; and for every $z$ , $U (t) Ψ_{z} = e^{- i ω t / 2} Ψ_{z e^{- i ω t}}$ , while the evolved coherent means and variances are $⟨ Q ⟩_{z (t)} = \sqrt{2} ℓ Re z (t)$ , $⟨ P ⟩_{z (t)} = \frac{\sqrt{2} ℏ}{ℓ} Im z (t)$ , $Var (Q)_{z (t)} = ℓ^{2} / 2$ , $Var (P)_{z (t)} = ℏ^{2} / (2 ℓ^{2})$ with $z (t) = z e^{- i ω t}$ (Coherent evolution and variance clauses (a), (b), (c), (e), (f); The quadratic-potential oscillator model of the framework clause (c)).

[F3]

Coherent coefficients and the energy law. $Ψ_{z} = e^{- ∣ z ∣^{2} / 2} \sum_{n} \frac{z^{n}}{\sqrt{n!}} ψ_{n}$ with coefficients $⟨ Ψ_{z}, ψ_{n} ⟩ = e^{- ∣ z ∣^{2} / 2} \frac{z^{n}}{\sqrt{n!}}$ ; the mean of the assigned excitation-number outcome law in unit $1$ is $⟨ N_{ψ} Ψ_{z}, Ψ_{z} ⟩ = ∣ z ∣^{2}$ , since $Ψ_{z} \in D (N_{ψ})$ by The coherent states of the oscillator model clause (c), the numerical value is given by Coherent-state convergence clause (e), and the outcome-law moment formula is Mean, second moment and variance of an assigned observable in a preparation clause (a). The energy outcome law of the assigned observable $(H_{1}, J)$ is $μ_{Ψ_{z}}^{H_{1}} (B) = \sum_{n : E_{n} \in B} ∣ ⟨ Ψ_{z}, ψ_{n} ⟩ ∣^{2}$ with $E_{n} = ℏ ω (n + \frac{1}{2})$ ; its mean is $⟨ H_{1} ⟩_{z} = ℏ ω (∣ z ∣^{2} + \frac{1}{2})$ by $H_{1} = ℏ ω (N_{ψ} + \frac{1}{2} I)$ and the same moment formula, using $D (H_{1}) = D (N_{ψ})$ from The energy observable of the oscillator model: spectrum, eigenbasis and outcome law clause (b). (Coherent-state convergence clauses (a), (e); The coherent states of the oscillator model clause (b); The energy observable of the oscillator model: spectrum, eigenbasis and outcome law clauses (b), (c); Mean, second moment and variance of an assigned observable in a preparation clause (a); Closed ladder domains clause (e)).

[F4]

Euler's formula, parity and derivatives. For real $θ$ , $e^{i θ} = \cos θ + i \sin θ$ , hence $e^{- i ω t} = \cos ω t - i \sin ω t$ and $z (t) = (z_{1} \cos ω t + z_{2} \sin ω t) + i (z_{2} \cos ω t - z_{1} \sin ω t)$ for $z = z_{1} + i z_{2}$ ; the trigonometric functions satisfy $\sin^{'} = \cos$ , $\cos^{'} = - \sin$ , $\sin (- θ) = - \sin θ$ and $\cos (- θ) = \cos θ$ ( $\exp (x + i y) = e^{x} (\cos y + i \sin y)$ , $∣ \exp (x + i y) ∣ = e^{x}$ , and $e^{i π} + 1 = 0$ , Complex sine, cosine, hyperbolic sine, and hyperbolic cosine are entire with their standard derivatives, Pythagorean and parity identities for all six trigonometric functions on their natural domains).

[F5]

Means 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 with mean $⟨ A f, f ⟩$ and variance $∥ (A - ⟨ A f, f ⟩ I) f ∥^{2}$ , and the preparation spread is its square root (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)).

[F6]

Coherent fluctuations. For every $w \in C$ one has $Δ_{Ψ_{w}} Q = ℓ / \sqrt{2}$ , $Δ_{Ψ_{w}} P = ℏ / (ℓ \sqrt{2})$ and $Δ_{Ψ_{w}} Q Δ_{Ψ_{w}} P = ℏ / 2$ (Fluctuations of the oscillator coherent states and the position-momentum relation of the model clauses (a), (b)).

