## Homework 7

1. A particle is in the state . Find the expected values of , , , and .

2. A particle is in the state . If a measurement of the component of angular momentum is made, what are the possilbe outcomes and what are the probabilities of each?

3. Calculate the matrix elements and

4. The Hamiltonian for a rotor with axial symmetry is where the are constant moments of inertia. Determine and plot the eigenvalues of for dumbbell-like case that .

5. Prove that is only possible for .

6. Write the spherical harmonics for in terms of the Cartesian coordinates , , and .

7. A particle in a spherically symmetric potential has the wave-function . A measurement of is made. What are the possible results and the probabilities of each? If the measurement of yields , what are the possible measured values of and what are the corresponding probabilities?

8. The deuteron, a bound state of a proton and neutron with , has a binding energy of -2.18 MeV. Assume that the potential is a spherical well with potential of for Fermis and zero potential outside. Find the approximate value of using numerical techniques.

Jim Branson 2013-04-22