Nonrelativistic Propagator Table

This table collects common real-time propagator kernels for nonrelativistic quantum mechanics. Each kernel is a coordinate-space transition amplitude for the time-evolution operator:

$$K(q_f,t_f;q_i,t_i) = \langle q_f\rvert U(t_f,t_i)\lvert q_i\rangle.$$

It is an amplitude density with respect to the specified coordinate measure. It must act on a normalizable initial wavefunction before position probabilities are formed. The formulas below use

$$T=t_f-t_i.$$

Unless stated otherwise, take $T>0$ and fix Fresnel phases by the forward free-particle limit. For the semibounded free and oscillator Hamiltonians, $T\to T-i0^+$ provides spectral damping. The constant-force Hamiltonian is unbounded below, so that notation is not a bounded imaginary-time operator prescription for it. For other configuration spaces, the integration measure and boundary conditions are part of the kernel.

Common Propagator Kernels

Kernel	Configuration space	Best use	Canonical page
Free particle	line	one-dimensional derivation, wave-packet spreading	Free-Particle Propagator
Free particle	$\mathbb R^d$	product structure, multidimensional short-time kernel	Propagators in Multiple Dimensions
Harmonic oscillator	line	quadratic dynamics, coherent states, Gaussian path integrals	Harmonic-Oscillator Propagator
Infinite square well	interval $0\lt x\lt L$	hard-wall spectral expansion	Infinite Square Well
Particle on a ring	circle $S^1$	winding sectors, angular motion	Particle on a Ring
Constant force	line	uniform-field motion, Airy-energy contrast	this table
Semiclassical Van Vleck	general smooth configuration space	large-action approximation	Semiclassical Propagator

General Identities

The kernel evolves wavefunctions by

$$\psi(q_f,t_f) = \int dq_i\, K(q_f,t_f;q_i,t_i)\psi(q_i,t_i),$$

where $dq_i$ denotes the correct configuration-space measure.

The equal-time limit is the identity kernel:

$$K(q_f,t_i;q_i,t_i) = \delta(q_f-q_i),$$

with the delta distribution understood relative to the same measure.

The composition law is

$$K(q_f,t_f;q_i,t_i) = \int dq\, K(q_f,t_f;q,t)K(q,t;q_i,t_i).$$

For a time-independent Hamiltonian with a complete orthonormal energy basis, the spectral form is

$$K(q_f,T;q_i,0) = \sum_n \psi_n(q_f)\psi_n^*(q_i) e^{-iE_nT/\hbar},$$

with integrals added for continuous spectra.

One-Dimensional Free Particle

For

$$H=\frac{p^2}{2m}$$

on the line,

$$K_0(x_f,T;x_i,0) = \left( \frac{m}{2\pi i\hbar T} \right)^{1/2} \exp\left[ \frac{im(x_f-x_i)^2}{2\hbar T} \right].$$

Use this kernel on the full line. On an interval, half-line, ring, or box, the same differential expression requires different boundary conditions and therefore a different kernel; see Propagators and Boundary Conditions.

D-Dimensional Free Particle

For a free particle on $\mathbb R^d$,

$$K_0^{(d)}(\mathbf r_f,T;\mathbf r_i,0) = \left( \frac{m}{2\pi i\hbar T} \right)^{d/2} \exp\left[ \frac{im\lvert\mathbf r_f-\mathbf r_i\rvert^2} {2\hbar T} \right].$$

The kernel factorizes into one-dimensional free kernels in Cartesian coordinates. Propagators in Multiple Dimensions derives the formula and develops its configuration-space measure and product structure.

Constant Force

Let

$$H = \frac{p^2}{2m}-Fx,$$

so the classical force is $+F$. The exact real-time kernel on the line is

$$K_F(x_f,T;x_i,0) = \left( \frac{m}{2\pi i\hbar T} \right)^{1/2} \exp\left[ \frac{i}{\hbar}S_F \right],$$

where

$$S_F = \frac{m(x_f-x_i)^2}{2T} + \frac{FT}{2}(x_f+x_i) - \frac{F^2T^3}{24m}.$$

Setting $F=0$ recovers the free-particle kernel. The energy eigenfunctions of a linear potential are Airy functions, but the time-domain propagator remains a simple classical-action Gaussian because the Lagrangian is quadratic plus a linear term.

Infinite Square Well

For a particle on $0\lt x\lt L$ with hard-wall boundary conditions,

$$\psi(0)=\psi(L)=0,$$

the normalized eigenfunctions are

$$\psi_n(x) = \sqrt{\frac{2}{L}} \sin\frac{n\pi x}{L}, \qquad n=1,2,\ldots,$$

and

$$E_n = \frac{n^2\pi^2\hbar^2}{2mL^2}.$$

The kernel is the spectral sum

$$K_{\rm box}(x_f,T;x_i,0) = \sum_{n=1}^{\infty} \frac{2}{L} \sin\frac{n\pi x_f}{L} \sin\frac{n\pi x_i}{L} \exp\left[ -\frac{iE_nT}{\hbar} \right].$$

This is not the free-particle kernel restricted to $0\lt x\lt L$. The boundary conditions are part of the Hamiltonian domain and change the kernel.

Harmonic Oscillator

For

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

away from caustic times $\sin\omega T=0$,

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

At

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

the distributional limit is $e^{-in\pi/2}\delta(x_f-(-1)^n x_i)$. For noncaustic $T>0$, the prefactor is $\sqrt{m\omega/(2\pi\hbar|\sin\omega T|)}e^{-i\pi/4-i\lfloor\omega T/\pi\rfloor\pi/2}$; do not take an independent principal square root at each time. The oscillator kernel is exact because the action is quadratic.

