A Modern Approach to Quantum Mechanics · Problem 10.17

Time-Independent Perturbations — John S. Townsend

Physics — Quantum Mechanics · two-dimensional-harmonic-oscillator

Problem

Consider the Hamiltonian for the two-dimensional motion of a particle of mass μ\mu in a harmonic oscillator potential: H^=p^x22μ+12μω2x^2+p^y22μ+12μω2y^2\hat{H} = \frac{\hat{p}_x^2}{2\mu} + \frac{1}{2}\mu\omega^2 \hat{x}^2 + \frac{\hat{p}_y^2}{2\mu} + \frac{1}{2}\mu\omega^2 \hat{y}^2 (a) Show that the energy eigenvalues are given by En=(n+1)ℏωE_n = (n+1)\hbar\omega, where the integer n=n1+n2n = n_1 + n_2, with n1,n2=0,1,2,…n_1, n_2 = 0, 1, 2, \ldots (b) Express the operator L^z=x^p^y−y^p^x\hat{L}_z = \hat{x}\hat{p}_y - \hat{y}\hat{p}_x in terms of the lowering operators a^1=μω2ℏ(x^+iμωp^x)anda^2=μω2ℏ(y^+iμωp^y)\hat{a}_1 = \sqrt{\frac{\mu\omega}{2\hbar}}\left(\hat{x} + \frac{i}{\mu\omega}\hat{p}_x\right) \quad \text{and} \quad \hat{a}_2 = \sqrt{\frac{\mu\omega}{2\hbar}}\left(\hat{y} + \frac{i}{\mu\omega}\hat{p}_y\right) and the corresponding raising operators a^1†\hat{a}_1^\dagger and a^2†\hat{a}_2^\dagger. Give a symmetry argument showing that [H^,L^z]=0[\hat{H}, \hat{L}_z] = 0. Evaluate this commutator directly and confirm that it indeed vanishes. (c) Determine the correct linear combination of the energy eigenstates with energy E1=2ℏωE_1 = 2\hbar\omega that are eigenstates of L^z\hat{L}_z by diagonalizing the matrix representation of L^z\hat{L}_z restricted to this subspace of states.

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