Harmonic-Oscillator Propagator Notebook

This notebook guide tests the harmonic-oscillator propagator in the representation from which it is most naturally reconstructed: the Hermite energy basis. It compares a regulated spectral sum with the exact Mehler kernel, evolves a coherent state by number-state phases, and visualizes the same motion as a rigidly rotating Wigner Gaussian.

The closed-form derivation, classical action, and caustic interpretation belong to Harmonic-Oscillator Propagator. The Hilbert-space construction of coherent states belongs to Coherent States, while Coherent States in Phase Space owns their Wigner geometry. This page is the numerical bridge among those canonical results.

The existing source notebook notebooks/wave-mechanics-canonical-systems/harmonic-oscillator/harmonic-oscillator-eigenstates.ipynb provides a basis-generation benchmark. A Dynamics-specific notebook should reuse its eigenstate checks and add spectral kernels, branch tracking, coherent evolution, and phase-space validation.

Purpose

The notebook should demonstrate that:

• a truncated Hermite sum converges to a regulated exact kernel;
• the real-time kernel is better tested through its action on smooth states than by an unregulated pointwise partial sum;
• coherent states remain Gaussian and follow the classical oscillator orbit;
• zero-point and caustic phases are visible in complex amplitudes even when densities look correct;
• the finite spectral propagator composes exactly within its retained subspace;
• coordinate-grid, basis-cutoff, regulator, and caustic errors require different convergence tests.

The central rule is to compare like with like. A damped spectral sum must be compared with a complex-time kernel, and a finite number-state sum represents a projector onto a finite oscillator subspace rather than the identity on every grid vector.

Physics Goal

Use

$$H = \frac{p^2}{2m} + \frac12m\omega^2x^2,$$

with oscillator length

$$\ell = \sqrt{\frac{\hbar}{m\omega}}.$$

The exact real-time kernel away from caustics is

$$\begin{aligned} K_{\rm ho}(x_f,T;x_i,0) &= \left[ \frac{m\omega} {2\pi i\hbar\sin\omega T} \right]^{1/2} \\ &\quad\times \exp\left\{ \frac{im\omega} {2\hbar\sin\omega T} \left[ (x_f^2+x_i^2)\cos\omega T -2x_fx_i \right] \right\}. \end{aligned}$$

The same operator has the spectral representation

$$K_{\rm ho}(x_f,T;x_i,0) = \sum_{n=0}^{\infty} \psi_n(x_f)\psi_n^*(x_i) e^{-i\omega T(n+1/2)}.$$

The notebook should establish the equivalence without pretending that an oscillatory real-time completeness series must converge pointwise like an ordinary function series.

Stable Hermite Basis

With $\xi=x/\ell$, write

$$\psi_n(x) = \ell^{-1/2}\varphi_n(\xi),$$

where the normalized dimensionless Hermite functions begin with

$$\varphi_0(\xi) = \pi^{-1/4}e^{-\xi^2/2},$$

and

$$\varphi_1(\xi) = \sqrt2\,\xi\varphi_0(\xi).$$

Generate higher functions by the normalized recurrence

$$\begin{aligned} \varphi_{n+1}(\xi) &= \sqrt{\frac{2}{n+1}}\, \xi\varphi_n(\xi) \\ &\quad- \sqrt{\frac{n}{n+1}}\, \varphi_{n-1}(\xi). \end{aligned}$$

This is preferable to separately evaluating $H_n(\xi)$, $2^n$, and $n!$. Those factors can overflow or underflow even when their normalized product is moderate. A trusted special-function routine is a useful independent comparison, but it does not remove the need to check whether the physicists’ Hermite convention and oscillator scaling match the analytic formulas.

On a coordinate grid, form the overlap matrix

$$S_{mn} = \Delta x \sum_j \psi_m^*(x_j)\psi_n(x_j).$$

For the subset used in the calculation,

$$S_{mn} \approx \delta_{mn}.$$

Loss of orthonormality at high $n$ usually means the box does not include the classical turning region, the grid is too coarse to resolve the nodes, or the recurrence has become numerically unstable.