Particle on a Ring

Let a particle move on a ring of radius $R$ with moment of inertia

$$I=mR^2.$$

For periodic wavefunctions, no flux, and angular measure $d\theta$ with $\theta\sim\theta+2\pi$, one spectral form is

$$K_{S^1}(\theta_f,T;\theta_i,0) = \frac{1}{2\pi} \sum_{n\in\mathbb Z} \exp\left[ in(\theta_f-\theta_i) - \frac{i\hbar n^2T}{2I} \right].$$

An equivalent winding-sector form is

$$K_{S^1}(\theta_f,T;\theta_i,0) = \sum_{w\in\mathbb Z} \left( \frac{I}{2\pi i\hbar T} \right)^{1/2} \exp\left[ \frac{iI(\theta_f-\theta_i+2\pi w)^2} {2\hbar T} \right],$$

with the same real-time prescription as the free particle. The winding sum makes the topology visible; the spectral sum makes angular-momentum quantization visible.

Semiclassical Van Vleck Form

For a smooth scalar potential in $d$ Cartesian coordinates, away from focal points, the leading semiclassical propagator is

$$K_{\rm sc}(q_f,t_f;q_i,t_i) = \sum_{\gamma:i\to f} \left( \frac{1}{2\pi i\hbar} \right)^{d/2} \lvert D_\gamma\rvert^{1/2} \exp\left[ \frac{i}{\hbar}S_\gamma - i\frac{\pi}{2}\nu_\gamma \right].$$

Here

$$D_\gamma = \det\left( - \frac{\partial^2S_\gamma} {\partial q_f\,\partial q_i} \right).$$

This is not an exact formula for arbitrary potentials. It is the leading isolated-trajectory stationary-phase approximation. It must be repaired near caustics and supplemented by uniform approximations when classical branches coalesce.

Boundary and Measure Reminders

Propagators and Boundary Conditions explains why the same differential expression can define different Hamiltonians on different domains:

Domain	Boundary data	Kernel consequence
full line	square-integrability	continuum free kernel
interval with hard walls	$\psi=0$ at endpoints	sine spectral sum
ring	periodicity in angle	winding or angular-momentum sum
half-line	self-adjoint boundary condition at endpoint	image or spectral kernel depends on boundary condition

Always check the measure. A kernel on a ring evolves wavefunctions with $d\theta$, while a radial kernel in three dimensions uses the radial measure or a transformed radial wavefunction convention.

Common Mistakes

• Treating propagators as probabilities rather than amplitudes.
• Reusing the full-line free kernel in a box or on a ring without imposing boundary conditions.
• Dropping the square-root phase in real-time Gaussian kernels.
• Forgetting that equal-time limits are distributions.
• Confusing time-domain propagator kernels with energy-domain Green functions.
• Using closed-form oscillator kernels at caustic times without the delta-function and Maslov phase.
• Ignoring the measure associated with angular, radial, or constrained coordinates.

References

• R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals, McGraw-Hill, 1965.
• M. C. Gutzwiller, Chaos in Classical and Quantum Mechanics, Springer, 1990.
• H. Kleinert, Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets, 5th ed., World Scientific, 2009.
• L. S. Schulman, Techniques and Applications of Path Integration, Wiley, 1981.
• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994.

Exercises

1. Recover the one-dimensional free kernel from the $d$-dimensional formula.

Solution

Set $d=1$ and $\mathbf r_f-\mathbf r_i=x_f-x_i$. Then

$$K_0^{(1)} = \left( \frac{m}{2\pi i\hbar T} \right)^{1/2} \exp\left[ \frac{im(x_f-x_i)^2}{2\hbar T} \right],$$

which is the one-dimensional free-particle kernel.

2. Check that the constant-force kernel reduces to the free kernel when $F=0$.

Solution

The constant-force action is

$$S_F = \frac{m(x_f-x_i)^2}{2T} + \frac{FT}{2}(x_f+x_i) - \frac{F^2T^3}{24m}.$$

Setting $F=0$ gives

$$S_F \to \frac{m(x_f-x_i)^2}{2T},$$

and the prefactor is already the free-particle prefactor. Thus $K_F\to K_0$.

3. Explain why the infinite-square-well kernel is not the full-line free kernel restricted to $0\lt x\lt L$.

Solution

The infinite square well imposes the hard-wall boundary conditions

$$\psi(0)=\psi(L)=0.$$

These boundary conditions change the domain of the Hamiltonian and force a sine eigenbasis. The full-line free kernel does not vanish at the walls and does not preserve the hard-wall domain. The correct kernel must be built from the well eigenfunctions or an equivalent image construction.

4. Show that the ring spectral kernel is periodic in $\theta_f$.

Solution

The spectral kernel is

$$K_{S^1} = \frac{1}{2\pi} \sum_{n\in\mathbb Z} e^{in(\theta_f-\theta_i)} e^{-i\hbar n^2T/(2I)}.$$

Replacing $\theta_f$ by $\theta_f+2\pi$ multiplies each term by

$$e^{i2\pi n}=1$$

because $n$ is an integer. Therefore the kernel is periodic in $\theta_f$.

5. Why does the Van Vleck formula need a caustic warning?

Solution

The Van Vleck prefactor contains

$$\lvert D_\gamma\rvert^{1/2}, \qquad D_\gamma = \det\left( - \frac{\partial^2S_\gamma} {\partial q_f\,\partial q_i} \right).$$

At a caustic, the classical map between endpoint data becomes singular, and the isolated-trajectory prefactor can diverge. The isolated-trajectory approximation has failed. A uniform approximation combines nearby branches where appropriate; the exact kernel can also be a distribution, as the oscillator parity delta at a half period demonstrates. A divergent prefactor alone is not a divergent transition probability for a normalized packet.