Quantum Mechanics Physics 215

Homework 4. Due in Class Friday February 25

1. A particle of mass is constrained to move on a circle of radius R. Show that the Hamiltonian is . 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 by . Let be the operator which is represented by multiplication by .

   (a) Compute and deduce the uncertainty relation: .

   (b) Since , we must have . However, an eigenstate of has the property that . (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 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 and , where j labels one of the three coordinates, x,y, or z. The position and momentum operators are defined via

   

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

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

   

   (c) Hence show that the eigenvalues of 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):

   

   where is positive. In the case of three dimensions. find the minimum value of 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 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 . Assume that has the following properties:

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

   Prove the following facts:

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

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

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

   

   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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