Finite Spectral Kernel

Retain the states $n=0,\ldots,M-1$ and define

$$K_M(x_f,T;x_i,0) = \sum_{n=0}^{M-1} \psi_n(x_f)\psi_n^*(x_i) e^{-i\omega T(n+1/2)}.$$

At $T=0$ this is

$$P_M(x_f,x_i) = \sum_{n=0}^{M-1} \psi_n(x_f)\psi_n^*(x_i),$$

the kernel of the finite-rank projector $P_M$. It is not a delta function. Narrower projector peaks and stronger truncation oscillations appear as $M$ increases, so a pointwise value at one coordinate is not a useful completeness test by itself.

Within the retained basis,

$$U_M(T) = \sum_{n=0}^{M-1} e^{-i\omega T(n+1/2)} \lvert n\rangle\langle n\rvert$$

obeys

$$U_M^\dagger(T)U_M(T)=P_M$$

and

$$U_M(T_2)U_M(T_1) = U_M(T_1+T_2).$$

Thus the spectral phases are exactly unitary on the retained subspace. Coordinate quadrature may spoil these identities slightly if the sampled basis is not orthonormal.

Why the Kernel Sum Needs a Regulator

For real $T$, the spectral terms have unit-modulus time phases. The exact kernel is an oscillatory distribution at caustics and has constant magnitude in the final coordinate away from them. An unregulated partial sum can therefore show persistent ringing without ordinary pointwise convergence.

Introduce a positive imaginary-time displacement

$$T_\eta=T-i\eta, \qquad \eta\gt0.$$

Then

$$\begin{aligned} e^{-iE_nT_\eta/\hbar} &= e^{-i\omega T(n+1/2)} e^{-\omega\eta(n+1/2)}. \end{aligned}$$

The regulated finite sum is

$$\begin{aligned} K_{M,\eta}(x_f,T;x_i,0) &= \sum_{n=0}^{M-1} \psi_n(x_f)\psi_n^*(x_i) \\ &\quad\times e^{-i\omega T(n+1/2)} e^{-\omega\eta(n+1/2)}. \end{aligned}$$

For fixed $\eta\gt0$, this converges to the complex-time exact kernel

$$K_\eta(x_f,T;x_i,0) = K_{\rm ho}(x_f,T-i\eta;x_i,0),$$

with the sine, cosine, and square-root branch all evaluated by analytic continuation from $\eta\gt0$. This is the numerical form of Abel summation and the Mehler formula.

The correct order of tests is:

1. hold $\eta$ fixed and increase $M$ until the regulated sum converges;
2. repeat at smaller $\eta$ with a larger basis;
3. infer the real-time limit only after convergence at each regulator.

Reducing $\eta$ at fixed $M$ removes the damping that made the sum numerically controlled and can make the result worse.

Exact-Kernel Comparison

Choose a fixed source point $x_i$ and evaluate both $K_{M,\eta}$ and $K_\eta$ on an output interval. Compare their complex values, not only their magnitudes. Useful errors are

$$\epsilon_{K,2} = \left[ \Delta x_f \sum_j \left\lvert K_{M,\eta}(x_{f,j},T;x_i,0) - K_\eta(x_{f,j},T;x_i,0) \right\rvert^2 \right]^{1/2},$$

and

$$\epsilon_{K,\infty} = \max_j \left\lvert K_{M,\eta} - K_\eta \right\rvert.$$

Also plot:

• real and imaginary parts of both kernels;
• absolute error;
• wrapped phase error where $\lvert K_\eta\rvert$ is above a stated threshold;
• error versus $M$ for several fixed values of $\omega\eta$.

Do not divide by $K_\eta$ where it is extremely small. A relative phase or relative-amplitude error becomes meaningless near zeros of the regulated kernel.

Coherent-State Benchmark

Let the initial state be

$$\lvert\alpha_0\rangle = e^{-\lvert\alpha_0\rvert^2/2} \sum_{n=0}^{\infty} \frac{\alpha_0^n}{\sqrt{n!}} \lvert n\rangle.$$

Its coordinate wavefunction is

$$\begin{aligned} \langle x\vert\alpha\rangle &= \frac{1}{(\pi\ell^2)^{1/4}} \exp\left[ - \frac{x^2}{2\ell^2} + \frac{\sqrt2\,\alpha x}{\ell} \right. \\ &\qquad\left. - \frac{\alpha^2}{2} - \frac{\lvert\alpha\rvert^2}{2} \right]. \end{aligned}$$

