A Modern Approach to Quantum Mechanics · Problem 10.11

Time-Independent Perturbations — John S. Townsend

Physics — Quantum Mechanics · cylindrical-potential-well

Problem

A particle of mass μ\mu is in the cylindrical potential well V(ρ)={0ρ<a∞ρ>aρ=x2+y2V(\rho) = \begin{cases} 0 & \rho < a \\ \infty & \rho > a \end{cases} \quad \rho = \sqrt{x^2 + y^2} (a) Determine the three lowest energy eigenvalues for states that also have pzp_z and LzL_z equal to zero. (b) Determine the three lowest energy eigenvalues for states with pzp_z equal to zero. The states may have nonzero LzL_z. Suggestion: Work out Problem 9.23 before attempting this problem. Watch out for the appearance of Bessel's equation and ordinary Bessel functions when solving the radial equation.

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