Here, and throughout the semester, please show your work on all problems. Organize your homework so the grader can follow your solution clearly. Explain (in words) what you're doing and what assumptions you make, whenever it seems appropriate.

These problems are all meant to be review (but not necessarily easy!) . Let me know if you feel there are some gaps in your preparation. Also, I'm curious about how long or hard this set "feels" to you.

1) Write down the usual time-dependent Schrodinger equation for a spinless particle in one dimension with potential energy V(x). This equation separates; that is, you can find solutions of the form . Do this and deduce the form of the function f(t) and the (hopefully familiar!) equation that determines u(x). Explain where the energy enters into your discussion.

2) A normalized wave function can always be expanded in the form , where the functions are a complete orthonormal set of eigenfunctions of the Hamiltonian, with eigenvalues .

a) Explain what all those words mean to you: "complete, orthonormal set", "eigenfunctions of the Hamiltonian", "eigenvalues ".

(Use equations, or words, as you wish. Be brief! No fancy or technical discussion required, just something simple and explanatory.)

b) Show how it follows from orthonormality that

c) Rewrite the eqns and in Dirac notation. (No calculations of any kind required here, I just want you to show me what those equations look like in Dirac notation)

d) Find . (Show your work! The answer is simple, but where does it come from mathematically?)

e) Explain in words what, e.g., tells you.

(How would your answer change if the functions were eigenfunctions of some other operator besides energy?)

3) An electron in hydrogen has the following wave fn:

(The 's are hydrogen radial wavefns, and 's are spherical harmonics)

a) What must "c" be, in the expression above? Why? (Is it unique? Explain...)

b) If you measure orbital ang. momentum, L^2, what value(s) might you find, and what is the probability of each?

c) Same, for z-component of orbital angular momentum, and also find the expectation value of Lz.

d) Same, for total energy. Also, find the expectation value of energy.

e) What is ?

f) Suppose you measure Lz and the total energy for the above electron, and you happen to find , and . Write the new wave function, after the measurement.

4a) Find the eigenvalues and normalized eigenvectors of the Hamiltonian , where are just some given constants.

(Hint: Your eigenvalues better all come out real. How do I know that?! )

Are your eigenvectors unique? (Explain...) Also, find the inverse of H.

b) A particle with the Hamiltonian from part a above starts out in the state . If you measured the energy of the particle, what values might you get, with what probabilities?

Also, what is the expectation value of energy?

5a) Griffiths problem 2.17 , but do only parts a and c.

b) Griff. problem 3.48, only do parts a and b.

For 5b: There's one slightly ugly integral here. See Griff. p. 27 for a very similar one, or just let Mathematica do it for you. (Prof. Taylor assigned this last semester, but give it a good try without peeking at your old solutions first!)