PHYS 424 Exam 2

November 21, 2001

1. Show that [ri, Jj] = Sk i hbar eijk rk

2. Two particles of mass m are attached to the ends of a massless rigid rod of length a. The system is free to rotate in three dimensions about the center, and the center point is fixed.
    (a) Show that the energies of this rigid rotor are 

                            hbar2 n(n+1)
                       En = ------------
                               ma2
where n is an integer.
    (b) What is the degeneracy of the nth energy level?

3. Two spin-1/2 Fermions are in an infinite one-dimensional square well. They are to be placed in a state with a total spin of one and a spin-projection of one. What are the two lowest-energy states available (specify both spin and space dependence)? What are the energies and degeneracies of those two states?

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PHYS 424 Exam 2

November 28, 2001

1. Find [pi, Jj]

2. Two particles of mass M are attached to the ends of a massless rigid rod of length a. The system is free to rotate in three dimensions about the center, and the center point is fixed.
    (a) Show that the energies of this rigid rotor are Elm = [hbar2/(Ma2)] l(l+1) where l and m are integers [m and M are different symbols].
    (b) What is the degeneracy of the lth energy level?

3. Two spin-1 Bosons are in an infinite one-dimensional square well. They are to be placed in a state with a total spin of two and a spin-projection of two. What are the two lowest-energy states available (specify both spin and space dependence)? What are the energies and degeneracies of those two states?

PHYS 424 Exam 2

November 25, 2002

1. (a) An electron is in a state with spin up along the z-axis, i.e. 

    1
    0  .
What is the probability of finding that electron in a state that is spin-up along the x-axis? Justify your answer.

    (b). If only electrons whose spin is up-along-x are kept in the apparatus after determining the spin direction, what is the wave function of the electron after the measurement?

    (c). What is the probability of now finding the electron in a state that has spin down along the z axis? Prove your answer.

For reference, the Pauli spin matrices for x, y, and z are 

      0  1     0  -i     1   0
      1  0  ,  i   0  ,  0  -1
respectively.

2. Suppose that we have three particles, which can each occupy any of the three states ya, yb, and yc. If the three states are different, how many different three-particle states are available if the particles are

    (a). Distinguishable

    (b). Indistinguishable Bosons (spin 0)

    (c). Indistinguishable Fermions (half-integer spin)

    (d). What changes would there be if two of the states were the same?

3. Calculate the average value of r2 in the state |n n-1> of the hydrogen atom.
The radial factor for the hydrogen wave function is Rn n-1 = Nn rn-1 exp[- r / (na) ] ,   and
ò0¥ rm e -b r = m! / b m+1

PHYS 424 Exam 2

November 24, 2003

1. Calculate the average value of r3 in the state |n n-1> of the hydrogen atom.
The radial factor for the hydrogen wave function is Rn n-1 = Nn rn-1 exp[- r / (na) ] ,   and
ò0¥ rm e -b r = m! / b m+1

2. An electron is in the spin state 

                                     4
                             y = A
                                    3i

    (a) Determine the constant A.

    (b) Find the expected value (average value of a measurement) of 

                                 0  -i
                  Sy = (hbar/2)
                                 i   0

    (c) What is the probability that a measurement will give - hbar/2 for the y-component of the spin?

    The unnormalized eigenvectors of Sy are 

          1        1
              and    
         -i        i

3. Show algebraically that the eigenvalues of the z-component of the angular momentum differ by integer multiples of h-bar. Recall that
[Ji , Jj ] = i hbar Sk eijk Jk .


PHYS 424 Exam 2

December 7, 2003

1. Calculate the average value of r4 in the state |n n-1> of the hydrogen atom.
The radial factor for the hydrogen wave function is Rn n-1 = Nn rn-1 exp[- r / (na) ] ,   and
ò0¥ rm e -b r = m! / b m+1

2. An electron is in the spin state 

                                    4i
                             y = A
                                     3

    (a) Determine the constant A.

    (b) Find the expected value (average value of a measurement) of 

                                 0  1
                  Sx = (hbar/2)
                                 1  0

    (c) What is the probability that a measurement will give - hbar/2 for the x-component of the spin?

    The unnormalized eigenvectors of Sx are 

          1        1
              and    
         -1        1

3. Show that [J2, Ji] = 0. Recall that [Ji , Jj ] = i hbar Sk eijk Jk . What is the physical importance of this result?
Last Revised 05/04/26