Introduction to Quantum Mechanics · Problem 7.52

Time-Independent Perturbation Theory — David J. Griffiths

Physics — Quantum Mechanics · 2d_oscillator_magnetic_field

Problem

Consider a spinless particle of charge qq and mass mm constrained to move in the xyxy plane under the influence of the two-dimensional harmonic oscillator potential V(x,y)=12mω2(x2+y2).V(x,y) = \frac{1}{2}m\omega^2\left(x^2 + y^2\right). (a) Construct the ground state wave function, ψ0(x,y)\psi_0(x,y), and write down its energy. Do the same for the (degenerate) first excited states. (b) Now imagine that we turn on a weak magnetic field of magnitude B0B_0 pointing in the zz-direction, so that (to first order in B0B_0) the Hamiltonian acquires an extra term H′=−μ⋅B=−q2m(L⋅B)=−qB02m(x py−y px).H' = -\boldsymbol{\mu}\cdot\mathbf{B} = -\frac{q}{2m}(\mathbf{L}\cdot\mathbf{B}) = -\frac{qB_0}{2m}\left(x\,p_y - y\,p_x\right). Treating this as a perturbation, find the first-order corrections to the energies of the ground state and first excited states.

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