Physics 402, Fall 2008


Official course description
PHYS402 Quantum Physics II; (4 credits) 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 PHYS422. Quantum states as vectors; spin and spectroscopy, multiparticle systems, the periodic table, perturbation theory, band structure, etc.

0101(55509) D. Brill (Seats=45, Open=33, Waitlist=0) Books
MWF.......10:00am-10:50am (PHY 1402)
W.........11:00am-11:50am (PHY 1402)

Textbook
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


Syllabus
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.

Assignments
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.

  1. A one-dimensional simple harmonic oscillator is governed by the Hamiltonian
    H = p2

    2m
    + 1

    2
    mw2x2
    (*)
    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) = 1

    2
    (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 x oscillate in time? Note that for many values of n and m, x 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 oscillation.
    Here n 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.

  2. A quantum mechanical system can exist in 3 states only, |0,  |1 and |2. They are orthonormal i|j = dij, and the Hamiltonian is
    H = E( |11| + |22| ) + l( |01+2| +|1+20| ).
    Here |1+2 means [1/(2)](|1+|2). l is a real constant.
    If there is no interaction between the states, that is if l = 0, then |0 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 |i are not stationary.

    a. Evaluate H|1-2 = [1/(2)]H(|1-|2) and thus show that |1-2 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 |1+2

    c. Write H in matrix form and find the energies of the other two stationary states of this Hamiltonian.

  3. Evaluate the commutator [x, p]. 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].