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Quantum Mechanics Physics 215

Homework 4. Due in Class Friday February 25

  1. A particle of mass tex2html_wrap_inline53 is constrained to move on a circle of radius R. Show that the Hamiltonian is tex2html_wrap_inline57 . What are the allowed energy levels of the system? Are the energy levels degenerate or non-degenerate? If degenerate, interpret the degeneracy.
  2. In polar coordinates, we may represent tex2html_wrap_inline59 by tex2html_wrap_inline61 . Let tex2html_wrap_inline63 be the operator which is represented by multiplication by tex2html_wrap_inline65 .

    (a) Compute tex2html_wrap_inline67 and deduce the uncertainty relation: tex2html_wrap_inline69 .

    (b) Since tex2html_wrap_inline71 , we must have tex2html_wrap_inline73 . However, an eigenstate of tex2html_wrap_inline59 has the property that tex2html_wrap_inline77 . (Why?) Such as state therefore violates the uncertainty relation derived in part (a). Explain this apparent paradox.

  3. In this problem, we use algebraic methods to deduce that the eigenvalues of tex2html_wrap_inline59 are integers. Hence the orbital angular momentum quantum number l must be a non-negative integer. (i.e. half-integer values are not given by this method. The reason is that we will use the coordinate basis and half-integer spins do not have a coordinate basis).

    (a) Taking inspiration from the algebraic solution of the harmonic oscillator, introduce creation and annihilation operators tex2html_wrap_inline83 and tex2html_wrap_inline85 , where j labels one of the three coordinates, x,y, or z. The position and momentum operators are defined via

    eqnarray26

    where m and tex2html_wrap_inline95 are arbitrary parameters. Compute the operator tex2html_wrap_inline59 in terms of these creation and annihilation operators.

    (b) Show, by means of a linear transformation on tex2html_wrap_inline85 and tex2html_wrap_inline83 , that tex2html_wrap_inline59 can be written in terms of new creation and annihilation operators, tex2html_wrap_inline105 and their hermitian conjugates as follows:

    displaymath107

    (c) Hence show that the eigenvalues of tex2html_wrap_inline59 must be integers.

    Note: You may find the discussion in Baym pp. 380-383 to be helpful.

  4. Solve for the lowest energy state in a square well potential in two and three dimensions with zero angular momentum (i.e. l = 0):

    displaymath113

    where tex2html_wrap_inline115 is positive. In the case of three dimensions. find the minimum value of tex2html_wrap_inline115 which is necessary in order that there be at least one bound state. This is in contrast to the situation in one dimension where there is always binding no matter how small tex2html_wrap_inline115 is. In two dimensions, is the situation analogous to the three dimensional case or to the one dimensional case?

  5. Let us denote the time reversal operator by tex2html_wrap_inline121 . Assume that tex2html_wrap_inline121 has the following properties:

    (i) tex2html_wrap_inline125 and
    (ii) tex2html_wrap_inline121 is anit-unitary,
    i.e. tex2html_wrap_inline129 , and tex2html_wrap_inline131 , where c is a complex number.

    Prove the following facts:

    (a) tex2html_wrap_inline135
    where tex2html_wrap_inline137 depends on j but not on m. [Hint: use (i) above, where you consider the action of tex2html_wrap_inline121 on tex2html_wrap_inline145 and tex2html_wrap_inline147 respectively.]

    (b) tex2html_wrap_inline149
    using the result of part (a) and property (ii) above.

    (c) Show that by appropriate choice of tex2html_wrap_inline137 in part (a), one can represent tex2html_wrap_inline121 by:

    displaymath155

    where K is the (anti-unitary) complex conjugation operator.

    (d) Consider an atomic system with an odd number of electrons (so that the total angular momentum of the electrons is half-integral). Show that the energy levels of the system must be at least twofold degenerate. (This is called Kramers degeneracy).




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Peter Young
Tue Feb 8 16:13:52 PST 2000