Coherent-State Semiclassics Preview

Coherent-state semiclassics represents quantum evolution with states localized in phase space and then uses classical or near-classical trajectories to organize the answer. Depending on the approximation, one may propagate a single packet center, allow its covariance to evolve, solve a coherent-state boundary-value problem, or integrate over an ensemble of initial packets.

Those procedures are related, but they are not interchangeable. A single coherent state can remain exact under harmonic motion, fail immediately under a nonlinear Hamiltonian, and still serve as one element of a successful many-packet semiclassical representation.

Required background. Coherent states define oscillator packets; coherent-state dynamics gives their exact harmonic evolution; TDVP explains projection onto a chosen state family.

Canonical coherent-state phase space

For one canonical degree of freedom, choose a positive width scale $b$ and define

$$z = \frac{1}{\sqrt{2}} \left( \frac{q}{b} + i\frac{bp}{\hbar} \right).$$

The corresponding normalized oscillator coherent states satisfy

$$\hat a\lvert z\rangle = z\lvert z\rangle$$

and resolve the identity:

$$I = \int_{\mathbb C} \frac{d^2z}{\pi} \lvert z\rangle \langle z\rvert.$$

Their overlap is

$$\langle z'\vert z\rangle = \exp\left[ -\dfrac{\lvert z'\rvert^2}{2} -\dfrac{\lvert z\rvert^2}{2} +z'^*z \right],$$

so

$$\lvert\langle z'\vert z\rangle\rvert^2 = e^{-\lvert z'-z\rvert^2}.$$

The states are localized but nonorthogonal. Their canonical variances are

$$\Delta q = \frac{b}{\sqrt2}, \qquad \Delta p = \frac{\hbar}{\sqrt2\,b},$$

and therefore

$$\Delta q\,\Delta p = \frac{\hbar}{2}.$$

The freedom to choose $b$ is part of the approximation whenever the Hamiltonian does not select a natural oscillator width. A poor width can make a frozen packet inefficient even when a broader Gaussian or many-packet method works well.

The Semiclassical Scaling

In a standard canonical limit, hold the physical phase-space center $(q,p)$ fixed and choose a width proportional to $\sqrt{\hbar}$. For example, with fixed positive reference mass $m_{\mathrm{ref}}$ and frequency $\omega_{\mathrm{ref}}$, take [Baranger et al., §2.1]

$$b = \sqrt{\frac{\hbar}{m_{\mathrm{ref}}\omega_{\mathrm{ref}}}}.$$

Then

$$\Delta q = O(\sqrt{\hbar}), \qquad \Delta p = O(\sqrt{\hbar}),$$

while

$$\lvert z\rvert^2 = O(\hbar^{-1})$$

away from the phase-space origin. The packet occupies a shrinking phase-space neighborhood even though its coherent-state label becomes large. An upper bound $b=O(\sqrt{\hbar})$ alone would not guarantee this: if $b$ shrinks too quickly, $\Delta p=\hbar/(\sqrt2\,b)$ need not shrink at all. The stated scaling controls both quadratures.

At fixed $\hbar$, large occupation can provide a related limit. For a canonical coherent state,

$$\langle N\rangle = \lvert z\rvert^2, \qquad \frac{\Delta N}{\langle N\rangle} = \frac{1}{\sqrt{\langle N\rangle}}.$$

Small relative number fluctuations help explain classical-looking field amplitudes, but they do not imply zero absolute noise, exact orthogonality, or immunity to nonlinear quantum evolution.

Three Different Approximations

The phrase “propagate a coherent state semiclassically” can refer to three distinct levels.

Exact invariant motion

For a Hamiltonian that is at most a phase rotation and displacement in $\hat a$ and $\hat a^\dagger$, an initial coherent state remains in the coherent-state family:

$$e^{-i\hat Ht/\hbar} \lvert z_0\rangle = e^{i\chi(t)} \lvert z(t)\rangle.$$

The packet center follows a phase-space rotation and displacement preserving the chosen covariance, its covariance remains the coherent-state covariance, and only the state-vector phase requires additional bookkeeping. This is exact quantum dynamics, not merely a leading semiclassical estimate.

One projected packet

For a generic Hamiltonian, impose an ansatz

$$\lvert\psi(t)\rangle \approx e^{i\chi(t)} \lvert z(t)\rangle$$

and determine $z(t)$ variationally. This projects the Schrödinger velocity onto the tangent space of the coherent-state manifold. The result can track a packet center while missing squeezing, spreading, skewness, splitting, or interference.

