Overview#

Section 3.1.3 ended with the core idea: quantization means action becomes the phase exponent, and quantum mechanics is a sum over all paths. This section makes that precise. We introduce the propagator — the fundamental transition amplitude — express it as Feynman’s path integral (sum over paths weighted by \(\mathrm{e}^{\mathrm{i}S/\hbar}\)), and decode the functional integral via time-slicing into concrete integrals over single time steps. The result: a complete, equivalent reformulation of quantum mechanics that makes the classical limit transparent.

The Propagator: Transition Amplitude#

• It determines everything: Given any initial state \(\psi(x', t')\), the state at any later time is

  \[ \psi(x, t) = \int K(x, t; x', t') \, \psi(x', t') \, \mathrm{d}x' \]

• The propagator also satisfies a composition property: you can split a path at any intermediate time \(t_m\):

  \[ K(x, t; x', t') = \int K(x, t; x_m, t_m) \, K(x_m, t_m; x', t') \, \mathrm{d}x_m \]

  This follows from inserting the identity \(\int \vert x_m \rangle \langle x_m \vert \, \mathrm{d}x_m\) into the product \(\hat{U}(t, t') = \hat{U}(t, t_m) \hat{U}(t_m, t')\).

Feynman’s Path Integral#

Each path contributes with unit probability weight \(|\mathrm{e}^{\mathrm{i}S/\hbar}|=1\) but different phase \(S/\hbar\). Paths with similar action interfere constructively; paths with wildly different action oscillate and cancel.

This is the stationary-phase principle from §3.1.3 realized exactly: the classical path (where \(\delta S = 0\)) is surrounded by neighboring paths with nearly equal phase, so they all add coherently. Off the classical path, the phase varies rapidly and neighboring paths cancel. Thus the classical path dominates, and quantum effects emerge from the sum of near-classical paths.

The Infinitesimal Propagator: One Time Slice#

For a single small time step \(\delta t\), we focus on the short-time propagator \(K_{\delta t}(x, x')\) from \(x'\) to \(x\) in time \(\delta t\).

For a nonrelativistic particle with Lagrangian \(L = \tfrac{1}{2}m\dot{x}^2 - V(x)\), the action over one slice (midpoint rule for the potential) is

For a free particle (\(V=0\)), this reduces to

Time-Slicing: Turning Paths into Integrals#

Divide the total time interval from \(t'\) to \(t\) into \(N\) small steps of duration \(\delta t = (t-t')/N\). Insert complete sets of position eigenstates at each intermediate time \(t_0 < t_1 < \cdots < t_N\):

The full propagator factorizes:

Each factor \(K_{\delta t}\) is the one-step kernel (64). Multiplying the exponentials builds up the total phase along a discretized path; in the limit \(N \to \infty\) (\(\delta t \to 0\)), this becomes the continuum path integral.

Summary#

• The propagator \(K(x, t; x', t') = \langle x \vert \hat{U} \vert x' \rangle\) is the central transition amplitude; it encodes the full time evolution.

• Feynman’s path integral expresses \(K\) as a sum over all histories, each weighted by \(\mathrm{e}^{\mathrm{i}S/\hbar}\). The classical path dominates via stationary phase; quantum paths contribute through interference.

• The one-step propagator \(K_{\delta t}\) is given explicitly in (64) (free-particle limit and \(d\)-dimensional prefactor noted there).

• Time-slicing converts the functional integral into a sequence of ordinary integrals over intermediate positions, which converges to the continuum path integral as \(N \to \infty\).