Physics 4410 Sample Midterm
1) (10 points) Consider a hydrogen atom in its ground hyperfine state with total spin f
state lies an energy
above the f
1 . This
0 hyperfine state. Now apply a weak magnetic field B ,
which adds the Hamiltonian
H'
2
B
S B
To
Physics 4410 Quantum Mechanics 2 2014
Problem Set #1
Due Friday, January 24 at 2 pm
Problem 1.1: Larmor precession. (15 points)
Consider a spin-1/2 particle with gyromagnetic ratio in a uniform magnetic eld B = B z .
In class we found the solution for all
Physics 4410 Quantum Mechanics 2 Spring 2014
Problem Set #6
Due Friday, February 28 at 2pm
Problem 6.1: Generic good eigenstates. (15 points)
If you believe the examples of previous homework, it seems like whenever there is degena
eracy between states |A
Physics 4410 Quantum Mechanics 2 Spring 2014
Problem Set #4
Due Friday, February 14 at 2 pm
Problem 4.1: Two-dimensional harmonic oscillator. (15 points)
Consider as the unperturbed Hamiltonian the two-dimensional harmonic oscillator:
H0 =
p2
1
p2
x
+ y +
Physics 4410 Quantum Mechanics 2 Spring 2014
Problem Set #3
Due Friday, February 7 at 2pm
Problem 3.1: Second order energy. (15 points)
Consider a perturbed Hamiltonian
H = H 0 + H = H 0 + H ,
(1)
0
0
where as usual H 0 is the unperturbed part, with eigen
Physics 4410 Quantum Mechanics 2 Spring 2014
Problem Set #5
Due Friday, February 21 at 2 pm
Problem 5.1: Fine structure calculations. (10 points)
The rst-order correction to the energy levels of the hydrogen atom was found to be,
1
Enj =
0
(En )2
4n
3
2
2
Physics 4410 Quantum Mechanics 2 Spring 2014
Problem Set #9
Due Monday, April 7 at 2pm
Problem 9.1: Semiclassical WKB derivation. (15 points)
The WKB approximation is generally termed a semiclassical limit. For another way to
understand this, we will deri
Physics 4410 Quantum Mechanics 2 Spring 2014
Problem Set #10
Due Monday, April 14 at 2pm
Problem 10.1: Connection formulas. (30 points)
WKB approximations can be made in both classically allowed and classically forbidden
regions, but break down at the cla
Physics 4410 Quantum Mechanics 2 Spring 2014
Problem Set #7
Due Friday, March 7 at 2pm
Problem 7.1: Strong and weak eld Zeeman. (20 points)
In this problem, consider only the n = 2 level of a hydrogen atom placed in a uniform static
magnetic eld B = B z .
Physics 4410 Quantum Mechanics 2 Spring 2014
Problem Set #11
Due Monday, April 21 at 2pm
Problem 11.1: Rotating wave potential. (20 points)
The sinusoidal perturbation Hba = Vba cos t can cause both absorption up to a higher energy
level and stimulated em
Physics 4410 Quantum Mechanics 2 2014
Problem Set #2
Due Friday, January 31 at 2 pm
Problem 2.1:
Here we will explore some properties of angular momentum eigenstates (20 pts: 10 each):
(a) Show that for eigenstates of Lz , denoted as |l, m , the expecta
Physics 4410 Quantum Mechanics 2 Spring 2014
Problem Set #12
Due Monday, April 28 at 2pm
Problem 12.1: Rutherford scattering. (15 points)
The scattering of a charged particle of charge q against a charged particle with charge Q
(the second of which we wil
Physics 4410 Quantum Mechanics 2 Spring 2014
Problem Set #8
Due Monday, March 31 at 2pm
Problem 8.1: Two-state variational method. (20 points)
Consider a quantum system with only two states; for example, they could be the spin states
of an electron. Call