Propagator Kernel

The propagator kernel is the coordinate-space matrix element of the time-evolution operator:

K(x_f,t_f;x_i,t_i) = \langle x_f\rvert U(t_f,t_i)\lvert x_i\rangle.

It is a transition amplitude, not a probability.

Wavefunction Evolution

Given an initial wavefunction $\psi(x_i,t_i)$ , the later wavefunction is

\psi(x_f,t_f) = \int_{-\infty}^{\infty} K(x_f,t_f;x_i,t_i)\psi(x_i,t_i)\,dx_i.

For other configuration spaces, the integration domain and measure must match the Hilbert-space inner product.

Initial Condition

At equal times, the propagator reduces to a delta distribution:

K(x_f,t_i;x_i,t_i) =\delta(x_f-x_i).

This says that no time evolution maps a position amplitude to itself by the identity kernel.

Differential Equation

As a function of the final variables, the kernel satisfies the time-dependent Schrödinger equation:

i\hbar\frac{\partial}{\partial t_f} K(x_f,t_f;x_i,t_i) = H_{x_f} K(x_f,t_f;x_i,t_i),

with the delta-function initial condition above. Here $H_{x_f}$ means that the coordinate-space Hamiltonian acts on the final coordinate.

Composition Law

Time-evolution operators compose:

U(t_b,t_a) =U(t_b,t_c)U(t_c,t_a).

In coordinate representation this becomes

K(x_b,t_b;x_a,t_a) = \int dx_c\, K(x_b,t_b;x_c,t_c) K(x_c,t_c;x_a,t_a).

This identity is the Composition Law and is the direct route to path-integral time slicing.

Spectral Form

For a time-independent Hamiltonian with discrete eigenstates,

K(x_f,t;x_i,0) = \sum_n \psi_n(x_f)\psi_n^*(x_i) e^{-iE_nt/\hbar}.

Continuous spectra add integrals over continuum labels. Spectral Decomposition of the Propagator develops the discrete, continuous, and mixed cases and makes clear how bound states and scattering states contribute to propagation.

Free-Particle Kernel

For $H=p^2/(2m)$ on the real line and $T=t_f-t_i>0$ , the momentum resolution of the identity gives

K_0(x_f,T;x_i,0) = \int_{-\infty}^{\infty}\frac{dp}{2\pi\hbar} \exp\!\left[ \frac{i}{\hbar}p(x_f-x_i)-\frac{iT}{\hbar}\frac{p^2}{2m} \right].

The oscillatory Gaussian is defined by the causal convergence prescription $T\mapsto T-i0^+$ , or equivalently by first evaluating it with a small damping factor and then taking the limit. The result is

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

The square-root branch is fixed by continuity from the damped integral and by the distributional condition $K_0\to\delta(x_f-x_i)$ as $T\to0^+$ . Its phase is the classical action,

S_{\mathrm{cl}} = \frac{m(x_f-x_i)^2}{2T},

so stationary phase selects the classical momentum $p_{\mathrm{cl}}=m(x_f-x_i)/T$ . In $d$ Cartesian dimensions the independent Gaussian integrals give

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

Convolving this kernel with an initial packet displays free spreading directly. The Free-Particle Propagator: First Encounter keeps the coordinate-space interpretation close to the canonical system; this page owns the derivation and general kernel structure.

Relation to Path Integrals

The path integral begins by applying the composition law many times, inserting intermediate positions, and taking a formal continuum limit. The result is a sum over histories weighted by $e^{iS/\hbar}$ .

Relation to Green Functions

Propagators evolve states in time. Green functions often solve inhomogeneous differential equations or resolvent equations with specified boundary conditions. The two are related, but not identical; boundary conditions and time-ordering conventions matter.

Common Mistakes

Treating $K$ as a probability instead of an amplitude.
Forgetting the integration over the initial coordinate when evolving a wavefunction.
Ignoring the measure and boundary conditions of the configuration space.
Confusing the propagator kernel with an energy-domain Green function.
Forgetting that the equal-time limit is a distribution.
Quoting the free-particle square root without its convergence and branch prescription.
Using the infinite-line kernel when walls, periodicity, or another configuration-space boundary condition changes the kernel.

Cross-Links

References

R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994.
L. S. Schulman, Techniques and Applications of Path Integration, Wiley, 1981.
R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals, McGraw-Hill, 1965.
D. J. Griffiths and D. F. Schroeter, Introduction to Quantum Mechanics, 3rd ed., Cambridge University Press, 2018.

Exercises

Show that the composition law for $K$ follows from inserting a position resolution of identity at the intermediate time.

Solution

Start from

K(x_b,t_b;x_a,t_a) = \langle x_b\rvert U(t_b,t_c)U(t_c,t_a)\lvert x_a\rangle.

Insert

I=\int dx_c\,\lvert x_c\rangle\langle x_c\rvert.

This gives

K(x_b,t_b;x_a,t_a) = \int dx_c\, K(x_b,t_b;x_c,t_c) K(x_c,t_c;x_a,t_a).

Use dimensional analysis to verify that the one-dimensional free kernel has units of inverse length.

Solution

Because $\hbar T/m$ has units of length squared, the prefactor $\sqrt{m/(\hbar T)}$ has units of inverse length. The exponential is dimensionless. This is also required by $\psi(x_f,T)=\int dx_i\,K\psi(x_i,0)$ .

Apply stationary phase to the momentum integral for $K_0$ and identify the stationary momentum.

Solution

The phase is $\Phi(p)=p(x_f-x_i)-p^2T/(2m)$ . Solving $d\Phi/dp=0$ gives

p_{\mathrm{cl}}=\frac{m(x_f-x_i)}{T},

the momentum of the classical straight path joining the endpoints in time $T$ .

Derive the three-dimensional free kernel by factorizing the Cartesian momentum integrals.

Solution

Since $p^2=p_x^2+p_y^2+p_z^2$ and the measure factorizes, the integral is the product of three one-dimensional kernels. Therefore

K_0(\mathbf x_f,T;\mathbf x_i,0) = \left(\frac{m}{2\pi i\hbar T}\right)^{3/2} \exp\!\left[ \frac{im\lvert\mathbf x_f-\mathbf x_i\rvert^2}{2\hbar T} \right].