A Modern Approach to Quantum Mechanics · Problem 11.1

Identical Particles — John S. Townsend

Physics — Quantum Mechanics · anharmonic-oscillator-perturbation

Problem

Consider a perturbation H^1=bx^4\hat{H}_1 = b\hat{x}^4 to the simple harmonic oscillator Hamiltonian H^0=p^x22m+12mω2x^2\hat{H}_0 = \frac{\hat{p}_x^2}{2m} + \frac{1}{2}m\omega^2\hat{x}^2 This is an example of an anharmonic oscillator, one with a nonlinear restoring force. (a) Show that the first-order shift in the energy is given by En(1)=3ℏ2b4m2ω2(1+2n+2n2)E_n^{(1)} = \frac{3\hbar^2 b}{4m^2\omega^2}(1 + 2n + 2n^2) (b) Argue that no matter how small bb is, the perturbation expansion will break down for some sufficiently large nn. What is the physical reason?

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