Principles of Quantum Mechanics · Problem 2.8.7

The Postulates—a General Discussion — R. Shankar

Physics — Quantum Mechanics · harmonic-oscillator-classical-action

Problem

Consider the harmonic oscillator, for which the general solution is

x(t) = A\cos\omega t + B\sin\omega t.

Express the energy in terms of

A

and

B

and note that it does not depend on time. Now choose

A

and

B

such that

x(0)=x_1

and

x(T)=x_2

. Write down the energy in terms of

x_1

x_2

, and

T

. Show that the action for the trajectory connecting

x_1

and

x_2

S_\text{cl}(x_1, x_2, T) = \frac{m\omega}{2\sin\omega T}\left[(x_1^2+x_2^2)\cos\omega T - 2x_1 x_2\right].

Verify that

\partial S_\text{cl}/\partial T = -E