PHYS 471 - Quantum Mechanics
Exam 2 - Spring 2001



  1. Operators (10 points)
    1. Define a Hermitian operator and explain why observable quantities are always represented by Hermitian operators.


      2. A Hermitian operator is an operator whose complex transpose or adjoint is equal to the operator. Eigenvalues of a Hermitian operator are real and represent the values we would get when we make a measurement on the system. Quantities we observe must be real quantities and thus Hermitian operators represent real observable quantities.

    2. Two operators commute. Can we simultaneously determine the eigenvalues of these operators?

      3. Yes.

  2. Commutators (20 points)


    1. Determine the following:






      2. The middle two terms in the above expansion will be zero because Ly and y commute as do Ly and py.

      Now we need to evaluate [pz, Ly] and [z, Ly]



      Putting these expressions into the earlier expression gives







  3. Angular Momentum Operators (20 points)
    1. Calculate the following




      2. Remember the spherical harmonics form an orthonormal set and that Lx can be expressed in terms of the raising and lowering operators. Make use of (4.121)

      Lx can be written as 1/2(L+ + L-) and thus we can replace Lx with a sum of the raising an lowering operator.

      K± is a constant given by equation 4.121, with l = 3 and m=1. Because the spherical harmonics form an orthonormal set, the second term above will be zero and the first term will be ½ K+ .



      Following the same arguments as in part a),

      The first term is equal to zero because Y33 has m equal to its maximum possible value and when we try and raise it we get zero the second term will be 1/2K- with l = 3 and m = 3.





      because the spherical harmonics are orthogonal.

  4. The Hydrogen Atom (20 points)

    5. An electron in the Coulomb field of a proton is in a state described by the following spatial wave function



    1. Determine the value of the normalization constant A? (Remember the hydrogen wave functions constitute an orthonormal set.)

      2. The normalization condition is:



      Because the hydrogen wave functions form an orthonormal set this is equivalent to

    2. What is the expectation value of the energy?


      3. The coefficients in the above expansion represent the probability of making the measurement associated with each of those states. The general expression for the hydrogen wave function is ynlm.

      n Pn En
      1 -13.6 eV
      2 -13.6/4 eV
      3 -13.6/9 eV




    3. What is the expectation value of L2?


      4. We know that the hydrogen wave functions are eigenfunctions of both L2 and Lz.

      l Pl L2
      0 0 2
      1 2 2




    4. What is the expectation value of Lz?


      5. m Pm Lz
      0 0
      1
      -1 -




  5. Spin States (20 points)

    6. For the spinor , calculate the following .

    First we need to normalize the state. The normalization condition is that








  6. Indistinguishable Particles (10 points)

    7. Describe the differences between fermions and bosons.

    Fermions have odd half integer multiples of spin, obey the Pauli exclusion principle and have anti-symmetric wave functions. Bosons have integer spin, do not obey the Pauli exclusion principle and have symmetric wave functions.