Allowing a time-dependent covariance produces a thawed Gaussian manifold. It is larger than the ordinary coherent-state family and is exact for general quadratic Hamiltonians when the phase and width equations are treated consistently.

A trajectory sum

Instead of claiming that the state remains one packet, represent a propagator or wavefunction as a coherent sum over many trajectory contributions:

$$\mathcal A_{\mathrm{sc}} \sim \sum_\gamma \mathcal C_\gamma^{1/2} e^{iS_\gamma/\hbar+i\Delta_\gamma}.$$

The labels $\gamma$ may denote coherent-state boundary-value saddles or trajectories launched from initial phase-space points. The prefactor $\mathcal C_\gamma$ carries stability information, and $\Delta_\gamma$ includes convention-dependent phase corrections.

This level can represent deformation and interference even when every basis packet is frozen, because the total state is a superposition of many packets.

Variational Phase-Space Dynamics

For normalized canonical coherent states, define the covariant or $Q$ symbol

$$H_Q(z^*,z) = \langle z\vert \hat H \vert z\rangle.$$

Restricting the quantum action to the coherent-state manifold gives

$$S_{\mathrm{var}} ={} \int_{t_i}^{t_f}dt \Bigg[ \frac{i\hbar}{2} \left( z^*\dot z - \dot z^*z \right) - H_Q(z^*,z) \Bigg].$$

Stationary variation yields

$$i\hbar\dot z = \frac{\partial H_Q}{\partial z^*},$$

and its conjugate equation when the trajectory remains on the real phase-space section. In $(q,p)$ coordinates these are Hamilton equations generated by the restricted energy $H_Q(q,p)$.

This classical-looking form does not make the approximation exact. It only states that the chosen variational manifold inherits a symplectic structure.

Which Hamiltonian function appears?

For

$$\hat H = \frac{\hat p^2}{2m} + V(\hat q),$$

a smooth-potential expansion around the coherent-state center gives

$$H_Q(q,p) ={} \frac{p^2}{2m} + \frac{\hbar^2}{4mb^2} + V(q) + \frac{b^2}{4}V''(q) + O\left( b^4V^{(4)} \right).$$

With $b^2$ proportional to $\hbar$ as above, the difference between $H_Q$ and the point-particle Hamiltonian begins at order $\hbar$. Such corrections may be small in the trajectory equations yet still contribute an order-one phase after division by $\hbar$.

Different coherent-state symbols, including normal, antinormal, and Weyl symbols, differ by terms of this type. A calculation must state which symbol and time-slicing convention it uses.

Residual as the decisive test

For an optimally phased variational state, define

$$\lvert R(t)\rangle = \left( i\hbar\frac{d}{dt} - \hat H \right) \lvert\psi_{\mathrm{var}}(t)\rangle.$$

If $R=0$, the variational manifold is invariant along the trajectory and the motion is exact. If $R\ne0$, the discarded component drives the state out of the manifold. The integrated scale

$$\frac{1}{\hbar} \int_0^T \lVert R(t)\rVert\,dt$$

is an a posteriori diagnostic for accumulated state error. For a state in the operator domain and a well-defined unitary evolution with the same initial state, this integral bounds the Hilbert-space norm error. TDVP states the regularity and phase assumptions.

Frozen and Thawed Gaussians

A frozen Gaussian keeps its covariance fixed while its center and phase evolve. An ordinary canonical coherent state is a frozen Gaussian.

A thawed Gaussian also evolves a complex width or covariance matrix. Linearized classical stability then controls squeezing, rotation, and spreading. For a quadratic Hamiltonian, the thawed Gaussian family is invariant and reproduces exact Gaussian evolution.

For a nonlinear potential, expand around the packet center:

$$V(q+\xi) ={} V(q) + V'(q)\xi + \frac12V''(q)\xi^2 + \frac{1}{6}V'''(q)\xi^3 + \cdots.$$

The constant, linear, and quadratic terms preserve Gaussianity. Cubic and higher terms generate non-Gaussian structure. A thawed packet can delay failure by following the local curvature, but it cannot represent arbitrary skewness, splitting, or interference with one Gaussian.

This distinction prevents a common confusion:

• one frozen Gaussian cannot spread;
• a superposition or integral of many frozen Gaussians can spread and deform through changing centers, prefactors, and interference.

