Homework 8

  1. Calculate the $\ell=0$ phase shift for the spherical potential well for both and attractive and repulsive potential.

  2. Calculate the $\ell=0$ phase shift for a hard sphere $V=\infty$ for $r<a$ and $V=0$ for $r>a$. What are the limits for $ka$ large and small?

  3. Show that at large $r$, the radial flux is large compared to the angular components of the flux for wave-functions of the form $C{e^{\pm ikr}\over r}Y_{\ell m}(\theta,\phi)$.

  4. Calculate the difference in wavelengths of the 2p to 1s transition in Hydrogen and Deuterium. Calculate the wavelength of the 2p to 1s transition in positronium.

  5. Tritium is a unstable isotope of Hydrogen with a proton and two neutrons in the nucleus. Assume an atom of Tritium starts out in the ground state. The nucleus (beta) decays suddenly into that of He$^3$. Calculate the probability that the electron remains in the ground state.

  6. A hydrogen atom is in the state $\psi={1\over 6}\left(4\psi_{100}+3\psi_{211}-\psi_{210}+\sqrt{10}\psi_{21-1}\right)$. What are the possible energies that can be measured and what are the probabilities of each? What is the expectation value of $L^2$? What is the expectation value of $L_z$? What is the expectation value of $L_x$?

  7. What is $P(p_z)$, the probability distribution of $p_z$ for the Hydrogen energy eigenstate $\psi_{210}$? You may find the expansion of $e^{ikz}$ in terms of Bessel functions useful.

  8. The differential equation for the 3D harmonic oscillator $H={p^2\over 2m}+{1\over 2}m\omega^2r^2$ has been solved in the notes, using the same techniques as we used for Hydrogen. Use the recursion relations derived there to write out the wave functions $\psi_{n\ell m}(r,\theta,\phi)$ for the three lowest energies. You may write them in terms of the standard $Y_{\ell m}$ but please write out the radial parts of the wavefunction completely. Note that there is a good deal of degeneracy in this problem so the three lowest energies actually means 4 radial wavefunctions and 10 total states. Try to write the solutions $\psi_{000}$ and $\psi_{010}$ in terms of the solutions in cartesian coordinates with the same energy $\psi_{nx,ny,nz}$.

Jim Branson 2013-04-22