19.8 Homework
- Calculate theℓ= 0 phase shift for the spherical potential well for both and attractive and
repulsive potential. - Calculate theℓ= 0 phase shift for a hard sphereV=∞forr < aandV= 0 forr > a. What
are the limits forkalarge and small? - Show that at larger, the radial flux is large compared to the angular components of theflux
for wave-functions of the formCe
±ikr
r Yℓm(θ,φ).- Calculate the difference in wavelengths of the 2p to 1s transition inHydrogen and Deuterium.
Calculate the wavelength of the 2p to 1s transition in positronium. - 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. - A hydrogen atom is in the stateψ =^16
(
4 ψ 100 + 3ψ 211 −ψ 210 +√
10 ψ 21 − 1)
. What are the
possible energies that can be measured and what are the probabilities of each? What is the
expectation value ofL^2? What is the expectation value ofLz? What is the expectation value
ofLx?
7. What isP(pz), the probability distribution ofpzfor the Hydrogen energy eigenstateψ 210?
You may find the expansion ofeikzin terms of Bessel functions useful.
8. The differential equation for the 3D harmonic oscillatorH= p
2
2 m+1
2 mω(^2) r (^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 functionsψnℓm(r,θ,φ) for the three lowest energies. You
may write them in terms of the standardYℓm 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 10total states. Try to write
the solutionsψ 000 andψ 010 in terms of the solutions in cartesian coordinates with the same
energyψnx,ny,nz.