[F7]

The exponential series. For every real $s$ , $e^{s} = \sum_{n \geq 0} s^{n} / n!$ as the sum of an absolutely convergent series, and $\sum_{n \geq 1} n \frac{s^{n}}{n!} = s e^{s}$ , $\sum_{n \geq 2} n (n - 1) \frac{s^{n}}{n!} = s^{2} e^{s}$ by reindexing (The complex exponential by its power series, The complex exponential series converges absolutely for every complex argument).

[F8]

Status of laboratory identifications. P7 requires declared preparation, isolation, detection and sampling models for comparisons with recorded data; identifications of apparatus with the model are tested, uncertain physical assumptions, and agreement is evidence rather than proof (Preparation and apparatus postulate P7: declared preparations, isolation and sampling assumptions clauses (i), (iii), (v)). This supplies only the status qualification in clause (e), not a premise for clauses (a)-(d).

Proof

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

1.1givenF1F2

[The flow and its coherent action] By [F1] the model's flow is the strongly continuous unitary group generated by $- \frac{i}{ℏ} H_{1}$ ; by [F2] the explicit spectral family $U (t) f = \sum_{n} e^{- i ω (n + \frac{1}{2}) t} ⟨ f, ψ_{n} ⟩ ψ_{n}$ has the same generator, so by the uniqueness of the generator recorded in the autonomous-dynamics reading the two families agree, and $U (t) Ψ_{z} = e^{- i ω t / 2} Ψ_{z (t)}$ for every $z$ and $t$ .

1.2givenF3

[Energy law of the coherent preparation] By [F3] the coefficients of $Ψ_{z}$ in the eigenbasis are $⟨ Ψ_{z}, ψ_{n} ⟩ = e^{- ∣ z ∣^{2} / 2} \frac{z^{n}}{\sqrt{n!}}$ , so $∣ ⟨ Ψ_{z}, ψ_{n} ⟩ ∣^{2} = e^{- ∣ z ∣^{2}} \frac{∣ z ∣^{2 n}}{n!}$ and the energy law is the atomic law $μ_{Ψ_{z}}^{H_{1}} ({E_{n}}) = e^{- ∣ z ∣^{2}} \frac{∣ z ∣^{2 n}}{n!}$ with $E_{n} = ℏ ω (n + \frac{1}{2})$ ; and the mean energy is $⟨ H_{1} ⟩_{z} = ℏ ω (∣ z ∣^{2} + \frac{1}{2})$ by [F3].

1.3givenF4

[Euler and trigonometric calculus] By [F4], $e^{- i ω t} = \cos ω t - i \sin ω t$ , so for $z = z_{1} + i z_{2}$ one has $z (t) = z (\cos ω t - i \sin ω t) = (z_{1} \cos ω t + z_{2} \sin ω t) + i (z_{2} \cos ω t - z_{1} \sin ω t)$ , and the derivatives of $\sin$ and $\cos$ are $\cos$ and $- \sin$ .

2.1step 1.1F1

[Clause (a)] By step 1.1, $U (t) Ψ_{z} = e^{- i ω t / 2} Ψ_{z (t)}$ for every $t$ ; since $U (t)$ is a bijective linear isometry by [F1], the Schroedinger-picture state of the preparation $[Ψ_{z}]$ at time $t$ is the ray $[U (t) Ψ_{z}] = [Ψ_{z (t)}]$ , which is again a coherent preparation; the label $z (t) = z e^{- i ω t}$ rotates with angular frequency $ω$ .

2.2step 1.1F2F5

[Clause (b), complex form] By step 1.1 the evolved vector is $e^{- i ω t / 2} Ψ_{z (t)}$ , and multiplying by the phase $e^{- i ω t / 2}$ does not change the quadratic-form means; by [F2] and [F5], $⟨ Q ⟩_{z (t)} = \sqrt{2} ℓ Re z (t)$ and $⟨ P ⟩_{z (t)} = \frac{\sqrt{2} ℏ}{ℓ} Im z (t)$ .