Exact time evolution gives

$$U(T)\lvert\alpha_0\rangle = e^{-i\omega T/2} \lvert\alpha_0e^{-i\omega T}\rangle.$$

The finite spectral reconstruction is

$$\begin{aligned} \psi_M(x,T) &= e^{-\lvert\alpha_0\rvert^2/2} \sum_{n=0}^{M-1} \frac{\alpha_0^n}{\sqrt{n!}} \\ &\quad\times e^{-i\omega T(n+1/2)} \psi_n(x). \end{aligned}$$

The omitted norm is known before any grid calculation:

$$\begin{aligned} P_{\rm tail}(M) &= 1- e^{-\lvert\alpha_0\rvert^2} \sum_{n=0}^{M-1} \frac{\lvert\alpha_0\rvert^{2n}}{n!}. \end{aligned}$$

Choose $M$ so that this Poisson tail is below the target state error. Do not renormalize the truncated sum before reporting its norm; doing so conceals the omitted component.

Compare $\psi_M(x,T)$ with

$$\psi_{\rm coh}(x,T) = e^{-i\omega T/2} \langle x\vert \alpha_0e^{-i\omega T} \rangle.$$

This comparison checks the Hermite basis, number-state phases, zero-point phase, spatial grid, and coherent-state convention at once.

Classical Orbit and Fixed Width

Define

$$\alpha(T) = \alpha_0e^{-i\omega T}.$$

The exact center is

$$x_c(T) = \sqrt2\,\ell\, \operatorname{Re}\alpha(T),$$

and

$$p_c(T) = \frac{\sqrt2\hbar}{\ell} \operatorname{Im}\alpha(T).$$

Equivalently,

$$\begin{aligned} x_c(T) &= x_c(0)\cos\omega T + \frac{p_c(0)}{m\omega} \sin\omega T, \\ p_c(T) &= p_c(0)\cos\omega T - m\omega x_c(0) \sin\omega T. \end{aligned}$$

The variances stay fixed:

$$\Delta x^2 = \frac{\ell^2}{2}, \qquad \Delta p^2 = \frac{\hbar^2}{2\ell^2}.$$

Measure these moments from the reconstructed state at several times. A correct density can still accompany a wrong global phase, so moment checks supplement rather than replace the complex wavefunction error.

Phase-Space Visualization

Use dimensionless oscillator coordinates

$$Q=\frac{x}{\ell}, \qquad P=\frac{\ell p}{\hbar}.$$

The coherent-state center is

$$Q_c(T) = \sqrt2\,\operatorname{Re}\alpha(T), \qquad P_c(T) = \sqrt2\,\operatorname{Im}\alpha(T).$$

The dimensionless Wigner density is

$$\widetilde W_\alpha(Q,P;T) = \frac{1}{\pi} \exp\left[ - (Q-Q_c(T))^2 - (P-P_c(T))^2 \right].$$

Plot equal-aspect contours at several fractions of a period and overlay the classical orbit

$$Q_c^2+P_c^2 = 2\lvert\alpha_0\rvert^2.$$

The Gaussian should translate rigidly around the orbit without changing shape or area. Verify numerically that

$$\int dQ\,dP\, \widetilde W_\alpha(Q,P;T) = 1$$

and that its first moments agree with $(Q_c,P_c)$. Numerical Wigner transforms and negativity tests continue in the Wigner Function Notebook; this calculation uses the analytic coherent-state Wigner function only as an independent visualization.

Caustics and Global Phase

The closed kernel formula is singular when

$$\sin\omega T=0.$$

At

$$T_n=\frac{n\pi}{\omega},$$

its operator limit is

$$K_{\rm ho}(x_f,T_n;x_i,0) = e^{-in\pi/2} \delta\left( x_f-(-1)^n x_i \right).$$

Two especially useful state-level checks are

$$U\left(\frac{\pi}{\omega}\right) \psi(x) = -i\,\psi(-x),$$

and

$$U\left(\frac{2\pi}{\omega}\right) \psi(x) = -\psi(x).$$

The probability density returns after a full period, but the state acquires a minus sign. A test based only on $\lvert\psi\rvert^2$ misses this zero-point phase.

Do not evaluate the closed kernel formula exactly at a caustic. Compare the spectral action with the parity or identity limit, and approach the caustic from either side with a small positive $\eta$. Independently taking the principal square root at every real time can introduce discontinuous sign errors; the branch must be followed continuously or anchored to the spectral phases.

