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Quantum Mechanics Physics 215

Final Examination

Thursday, March 16, 2000, 8:00-11:00 am. Kerr 289.

Closed book; you may bring one page of notes if you wish.

  1. (25 points)
    A particle is in the ground state of a box with infinitely high walls of width L. Suddenly the walls are pulled apart symmetrically to twice the size. This expansion happens so fast that the wave function has no time to change while it is taking place.
    1. Show that the probability of finding the particle in the new ground state is tex2html_wrap_inline88 .
    2. Find the probability of finding the particle in first excited state of the expanded box.
  2. (20 points)
    Consider the commutation rules between the z-component of angular momentum, tex2html_wrap_inline92 , and position:

    displaymath94

    By sandwiching these commutation relations between states tex2html_wrap_inline96 and tex2html_wrap_inline98 , where n and n' are quantum numbers of some quantity other than angular momentum, and l and m have their usual meanings, show that

    1. tex2html_wrap_inline108 is zero unless tex2html_wrap_inline110 .
    2. tex2html_wrap_inline112 and tex2html_wrap_inline114 are zero unless tex2html_wrap_inline116 .
  3. (25 points)
    (For convenience set tex2html_wrap_inline118 )
    An electron is subject to a uniform, time-independent magnetic field of strength B in the positive z-direction so the Hamiltonian is given by tex2html_wrap_inline124 where g=2 and tex2html_wrap_inline128 is the Bohr magneton. At time t = 0, the electron is known to be in an eigenstate of tex2html_wrap_inline132 with eigenvalue 1/2, where tex2html_wrap_inline136 is a unit vector, lying in the yz-plane, that makes an angle tex2html_wrap_inline140 with the z axis.

    (a) Obtain the probability for finding the electron in the tex2html_wrap_inline144 state as a function of time.

    (b) Find the expectation value of tex2html_wrap_inline146 as a function of time.

    (c) Check explicitly that the extreme cases of tex2html_wrap_inline148 and tex2html_wrap_inline150 agree with your intuition.

    Note:

    displaymath152

  4. (30 points)
    (a) Show that for j=1,

    displaymath156

    Hint: Relate tex2html_wrap_inline158 to tex2html_wrap_inline160 .

    (b) Using the result of part (a) prove that

    displaymath162

    (c) Consider a spherical tensor operator of rank 1 (that is a vector)

    displaymath164

    Using the expression for tex2html_wrap_inline166 in part (b) evaluate

    displaymath168

    and show that your results are just what you expect from the transformation properties of a vector about the y-axis.

    Note: For j=1 the matrix for tex2html_wrap_inline160 is given by:

    displaymath176


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Next: About this document

Peter Young
Wed Mar 15 19:54:41 PST 2000