Assignment 3 [pdf] - Department of Physics - Carleton University

Carleton University Physics Department
PHYS 4708 (Fall 2012, H. Logan)
Homework assignment #3
Handed out Mon. Feb. 4, 2013; due Wed. Feb. 13 at the start of class.
Problems are worth 5 points each unless noted otherwise.
1. Compute the energy shift from the anomalous Zeeman perturbation,
∆EB = eB
2me
〈φℓjmj |Jz + Sz|φℓjmj 〉, (1)
for the n = 2 states of the hydrogen atom using the “brute force” method of expressing the
eigenstates |φℓjmj 〉 as a linear combination of |mℓms〉 states in order to evaluate the matrix
elements of Sz. (Use the table of Clebsch-Gordan coefficients to get the appropriate linear
combinations.) Compare your results to the shifts found using Eq. (12-24) of the textbook,
∆EB = eB¯h
2me
mj
1 +
j(j + 1) − ℓ(ℓ + 1) + 3/4
. (2)
2j(j + 1)
2. Work out the energies for the Lyman-α transition spectral lines in hydrogen (relative to the
zeroth-order energy ∆E2→1 = (3/4)EBohr 1 ) for the three following situations in the presence
of an external magnetic field B = Bˆz:
(a) A fake situation in which the electron has no spin.
(b) The case that the splitting due to the external B-field is very large compared to the fine
structure, so that the fine structure can be ignored (work in the eigenbasis of definite mℓ
and ms).
(c) The anomalous Zeeman case in which fine structure is a larger effect than the Zeeman
splitting (feel free to use the final results for the energy shifts from the text or lecture).
Note: Remember that this electric dipole transition obeys the selection rules ∆ℓ = ±1, ∆mℓ =
0, ±1 and that an electric dipole transition cannot flip the spin of the electron.
3. (Gasiorowicz 3rd edition problem 12-5) Consider a harmonic oscillator in three dimensions. If
the relativistic expression for the kinetic energy is used, what is the shift in the ground-state
energy?
Hint: treat the relativistic correction as a small perturbation as we did for the hydrogen atom.
Because the harmonic oscillator Hamiltonian can be written as H0 = |p| 2 /2m + mω 2 r 2 /2 =
|p| 2 /2m + V (x) + V (y) + V (z), where V (x) = mω 2 x 2 /2, etc., the three-dimensional harmonic
oscillator can be solved by separation of variables in Cartesian coordinates with the result
ψE(x, y, z) = un1(x)un2(y)un3(z), (3)
where the functions u are the 1-dimensional harmonic oscillator solutions in each direction.
The corresponding energy eigenvalues are E = ¯hω(n1 + n2 + n3 + 3/2).

4. (similar to Gasiorowicz 3rd edition problem 13-3) Consider two noninteracting electrons in a
one-dimensional infinite potential well.
(a) What is the ground-state wave function of the two-electron system if the two electrons
are in the same spin state?
(b) Re-express the ground-state wave function in terms of the relative coordinate x ≡ x1 −x2
and the centre-of-mass coordinate X ≡ (x1+x2)/2. Square to obtain the joint probability
density P (x, X) and integrate over X to find the probability density for the relative
coordinate P (x). Make a sketch of P (x). What is the average separation between the
two electrons?
5. Consider N spinless particles in a 3-dimensional infinite square well with sides of length L.
(a) Compute the total energy of the ground state of this system of N particles, assuming
that the particles are distinguishable. What is the total energy of the ground state if the
particles are identical bosons?
(b) Compute the total energy of the first excited state of the system. What is the degeneracy
of this state for the system of N distinguishable particles? What is the degeneracy if the
N particles are identical bosons?
(c) At finite temperature T , the probability of a system being in a particular quantum
state with total energy E is e −E/kBT (divided by a normalization factor), where kB is
Boltzmann’s constant. Compute the relative probability P (1st excited)/P (ground state)
for the system of N distinguishable particles and for the system of N identical bosons.
(Note: this statistical effect is why noninteracting bosons seem to like all being in the
same state.)