Baseline Parameters

A useful dimensionless baseline is

$$\hbar=m=\omega=1, \qquad \ell=1,$$

with

$$\alpha_0=1.5+0.5i, \qquad T=0.7.$$

For the regulated kernel comparison, begin with

$$\omega\eta=0.12, \qquad M=80,$$

and use a source point such as

$$x_i=-0.8.$$

A coordinate interval of approximately

$$-14\ell\le x\le14\ell$$

with several thousand points is a reasonable first run for the retained basis, but it is not a guaranteed universal choice. The largest retained Hermite functions extend toward turning points of order

$$\lvert x\rvert \sim \ell\sqrt{2M}.$$

The box and grid must therefore grow with the basis cutoff.

Notebook Workflow

Organize the notebook into these reproducible stages:

1. define physical scales, coordinate grid, basis cutoff, time, and regulator;
2. generate normalized Hermite functions by recurrence;
3. validate parity, normalization, and the overlap matrix;
4. form $K_{M,\eta}$ at a fixed source point;
5. compare it with the exact complex-time kernel;
6. run a two-parameter convergence study in $M$ and $\eta$;
7. construct a coherent state from analytic number coefficients;
8. evolve those coefficients and compare with the exact coherent wavefunction;
9. measure norm, center, covariance, fidelity, and complex error;
10. visualize the analytic Wigner Gaussian and classical orbit;
11. test half-period parity and full-period global phase;
12. run assertions before generating presentation plots.

Record the software versions, floating-point dtype, all grid parameters, the Hermite convention, the square-root branch strategy, and every tolerance.

Validation Checks

Use the coordinate-grid inner product

$$\langle f|g\rangle_{\Delta x} = \Delta x\sum_jf_j^*g_j.$$

For normalized analytic and numerical states, report

$$\epsilon_\psi = \lVert \psi_M-\psi_{\rm coh} \rVert_{\Delta x}$$

and

$$F_\psi = \left\lvert \langle \psi_{\rm coh}|\psi_M \rangle_{\Delta x} \right\rvert^2.$$

The minimum validation suite is:

Check	Target
Hermite recurrence	first few functions agree with analytic formulas
Basis overlap	$S_{mn}\approx\delta_{mn}$ over the validated subset
Parity	$\psi_n(-x)=(-1)^n\psi_n(x)$
Regulated kernel	$K_{M,\eta}\to K_\eta$ as $M$ increases at fixed $\eta$
Kernel symmetry	$K(x_f,T;x_i,0)=K(x_i,T;x_f,0)$
Spectral composition	$U_M(T_2)U_M(T_1)=U_M(T_1+T_2)$ within $P_M$
Coherent tail	numerical omitted norm agrees with $P_{\rm tail}(M)$
Coherent evolution	complex state error decreases with cutoff and grid refinement
First moments	$(x_c,p_c)$ follow the classical orbit
Covariance	$\Delta x^2$ and $\Delta p^2$ remain fixed
Wigner checks	normalization, center, and covariance are time independent in the co-moving frame
Half period	evolved state equals $-i$ times the parity-reflected state
Full period	evolved state equals $-\psi(0)$, not merely the same density

The regulated kernel is not unitary because $e^{-\eta H/\hbar}$ damps high energies. Test unitarity only for the real-time spectral phases, not for $K_{M,\eta}$.

Convergence Study

The relevant limits are coupled:

1. Basis cutoff. Increase $M$ at fixed $\eta$, box, and grid.
2. Regulator. Decrease $\eta$ only after the $M$ sequence has converged.
3. Box size. Increase $x_{\max}$ at approximately fixed $\Delta x$ to capture higher Hermite functions.
4. Grid spacing. Refine $\Delta x$ at fixed physical interval to resolve nodes and quadrature.
5. Coherent occupation. Increase $\lvert\alpha_0\rvert$ and choose $M$ from the Poisson tail rather than by habit.
6. Caustic distance. Approach $T_n$ from both sides and track the complex phase.

For each study, report basis-overlap residuals alongside the physical error. A kernel sum cannot be trusted when the numerical basis itself has stopped being orthonormal.

