Principles of Quantum Mechanics · Problem 21.14

Path Integrals—II — R. Shankar

Physics — Quantum Mechanics · path-integrals-anharmonic

Problem

Consider the anharmonic oscillator with potential

V(x) = \frac{1}{2}m\omega^2 x^2 + \lambda x^4

. Using the path integral perturbation expansion in

\lambda

, compute the first-order correction to the ground-state energy:

E_0 = \frac{\hbar\omega}{2} + \lambda \langle 0 | x^4 | 0 \rangle + O(\lambda^2).

Evaluate

\langle 0 | x^4 | 0 \rangle

using the harmonic oscillator ground state.