2.3step 1.1F6

[Clause (c)] By step 1.1 the preparation at time $t$ is the coherent preparation of label $z (t)$ , so [F6] applied with $w = z (t)$ gives $Δ_{Ψ_{z (t)}} Q = ℓ / \sqrt{2}$ , $Δ_{Ψ_{z (t)}} P = ℏ / (ℓ \sqrt{2})$ and product $ℏ / 2$ for every $t$ : the spreads are constant along the evolution.

2.4step 1.2F5F7algebra

[Clause (d), Poisson law and energy spread] Let $p_{n} : = e^{- ∣ z ∣^{2}} \frac{∣ z ∣^{2 n}}{n!}$ and $s : = ∣ z ∣^{2}$ . By step 1.2 the law is $μ_{Ψ_{z}}^{H_{1}} ({E_{n}}) = p_{n}$ ; by [F7], $\sum_{n} p_{n} = e^{- s} \sum_{n} \frac{s^{n}}{n!} = 1$ , $\sum_{n} n p_{n} = e^{- s} \sum_{n} n \frac{s^{n}}{n!} = e^{- s} s e^{s} = s$ , and $\sum_{n} n^{2} p_{n} = e^{- s} \sum_{n} n^{2} \frac{s^{n}}{n!} = e^{- s} (s^{2} e^{s} + s e^{s}) = s^{2} + s$ , because $n^{2} = n (n - 1) + n$ and the two reindexed series of [F7] apply. Hence the number-coordinate variance is $\sum_{n} (n - s)^{2} p_{n} = (s^{2} + s) - s^{2} = s = ∣ z ∣^{2}$ . Since $E_{n} - ℏ ω (s + \frac{1}{2}) = ℏ ω (n - s)$ , the energy-law variance is $(ℏ ω)^{2} s$ , and by the outcome-law spread definition [F5] its square root is $Δ_{Ψ_{z}} H_{1} = ℏ ω ∣ z ∣$ . Thus the energy law is the affine image of the Poisson law of parameter $∣ z ∣^{2}$ under $n \mapsto E_{n} = ℏ ω (n + \frac{1}{2})$ ; at $s = 0$ the convention $s^{0} = 1$ gives $p_{0} = 1$ and $p_{n} = 0$ for $n \geq 1$ .

3.1step 1.3step 2.2F4algebra

[Clause (b), classical form] By step 2.2 and step 1.3, with $z = z_{1} + i z_{2}$ , $x (t) = \sqrt{2} ℓ (z_{1} \cos ω t + z_{2} \sin ω t)$ and $p (t) = \frac{\sqrt{2} ℏ}{ℓ} (z_{2} \cos ω t - z_{1} \sin ω t)$ ; since $x_{0} = \sqrt{2} ℓ z_{1}$ and $\frac{p_{0}}{m ω} = \frac{\sqrt{2} ℏ z_{2}}{ℓ m ω} = \sqrt{2} ℓ z_{2}$ by $ℓ^{2} = ℏ / (m ω)$ , while $m ω x_{0} = \sqrt{2} m ω ℓ z_{1} = \frac{\sqrt{2} ℏ z_{1}}{ℓ}$ by the same identity, the displays $x (t) = x_{0} \cos ω t + \frac{p_{0}}{m ω} \sin ω t$ and $p (t) = p_{0} \cos ω t - m ω x_{0} \sin ω t$ follow. Differentiating with the trigonometric derivatives of step 1.3 gives $\dot{x} (t) = - x_{0} ω \sin ω t + \frac{p_{0}}{m} \cos ω t = \frac{p (t)}{m}$ and $\dot{p} (t) = - p_{0} ω \sin ω t - m ω x_{0} ω \cos ω t = - m ω^{2} (x_{0} \cos ω t + \frac{p_{0}}{m ω} \sin ω t) = - m ω^{2} x (t)$ for every $t$ .

4.1step 2.1step 2.3step 2.4step 3.1F8∎

[Clause (e)] Clauses (a), (b), (c) and (d) were obtained from the model's flow and energy observable, the explicit coherent evolution, the coherent fluctuation theorem and the exponential and trigonometric identities; no measurement record, apparatus model, damping or driving and no empirical value entered, and the identification of the mean motion with a laboratory oscillator is a tested identification under P7 as specified in [F8]. This is clause (e).

Evolution of the oscillator coherent states: classical mean motion and stationary fluctuations

Statement

Facts & Assumptions

Proof

Depends on

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