Expected Results

For a controlled calculation:

• the regulated spectral sum converges smoothly to the exact complex-time kernel;
• smaller regulators require larger basis cutoffs;
• unregulated pointwise partial sums exhibit truncation ringing;
• coherent-state spectral evolution agrees with the exact displaced Gaussian;
• the wavepacket center follows the classical ellipse while its covariance stays fixed;
• the Wigner Gaussian moves rigidly with no spreading;
• the half-period map performs parity with phase $-i$;
• the full-period density recurs while the state has phase $-1$;
• kernel formulas evaluated with an untracked principal square root fail across caustics.

These outcomes test distinct structures. No single plot substitutes for the full set.

Common Numerical Failures

• Comparing an unregulated partial sum pointwise with the real-time kernel. Use fixed positive $\eta$ or compare the operator action on a smooth state.
• Treating $P_M(x_f,x_i)$ as a delta function. It is a finite-rank projector kernel.
• Mixing Hermite conventions. The oscillator uses the physicists’ $H_n$, not the probabilists’ convention.
• Evaluating raw factorials and high-degree polynomials. Generate normalized Hermite functions by recurrence or use a validated special-function routine.
• Dropping the factor $\ell^{-1/2}$. Dimensionless Hermite functions need physical oscillator scaling.
• Omitting the zero-point phase. Densities can still look correct while full-period state recurrence is wrong.
• Using a principal square root independently at every time. This loses the continuous Maslov phase across caustics.
• Testing unitarity after imaginary-time damping. The regulator deliberately makes the evolution nonunitary.
• Renormalizing truncated coherent states before measuring tail error. This hides the basis cutoff.
• Keeping the box fixed while raising $M$. Higher basis functions extend farther and oscillate more rapidly.
• Plotting phase-space contours with unequal axis scales. A circular dimensionless Wigner Gaussian can be made to look squeezed.
• Confusing coherent-state recurrence with classicality. The state remains fully quantum and retains irreducible covariance.

Extensions

• Compare uniform-grid overlap integrals with Gauss–Hermite quadrature.
• Reconstruct the finite matrix $U_M(T)$ and inspect its eigenphases.
• Test a squeezed initial state and observe covariance rotation rather than rigid coherent translation.
• Add a linear drive and compare with the exact displaced solution.
• Compare the oscillator kernel with the free kernel in the short-time regime, using the same complex-time regulator.
• Study a weak quartic perturbation, where coherent shape preservation and exact semiclassical propagation both fail.

Cross-Links

• Harmonic-Oscillator Propagator
• Spectral Decomposition of the Propagator
• Composition Law
• Hermite Functions
• Hermite Polynomials
• Coherent States
• Coherent-State Dynamics
• Coherent States in Phase Space
• Wigner Function
• Wigner Function Notebook
• Convergence Tests
• Validation Tests

References

• R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals, McGraw-Hill, 1965.
• L. S. Schulman, Techniques and Applications of Path Integration, Wiley, 1981.
• D. J. Tannor, Introduction to Quantum Mechanics: A Time-Dependent Perspective, University Science Books, 2007.
• NIST Digital Library of Mathematical Functions, Section 18.18(vii), Hermite Poisson kernel and Mehler formula.
• W. P. Schleich, Quantum Optics in Phase Space, Wiley-VCH, 2001.
• SciPy Developers, scipy.special.eval_hermite documentation, consulted for the physicists’ Hermite convention and software cross-check.

Exercises

1. Derive the normalized Hermite recurrence

Starting from

$$H_{n+1}(\xi) = 2\xi H_n(\xi)-2nH_{n-1}(\xi),$$

derive the recurrence used for $\varphi_n(\xi)$.

Solution

Use

$$\varphi_n(\xi) = \frac{ e^{-\xi^2/2}H_n(\xi) }{ \pi^{1/4}\sqrt{2^n n!} }.$$

Divide the polynomial recurrence by

$$\pi^{1/4} \sqrt{2^{n+1}(n+1)!}$$

and include the common Gaussian. The first coefficient becomes

$$\frac{2\sqrt{2^n n!}} {\sqrt{2^{n+1}(n+1)!}} = \sqrt{\frac{2}{n+1}},$$

while the second becomes

$$\frac{2n\sqrt{2^{n-1}(n-1)!}} {\sqrt{2^{n+1}(n+1)!}} = \sqrt{\frac{n}{n+1}}.$$

Therefore

$$\varphi_{n+1} = \sqrt{\frac{2}{n+1}}\, \xi\varphi_n - \sqrt{\frac{n}{n+1}}\, \varphi_{n-1}.$$

2. Relate damping to complex time

Show that multiplying each spectral term by $e^{-\omega\eta(n+1/2)}$ is equivalent to evaluating the propagator at $T-i\eta$.

