Sample Test Problems

1. * We have shown that the Hermitian conjugate of a rotation operator is . Use this to prove that if the form an orthonormal complete set, then the set are also orthonormal and complete.

2. Given that is the one dimensional harmonic oscillator energy eigenstate: a) Evaluate the matrix element . b) Write the upper left 5 by 5 part of the matrix.

3. A spin 1 system is in the following state in the usual basis: . What is the probability that a measurement of the component of spin yields zero? What is the probability that a measurement of the component of spin yields ?

4. In a three state system, the matrix elements are given as , , , , and . Assume all of the matrix elements are real. What are the energy eigenvalues and eigenstates of the system? At the system is in the state . What is ?

5. Find the (normalized) eigenvectors and eigenvalues of the (matrix) operator for in the usual () basis.

6. * A spin particle is in a magnetic field in the direction giving a Hamiltonian . Find the time development (matrix) operator in the usual basis. If , find .

7. A spin system is in the following state in the usual basis: . What is the probability that a measurement of the component of spin yields ?

8. A spin system is in the state (in the usual eigenstate basis). What is the probability that a measurement of yields ? What is the probability that a measurement of yields ?

9. A spin object is in an eigenstate of with eigenvalue at t=0. The particle is in a magnetic field = which makes the Hamiltonian for the system . Find the probability to measure as a function of time.

10. Two degenerate eigenfunctions of the Hamiltonian are properly normalized and have the following properties.

What are the properly normalized states that are eigenfunctions of H and P? What are their energies?

11. What are the eigenvectors and eigenvalues for the spin operator ?

12. A spin object is in an eigenstate of with eigenvalue at t=0. The particle is in a magnetic field = which makes the Hamiltonian for the system . Find the probability to measure as a function of time.

13. A spin 1 system is in the following state, (in the usual eigenstate basis):

What is the probability that a measurement of yields 0? What is the probability that a measurement of yields ?

14. A spin object is in an eigenstate of with eigenvalue at t=0. The particle is in a magnetic field = which makes the Hamiltonian for the system . Find the probability to measure as a function of time.

15. A spin 1 particle is placed in an external field in the direction such that the Hamiltonian is given by

Find the energy eigenstates and eigenvalues.

16. A (spin ) electron is in an eigenstate of with eigenvalue at . The particle is in a magnetic field = which makes the Hamiltonian for the system . Find the spin state of the particle as a function of time. Find the probability to measure as a function of time.

Jim Branson 2013-04-22