Coherent-State Propagator

The canonical coherent-state propagator is

$$K(z_f^*,z_i;T) = \langle z_f\vert e^{-i\hat HT/\hbar} \vert z_i\rangle.$$

The notation emphasizes that a Bargmann-space propagator is holomorphic in the initial label $z_i$ and antiholomorphic in the final label $z_f^*$ after the standard normalization factors are separated.

Repeated insertion of the coherent-state resolution of identity gives a time-sliced phase-space path integral. A continuum action is often written schematically as

$$\mathcal S[v,u] ={} \int_0^Tdt \Bigg[ \frac{i\hbar}{2} \left( v\dot u - \dot v u \right) - H_{\mathrm{symb}}(v,u) \Bigg] + \mathcal B,$$

where $\mathcal B$ is an endpoint term fixed by the discretization and normalization convention.

Mixed boundary data

The saddle equations are first order:

$$i\hbar\dot u = \frac{\partial H_{\mathrm{symb}}}{\partial v}, \qquad - i\hbar\dot v = \frac{\partial H_{\mathrm{symb}}}{\partial u}.$$

For the propagator above, the natural boundary data are mixed:

$$u(0)=z_i, \qquad v(T)=z_f^*.$$

One does not generally fix both $u$ and $v$ at both endpoints. Along a complex saddle, $u(t)$ and $v(t)$ are independent complex variables; $v(t)$ need not equal $u(t)^*$ between the endpoints.

This is one reason coherent-state semiclassical propagation is not merely “follow the real classical orbit from $z_i$ to $z_f$.”

Saddle sum and stability

At leading semiclassical order,

$$K_{\mathrm{sc}} \sim \sum_\gamma \mathcal C_\gamma^{1/2} \exp\left[ \frac{i}{\hbar} \mathcal S_\gamma + i\Delta_\gamma \right].$$

The sum runs over relevant saddles satisfying the mixed endpoint data. The factor $\mathcal C_\gamma$ is built from the linearized map between endpoint variations, in close analogy with the Van Vleck Determinant.

The correction $\Delta_\gamma$ depends on the coherent-state family, Hamiltonian symbol, and discretization. In common spin and bosonic conventions it includes a Solari–Kochetov-type phase. Because an order-$\hbar$ shift in the action produces an order-one shift in $\mathcal S/\hbar$, this correction matters for phase-accurate interference.

Phase-space caustics

The coherent-state prefactor becomes singular when the relevant endpoint map loses local invertibility. This is a phase-space caustic, not a divergence of the exact propagator.

Near such a point:

• two or more saddles may coalesce;
• isolated Gaussian saddle terms cease to be uniform;
• a representation change or uniform approximation is required;
• branch and phase bookkeeping becomes essential.

The logic is the same as at WKB turning points and configuration-space caustics, although the singular map and canonical integral depend on the representation.

Initial-Value Representations

Boundary-value coherent-state saddles can require a difficult search for complex trajectories. An initial-value representation, or IVR, instead integrates over real initial phase-space points and propagates each point forward.

A frozen-Gaussian IVR has the schematic operator form

$$\hat U_{\mathrm{IVR}}(t) \sim{} \int \frac{d^nq_0\,d^np_0} {(2\pi\hbar)^n} \, C_t(q_0,p_0) \times e^{iS_t(q_0,p_0)/\hbar} \lvert g_{q_t,p_t}\rangle \langle g_{q_0,p_0}\rvert.$$

Here $(q_t,p_t)$ is the classical trajectory launched from $(q_0,p_0)$, $S_t$ is its action, and $C_t$ depends on the stability matrix and packet width. The Herman–Kluk propagator is the best-known example.

An IVR replaces trajectory root searches with an oscillatory phase-space integral. That trade is useful but not free:

• the stability prefactor may grow or fluctuate strongly;
• destructive interference causes a sampling or sign problem;
• chaotic dynamics can make long-time convergence difficult;
• width choices affect numerical efficiency;
• approximate propagation need not be exactly unitary;
• tunneling may require complexification or additional treatment.

The IVR is a trajectory-sum approximation. It should not be confused with a single frozen packet following one orbit.

