PC 441 2112 Quantum Mechanics I Block 2, 2007-08

Midblock Mindblock

Name:

Honor Pledge:

This is an approximately 3 hour closed book exam. You may use a table of integrals, a calculator, an 8.5” ´ 11” sheet with your formulas, etc. and your brain, but no other aids. Enjoy the experience!

1. The ground state of hydrogen is Y(r) = (1/pa³)^1/2exp(-r/a).

a. What is <r>? What is <r²>?

b. What is the most probable value of r? It’s not zero! First you need to find the probability that the electron will be found between r and r + dr.

c. What is <L_z>? Calculate it; don’t just give me an expression. Recall L_z = xp_y – p_yx.

2. The Hamiltonian of a two level system is

H = é a b ù

ë b a û where a and b are different real

numbers.

a. What are the possible eigenvaluess of this Hamiltonian? What are the units of a and b?

b. What are the eigenvectors of the energy?

c. If the system starts in the state é 1 ù

ë 0 û

what is the state of the system at time t?

Please staple your formula sheet to your completed exam. Thanks.