Homework 5#

This homework is due on Tuesday February 13th. You should submit a PDF of your solutions on myCourses.

  1. (a) Calculate \(\braket{r}\) and \(\braket{r^2}\) for an electron in a hydrogen atom with maximal angular momentum \(\ell=n-1\) and determine the uncertainty in position

\[\Delta r = \sqrt{\braket{r^2}-\braket{r}^2}.\]

(b) Use your result for \(\braket{r}\) to write the energy in terms of \(\braket{r}\).

(c) Do your results make sense in the classical limit? Explain why.

  1. This question gets you to work through the steps involved in going from equations (10.88) to (10.96) in Townsend (solution of 3D harmonic oscillator in spherical coordinates).

(a) Show that in terms of the variables \(\rho = r\sqrt{\mu\omega/\hbar}\) and \(\lambda= 2E/\hbar\omega\), the radial part of the wavefunction of the 3D harmonic oscillator is given by the differential equation

\[{d^2 u\over d\rho^2} -{\ell (\ell+1)\over \rho^2} u - \rho^2 u = -\lambda u,\]

where we have written \(\psi(r,\theta,\phi) = r^{-1} u(r) Y_{\ell m}\).

(b) Explain why it makes sense to look for a solution of the form

\[u = \rho^{\ell+1} e^{-\rho^2/2} f(\rho).\]

(c) Develop a power law solution for \(f(\rho)\), derive the recurrence relation for the polynomial coefficients, and explain how it results in quantization of the energy levels of the oscillator.

  1. Townsend 10.17 (2D harmonic oscillator with ladder operators)