Physics 212 – Problem Set 5 – Spring 2010
1. C-TDL Chapter 10 problem 2.
2. Numerical Effect of the Lowest Order Relativistic Correction on Hydrogen Energy Levels.
Evaluate the unperturbed energy levels n = 1, 2, and 3 for the hydrogen atom. Sketch and label them. Now
evaluate the change to these energy levels coming from the first relativistic correction
H(1) = −
1 p4
.
8 m 3 c2
Evaluate this in units of eV. Sketch the corrected energy levels for n = 1, 2, and 3, specifying any additional
quantum numbers needed.
3. Consider a hydrogenic (one-electron) atom with principal quantum number n = 2.
(a) If we neglect spin-orbit coupling, the states of this atom can be specified equally well either by the set
of quantum numbers (n ` m` ms ) or by the set (n ` j mj ). Write down the set of quantum numbers
(2 ` m` ms ) for each possible n = 2 eigenstate of this atom.
(b) Write down all possible eigenstates of this atom, using the alternative description (2 ` j mj ).
(c) Now you have specified the eight possible states of the atom in two different ways. Therefore each state
in one representation must be a linear combination of the states in the other representation. (Why)?
(0)
Expressing this fact in an equation, we write for each eigenfunction Ψ2`jmj ,
XX
(0)
Ψ2`jmj =
am` ms Ψ2`m` ms ,
m` ms
where am` ms are the expansion coefficients. Use the raising and lowering operators J± , L± , and S± , along
with the orthonormality property of the wave functions, to determine the expansion coefficients in each of
these expansions. [Hint: Define raising and lowering operators J± = Jx ± iJy . These operators obey the
same rules as do L± and S± .]
(d) Find the first-order correction to the n = 2 unperturbed energy of the hydrogen atom due to the spin-orbit
~ · L):
~
interaction, as follows (use H(1) = A(n, l)S
(0)
i. Using the zeroth-order eigenfunction Ψn`m` ms (where n = 2), set up the problem and calculate the
matrix elements needed for the secular equation; show how the secular determinant can be broken into
a block structure. (The largest block you will have to deal with is 2 × 2.) You will find it conv enient
to prove and use the identity
S · L = 21 (S− L+ + L− S+ + 2Lz Sz ).
ii. Find all eight roots of the secular determinant. (Some are very simple to write down.) Verify that
your results agree with those given in the text.
iii. Find all eight new zeroth-order eigenfunctions corresponding to the energies obtained in part (ii). Show
(0)
that these functions are precisely the same as the eigenfunctions Ψn`jmj (r) for n = 2 and thus are
eigenfunctions of L2 , S 2 , J 2 and Jz .
iv. Write the expression for the second-order correction to the energies of part (ii) due to the 3p states.
Leave your answer in terms of the radial integrals. Indicate explicitly which of the 3p states are
important in the perturbation.
4. Sakurai Ch. 3 Problem 18:
Consider an orbital angular momentum eigenstate |l = 2, m = 0i. Suppose this state is rotated by an angle β
about the y–axis. Find the probability for the new state to be found in m = 0, ±1, and ±2. [You will need the
spherical harmonics for l = 0, 1, and 2. These are on pg. 337 of Shankar, Appendix A of Sakurai, and pg. 248
of Merzbacher. Also note discussion on pg 202-203 of Sakurai.]
5. C-TDL Chapter XI problem 6.