Physics 402, Fall 2008
Official course description
Quantum Physics II;
Grade Method: REG/P-F/AUD.
Prerequisites: PHYS401, and PHYS374, and MATH240. Credit will be granted
for only one of the following: PHYS402 or former PHYS422. Formerly
Quantum states as vectors; spin and spectroscopy, multiparticle systems,
the periodic table, perturbation theory, band structure, etc.
D. Brill (Seats=45, Open=33, Waitlist=0) Books
- MWF.......10:00am-10:50am (PHY 1402)
- W.........11:00am-11:50am (PHY 1402)
David J. Griffiths, Quantum Mechanics (second edition preferred but first edition acceptable), Pearson/Prentice Hall, ISBN 0-13-111842-7
Added December 08: Solution to Final Exam
In Phys401 we covered chapters 1-4 of Griffiths. The plan for Phys402 is to cover Chapters 5-11 of that text.
Assuming a linear rate of chapters/week leads to the follwing preliminary schedule:
|Week of|| ||Chapter or event|
|Sept 1|| ||Recall of Ch. 1-4|
|Sept 8, 15|| ||Chapter 5|
|Sept 22, 29|| ||Chapter 6|
|Oct 6, 13|| ||First Exam to Ch 6; Chapter 7|
|Oct 20, 27|| ||Chapter 8|
|Nov 3, 10|| ||Chapter 9|
|Nov 17, 24|| ||Second Exam, to Ch 9. Chapter 10; Thanksgiving Nov 27|
|Dec 1, 8|| ||Chapter 11; last day of classes Dec 12|
|Dec 20|| ||Final Exam 8 am*|
*this is the last day of the exam period. Many may want to have the exam earlier. Since we meet W at 11, Dec 17 at 8 am may be a possibility. Let me know if you would have a conflict with this.
Actually we will run behind this schedule because chapters 5 and 6 have more material than the other chapters.
There will be approximately weekly assignments, mainly from the text, that count approximately 20% of your grade. See below.
There will be two exams during the term, one of them a take-home, and a Final Exam that counts about as much as the two in-term exams.
and other information will generally be posted on ELMS. However, the first assignment is due during the second week of classes, so it is reproduced here:
First Assignment due Wednesday September 10:
Two problems are taken from the Final Exam of Physics401 (spring 08), namely problems 2 and 4. The solution to that exam -- with hints only for those two problems -- wronlgy identifies both as the first problem of the current assignment; but do use that link if you find the equations below illegible (particularly on Firefox). It is, of course, a good idea to review the entire Final Exam.
- A one-dimensional simple harmonic oscillator is governed by the
In classical physics this oscillator oscillates at only one frequency w. What is the situation in quantum mechanics?
Let yn be the time-independent nth excited state of
the oscillator. The corresponding time-dependent Yn is astationary state, so the probability density Y*nYn is
of course constant and does not oscillate. Now suppose the oscillator starts out in the superposition state
Y(x, 0) = ||
(yn(x) + ym(x)) at t=0.|
a. Construct Y(x,t) and |Y(x,t)|2.
b. With what frequency (or frequencies) does its probability
density |Y(x,t)|2 oscillate in time? Express your answer in
terms of the w in H of (*), and n and m.
c. With what frequency (or frequencies) will the expectation value
áxñ oscillate in time? Note that for many values
of n and m, áx ñ vanishes for all times, hence
it does not have an oscillation frequency. Do not include such
"zero-amplitude" frequencies in your answer. That is, find a
condition on n and m that guarantees a non-zero amplitude of
Here n ¹ m or n = m+1 (mod 2) (that is, n and m must have different parity) is not enough as the condition. Express x in terms of raising and lowering operators.
- A quantum mechanical system can exist in 3 states only,
|0ñ, |1ñ and |2ñ. They are orthonormal
ái|jñ = dij, and the Hamiltonian is
H = E( |1ñá1| + |2ñá2| ) + l( |0ñá1+2| +|1+2ñá0| ).
Here |1+2ñ means
[1/(Ö2)](|1ñ+|2ñ). l is a real constant.
If there is no interaction between the states, that is if l = 0, then |0ñ is the ground or vacuum state, the other
two are degenerate excited states of energy E. Because of the
interaction term with l the states |iñ are not
a. Evaluate H|1-2ñ = [1/(Ö2)]H(|1ñ-|2ñ) and thus show that
|1-2ñ is stationary (that is, it's an eigenstate
of H). What is its energy?
b. Find the expectation value of the energy in the state
c. Write H in matrix form and find the energies of the other two
stationary states of this Hamiltonian.
- Evaluate the commutator [x, p²]. You may first do it via the representation of p we have been using in terms of a differential operator, so you know what the result is. But then do it "abstractly", that is, use only the canonical commutation relation [2.51].