Worked Contrast: Harmonic and Kerr Evolution

For the harmonic oscillator,

$$\hat H_0 = \hbar\omega \left( \hat N+\frac12 \right),$$

the number-state phases are linear in $n$. The coherent-state expansion therefore reorganizes into another coherent state:

$$e^{-i\hat H_0t/\hbar} \lvert z_0\rangle = e^{-i\omega t/2} \lvert z_0e^{-i\omega t}\rangle.$$

Now add a Kerr-type nonlinearity:

$$\hat H_K = \hbar\omega\hat N + \frac{\hbar\chi}{2} \hat N(\hat N-1).$$

Exact evolution gives

$$\lvert\psi(t)\rangle = e^{-\lvert z_0\rvert^2/2} \sum_{n=0}^{\infty} c_n(t) \lvert n\rangle.$$

Here

$$c_n(t) = \frac{z_0^n}{\sqrt{n!}} e^{-it\Omega_n},$$

with

$$\Omega_n = \omega n + \frac{\chi}{2}n(n-1).$$

The phase is quadratic in $n$, so the coefficients cannot generally be written as one global phase times $z(t)^n$. The state shears, collapses, revives, and can form nonclassical superpositions.

A coherent-state variational trajectory may still approximate the center for a limited time. It does not reproduce the full state once the number-dependent phase varies appreciably across the occupied range

$$\Delta n \sim \lvert z_0\rvert.$$

In fact, with $\bar n=|z_0|^2$,

$$\langle a(t)\rangle=z_0e^{-i\omega t} \exp\!\left[\bar n(e^{-i\chi t}-1)\right].$$

Its magnitude initially decays as $|z_0|e^{-\bar n\chi^2t^2/2}$: the centroid collapses on scale $1/(|\chi|\sqrt{\bar n})$. The one-packet variational equation instead gives $z(t)=z_0e^{-i(\omega+\chi\bar n)t}$ with constant magnitude. State deformation can matter even before the centroid has noticeably collapsed.

Spin and Generalized Coherent States

Coherent-state semiclassics is not confined to the oscillator plane.

Family	Classical phase space	Semiclassical scale	Background
Canonical oscillator states	plane	action over $\hbar$ or large occupation	Coherent States
Spin coherent states	sphere	large-$s$ limit	Spin Coherent States
Bosonic multimode states	product of mode planes	large mode occupations or field action	Harmonic Oscillator to Fields

Use the north-patch aligned spin state $|\hat{\mathbf n};s\rangle=e^{is\phi}e^{-i\phi S_z/\hbar}e^{-i\theta S_y/\hbar}|s,s\rangle$. With amplitude convention $e^{iS/\hbar}$, $i\hbar\langle\hat{\mathbf n};s|\partial_t|\hat{\mathbf n};s\rangle=-\hbar s(1-\cos\theta)\dot\phi$. Thus

$$S[\theta,\phi] = \int dt \Big[ -\hbar s (1-\cos\theta)\dot\phi - H_Q(\theta,\phi) \Big].$$

The sphere, its symplectic area, and the large-$s$ fluctuation scaling replace the oscillator plane and large occupation. The local Berry connection depends on a gauge patch; in this aligned-state convention the geometric phase is $-s\Omega$ for oriented solid angle $\Omega$, modulo $2\pi$. A gauge change cannot reverse the curvature.

Generalized coherent states share localization, overcompleteness, and group-orbit geometry, but their measures, symbols, curvatures, and semiclassical corrections are family-specific.

Quantum Optics and QFT Boundary

For retained bosonic modes,

$$\hat a_{\mathbf k} \lvert\{\alpha\}\rangle = \alpha_{\mathbf k} \lvert\{\alpha\}\rangle.$$

Under a free or linearly driven field Hamiltonian, the mode amplitudes obey the corresponding classical linear equations, and a product coherent state remains coherent. This underlies the use of coherent states for idealized laser fields and classical radiation.

Several boundaries must remain explicit:

• a coherent state retains vacuum and shot-noise fluctuations;
• large occupation suppresses relative fluctuations, not all quantum effects;
• interactions can squeeze modes, entangle them, and generate non-Gaussian correlations;
• a coherent-state path-integral variable is an integration label, not proof that the exact state stays coherent;
• continuum QFT requires regularization and renormalization;
• gauge fields require constraints, gauge fixing, or a physical-mode construction;
• inequivalent representations and infrared issues can arise with infinitely many modes.

The finite-mode oscillator picture is therefore a bridge, not a substitute for field-theoretic analysis. See Path Integrals and Harmonic Oscillator to Fields.

Exercises

1. Phase-space distance and overlap

Using

$$z = \frac{1}{\sqrt2} \left( \frac{q}{b} + i\frac{bp}{\hbar} \right),$$

express

$$\lvert\langle z'\vert z\rangle\rvert^2$$

in terms of $\Delta q=q'-q$ and $\Delta p=p'-p$.

