Physics 4410 1st Exam (1 1/2 hours) Feb 19, 1998
THIS IS LAST YEAR'S EXAM. Conversion to HTML didn't work so great, but I think you can read most of it. There's a copy outside my office as well.
Your name:
Please read these directions BEFORE beginning.
As always, please show all your work, and explain reasoning clearly where necessary.
You may have a normal 8.5 x11 sheet of paper with your own notes, written on one side only please.
There are 4 problems. Note that they are not worth equal amounts!!
Throughout this exam, we work (as we have all semester) in the usual "Sz basis" where Sz is diagonal, i.e. the basis states are eigenstates of Sz.
Feel free to look over my crib sheet before the exam begins. Don't hesitate to ask me if you have any questions!
Time dependent Schrodinger eqn:
Time dependence of energy eigenfunctions:
Pauli spin matrices:
,
(All have the same two eigenvalues, +/- 1)
The eigenvectors of
are 
The eigenvectors of
(Thats sigma_y) are 
The eigenvectors of
are 
The spin operator (for spin 1/2) is
.
Total angular momentum J=L+S.
The Hamiltonian for a particle with angular momentum S in a magnetic field B is
. If B is in the z direction, the magnetic moment of an electron precesses about the z axis with frequency 
, called the "cyclotron frequency".
Notation for eigenfunctions of angular momentum:
1a) Find the eigenvalues and normalized eigenvectors of the operator
(That's supposed to be 3 sigma_x + 4 sigma_y) (6 pts)Note: your eigenvalues must come out real. Why? (1 pt)
Are your eigenvectors orthogonal? (Should they be?) (1 pt)
b) Suppose you have some particle and measure 
and find 6
^2. Now you measure 
.
(Note: not 
) What possible values might you measure? (2 pts)
1c) Find the eigenvalues of 
(4 pts) (Show your work)
Find the normalized eigenvector of the above matrix corresponding to the eigenvalue 0. (2 pts)
2) A beam of electrons is prepared in the state 
at t=0.
a) At time 0, what are the possible outcomes of a measurement of Sz, with what probabilities? (2 pts)
b) Same as part a, but for Sy. (Please show /explain your work clearly. Feel free to use anything useful from my crib sheet) (4 pts)
c) Find 
and 
, at time 0. (5 pts)
2d) Suppose you started in state 
, measured Sz, and got (say) +/2. Then you go and measure Sx. What would the possible outcomes of this measurement be, with what probabilities? (2 pts)
2e) Again, start in the state (given at the top of the previous page) but put the electron into a constant uniform B field, 
(note! B is in the x direction, not in the z direction.)
Find the probability, at time T, that the electron spins in the positive x direction? (5 pts)
(Please show /explain your work clearly)
Describe briefly what you expect qualitatively for the time dependent behaviour of the spin vector in this B field. (Does your answer above make some physical sense?) (2 pts)
3a) An electron in hydrogen is in the state 
i) What values for 
(total angular momentum squared of the electron) could you measure, with what probabilities? (5 pts)
ii) Suppose your total Hamiltian for the electron is 
.
Find all possible values for the energy of the electron of part a, in terms of A, B, and C. (5 pts)
3b) A 3He nucleus is made of 2 protons and a neutron. In the ground state, all the particles are at rest relative to one another. (Thus, there is no orbital angular momentum between any of the particles) (Protons and neutrons are particles with intrinsic spin 1/2)
i) What possible eigenvalues of total angular momentum squared, 
, can the 3He nucleus have, if you ignore the Pauli principle? (4 pts)
ii) How does the Pauli principle change your answer? (You can thus deduce the spin of 3He in this way, with no ambiguity!) (2 pts)
4) CHOOSE: Either do part a (i.e. All the questions on this page) or part b (the question on the next page) Do NOT do both!!
a-i) Explain (with words, sketches, and/or eq'ns, but try to keep to the space given) how you might go about physically measuring Sz of a single particle. Don't just tell me the name of the apparatus, although that would be nice to mention, but explain clearly how it works. (Would it work with a neutral particle?) (10 pts)
a-ii) If you have a beam of particles incident on your apparatus described above, would it also tell you what 
is for the particles? (Briefly, how, or else why not?) (3 pts)
a-iii) Assume the beam is known to be electrons, all in exactly the same spin state, 
.
Can your apparatus tell us what 
are? How? (or, why not?) (3 pts)
4-b) AS EXPLAINED ON THE PAGE BEFORE, either do the question below, OR the 3 on the previous page. (You choose, don't do both!)
Explain (with words, sketches, and/or eq'ns, but keep to the space given below) the physics of paramagnetic resonance. What does it measure, and how? Why might you care? (How does it relate to MRI?)
Write out explicitly the time dependent Schrodinger Equation for a PMR experiment on an electron. Explain how you get it (just copying the equation from your crib sheet is worth no credit) Define any ambiguous symbols you introduce. Don't try to solve it, but do discuss qualitatively the important features of the solution, physically and/or mathematically. Why is the phenomenon called "resonance"? ( 16 pts)
Your name ____________ Problem number for this page ______