Simple Applications of Propagator Functions

Abstract

Let us first summarize what we know until now about the Feynman propagator , thinking first, for simplicity, of a one-dimensional system, described by the following Lagrangian:

$$\displaystyle{ L(x,\dot{x}) = \frac{m} {2} \dot{x}^{2} - V (x)\;. }$$

(23.1)

Then we know that

$$\displaystyle{ \begin{array}{ll} 1)\quad K{\bigl (x_{f }, t_{f }; x_{i },t_{i}\bigr )} = &\int _{x(t_{i})=x_{i}}^{x(t_{f})=x_{f} }[dx(t)] \\ &\, \times \text{exp}\left \{ \frac{\text{i}} {\hslash }\int _{t_{i}}^{t_{f} }dt\left [\frac{m} {2} \dot{x}^{2} - V (x)\right ]\right \}\;. \end{array} }$$

(23.2)

$$\displaystyle\begin{array}{rcl} 2)\quad & & \text{i}\hslash \frac{\partial } {\partial t_{f}}K{\bigl (x_{f},t_{f};x_{i},t_{i}\bigr )} = \left [-\frac{\hslash ^{2}} {2m}\, \frac{\partial ^{2}} {\partial x_{f}^{2}} + V {\bigl (x_{f}\bigr )}\right ]K{\bigl (x_{f},t_{f};x_{i},t_{i}\bigr )}\;. \\ & & K{\bigl (x_{f},t_{i};x_{i},t_{i}\bigr )} =\delta {\bigl ( x_{f} - x_{i}\bigr )}\;. {}\end{array}$$

(23.3)

$$\displaystyle{ \begin{array}{@{\hspace *{-5pc}}l@{\hspace *{1pc}}l@{}} \hspace{-60.0pt}3)\quad K{\bigl (x_{f }, t_{f }; x_{i },t_{i}\bigr )} =\sum _{ n=0}^{\infty }\phi _{n}{\bigl (x_{f}\bigr )}\phi _{n}^{{\ast}}{\bigl (x_{i}\bigr )}\,\text{e}^{-(\text{i}/\hslash )E_{n}(t_{f}-t_{i})}\;.\hspace{12.0pt}\end{array} }$$

(23.4)

We have already seen in some examples (particle in a square well, or constrained to move on a ring) that the representation (23.4) exists. More generally, (23.4) can be shown as follows: we know that the propagator for fixed x _i , t _i solves the Schrödinger equation.