Solution

The label separation is

$$z'-z = \frac{1}{\sqrt2} \left( \frac{\Delta q}{b} + i\frac{b\Delta p}{\hbar} \right).$$

Therefore

$$\lvert z'-z\rvert^2 = \frac{(\Delta q)^2}{2b^2} + \frac{b^2(\Delta p)^2}{2\hbar^2}.$$

Using the canonical overlap,

$$\boxed{ \lvert\langle z'\vert z\rangle\rvert^2 = \exp\left[ - \frac{(\Delta q)^2}{2b^2} - \frac{b^2(\Delta p)^2}{2\hbar^2} \right] }.$$

The overlap falls when the centers are separated by more than the packet resolution in either quadrature. It never vanishes exactly at finite separation.

2. Why the endpoint data are mixed

Vary

$$\mathcal S[v,u] ={} \int_0^Tdt \frac{i\hbar}{2} \left( v\dot u-\dot v u \right) - \int_0^Tdt\, H(v,u) + \mathcal B$$

and explain why a coherent-state propagator naturally fixes $u(0)$ and $v(T)$ rather than both variables at both times.

Solution

Integrating the kinetic variation by parts gives a bulk term

$$\delta\mathcal S_{\mathrm{bulk}} ={} \int_0^Tdt \Bigg[ \left( i\hbar\dot u - \frac{\partial H}{\partial v} \right)\delta v + \left( - i\hbar\dot v - \frac{\partial H}{\partial u} \right)\delta u \Bigg],$$

plus endpoint terms. The convention-dependent $\mathcal B$ is chosen so that the remaining endpoint variation vanishes when

$$\delta u(0)=0, \qquad \delta v(T)=0.$$

The bulk equations are first order in time. Fixing $u(0)$ and $v(T)$ supplies the required mixed data. Fixing both $u$ and $v$ at both endpoints would generally overdetermine the saddle equations.

3. Kerr evolution leaves the coherent manifold

For the Kerr Hamiltonian in the worked contrast and an initial label $z_0\ne0$, show that the evolved state cannot generally equal

$$e^{i\chi(t)} \lvert z(t)\rangle.$$

Solution

If the state remained coherent, the ratio of adjacent number-basis coefficients would be

$$\frac{c_{n+1}(t)}{c_n(t)} = \frac{z(t)}{\sqrt{n+1}},$$

apart from one global phase. Exact Kerr evolution instead gives

$$\frac{c_{n+1}(t)}{c_n(t)} ={} \frac{z_0}{\sqrt{n+1}} \times \exp\left[ - it \left( \omega+\chi n \right) \right].$$

The extra factor depends on $n$ when $\chi t$ is not an integer multiple of $2\pi$. No single label $z(t)$ can reproduce all adjacent ratios. The state has left the ordinary coherent-state manifold.

References

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• R. J. Glauber, “Coherent and incoherent states of the radiation field”, Physical Review 131, 2766–2788 (1963). Canonical field-mode coherent states and optical coherence.
• G. A. Hagedorn, “Semiclassical quantum mechanics. I. The $\hbar\to0$ limit for coherent states”, Communications in Mathematical Physics 71, 77–93 (1980). Controlled coherent-state propagation in the semiclassical limit.
• E. J. Heller, “Time-dependent approach to semiclassical dynamics”, Journal of Chemical Physics 62, 1544–1555 (1975). Gaussian wave-packet dynamics and local quadratic propagation.
• M. F. Herman and E. Kluk, “A semiclasical justification for the use of non-spreading wavepackets in dynamics calculations”, Chemical Physics 91, 27–34 (1984). Frozen-Gaussian initial-value representation.
• J. R. Klauder and B.-S. Skagerstam, eds., Coherent States: Applications in Physics and Mathematical Physics (World Scientific, 1985). Generalized coherent states and path-integral methods.
• E. A. Kochetov, “SU(2) coherent-state path integral for the Heisenberg ferromagnet”, Physical Review B 52, 4402–4408 (1995). Spin coherent-state semiclassics and extra-phase structure.
• A. Perelomov, Generalized Coherent States and Their Applications (Springer, 1986). Group-orbit coherent states and geometry.
• H. G. Solari, “Semiclassical treatment of spin system by means of coherent states”, Journal of Mathematical Physics 28, 1097–1102 (1987). Discretization-sensitive phase correction.
• M. Stone, K.-S. Park, and A. Garg, “The semiclassical propagator for spin coherent states”, Journal of Mathematical Physics 41, 8025–8049 (2000). Fluctuation determinants and the Solari–Kochetov correction.