Vol 3 - Landau, Lifshitz - Quantum mechanics - non-relativistic theory (3ed., 1991) · Problem 51.3

Spin

Physics — Landau & Lifshitz · quasi-classical-matrix-elements-anharmonic-oscillator

Problem

Calculate the quasi-classical matrix elements (exponential factor only) for an anharmonic oscillator with potential energy

U(x) = \frac{1}{2}m\omega^2 x^2 + \beta x^4

, under the condition

\hbar\omega \ll E_1, E_2 \ll m^2\omega^4/\beta. \tag{1}