Solution

Since

$$E_n = \hbar\omega\left(n+\frac12\right),$$

one has

$$\begin{aligned} \exp\left[ - \frac{iE_n(T-i\eta)}{\hbar} \right] &= \exp\left[ - \frac{iE_nT}{\hbar} \right] \exp\left[ - \frac{E_n\eta}{\hbar} \right] \\ &= e^{-i\omega T(n+1/2)} e^{-\omega\eta(n+1/2)}. \end{aligned}$$

The damping is therefore imaginary-time evolution appended to the real-time unitary.

3. Prove finite-subspace composition

Show that $U_M(T_2)U_M(T_1)=U_M(T_1+T_2)$ and explain why $U_M^\dagger U_M=P_M$ rather than the full identity.

Solution

Using orthonormal number states,

$$\begin{aligned} U_M(T_2)U_M(T_1) &= \sum_{m,n=0}^{M-1} e^{-iE_mT_2/\hbar} e^{-iE_nT_1/\hbar} \\ &\quad\times \lvert m\rangle \langle m\vert n\rangle \langle n\vert \\ &= \sum_{n=0}^{M-1} e^{-iE_n(T_1+T_2)/\hbar} \lvert n\rangle\langle n\vert. \end{aligned}$$

This is $U_M(T_1+T_2)$. Similarly,

$$U_M^\dagger U_M = \sum_{n=0}^{M-1} \lvert n\rangle\langle n\vert = P_M.$$

The omitted number states are outside the retained computational subspace, so the operator cannot equal the identity on the full Hilbert space.

4. Quantify coherent-state truncation

Derive the omitted norm of an $M$-state coherent expansion and explain how it should set the basis cutoff.

Solution

The number probability is Poisson:

$$p_n = e^{-\lvert\alpha_0\rvert^2} \frac{\lvert\alpha_0\rvert^{2n}}{n!}.$$

The retained norm is

$$\sum_{n=0}^{M-1}p_n,$$

so the omitted norm is

$$P_{\rm tail}(M) = 1- e^{-\lvert\alpha_0\rvert^2} \sum_{n=0}^{M-1} \frac{\lvert\alpha_0\rvert^{2n}}{n!}.$$

Choose the smallest $M$ for which this tail is below the desired state-norm budget, then verify that coordinate-grid error is smaller still. Raising $M$ beyond what the grid can resolve does not improve the calculation.

5. Recover the half- and full-period maps

Use the number-state phases to derive the state transformations at $T=\pi/\omega$ and $T=2\pi/\omega$.

Solution

At half a period,

$$\begin{aligned} e^{-i\omega T(n+1/2)} &= e^{-i\pi(n+1/2)} \\ &= -i(-1)^n. \end{aligned}$$

Because the parity operator has eigenvalue $(-1)^n$ on $\lvert n\rangle$,

$$U(\pi/\omega) = -i\,\Pi.$$

In position space this gives

$$\psi(x,\pi/\omega) = -i\,\psi(-x,0).$$

At a full period,

$$e^{-i2\pi(n+1/2)} = -1$$

for every $n$, so

$$U(2\pi/\omega)=-I.$$

The density recurs exactly, while the state carries a global minus sign.

6. Verify the Wigner covariance

For the dimensionless coherent-state Wigner density, compute the variances of $Q$ and $P$ and show that they do not depend on the center.

Solution

The density factorizes:

$$\widetilde W_\alpha(Q,P) = \frac{e^{-(Q-Q_c)^2}}{\sqrt\pi} \frac{e^{-(P-P_c)^2}}{\sqrt\pi}.$$

Each factor is a normalized Gaussian with variance $1/2$. Therefore

$$\langle(Q-Q_c)^2\rangle = \frac12, \qquad \langle(P-P_c)^2\rangle = \frac12,$$

and the cross covariance vanishes. Time evolution changes $(Q_c,P_c)$ but not these centered moments.