Name:
Exam 2 - PHY 4604 - Fall 2013
November 14, 2013
8:20-10:20PM, NPB 1002
Directions: Please clear your desk of everything except for pencils and pens. The exam is
closed book, and you are not allowed calculators or formula sheets. Leave substantial space
between you and your neighbor. Show your work on the space provided on the exam. I can
provide additional scratch paper if needed.
Unless otherwise noted, all parts (a), (b), . . . are worth 5 points, and the entire exam is
100 points.
Y00
Y11 (θ, ϕ)
1
=
2
s
−1
=
2
1
Y10 (θ, ϕ) =
2
1
π
s
s
3
· eiϕ · sin θ
2π
3
· cos θ
π
s
1 3
· e−iϕ · sin θ
Y1−1 (θ, ϕ) =
2 2π
s
1 15 2iϕ
2
Y2 (θ, ϕ) =
· e · sin2 θ
4 2π
−1
Y21 (θ, ϕ) =
2
s
15 iϕ
· e · sin θ · cos θ
2π
1
=
4
s
5
· (3 cos2 θ − 1)
π
1
Y2−1 (θ, ϕ) =
2
s
15 −iϕ
·e
· sin θ · cos θ
2π
1
=
4
s
15 −2iϕ
·e
· sin2 θ
2π
Y20 (θ, ϕ)
Y2−2 (θ, ϕ)
1
1. Short answer section
(a) What are the momentum eigenvectors, fp (x)? How are they normalized? Specifically, what is hfp |fp′ i?
(b) What is the general uncertainty relation for two operators, A and B, which do
not commute?
(c) If A, B, and C are Hermitian operators, what is the Hermitian conjugate of
A(B + iC)?
(d) How is the radial part of the wave function, R(r), related to u(r)?
2
2. Formalism
Consider the operator A below expressed as a matrix.
3 4
4 −3
A=
(a) What are the eigenvalues of A?
(b) What are the normalized eigenvectors of A?
3
!
(c) Labeling the eigenvectors as |ψa i and |ψb i, what are the projection operators,
|ψa ihψa | and |ψb ihψb |, for the eigenstates?
(d) Can one find simultaneous eigenvectors of A and B, where
0 1
1 0
B=
4
!
? Why?
3. Measurements
The Hamiltonian of a three state system is given by


0 1 1


H = h̄ω  1 0 1  .
1 1 0
It has eigenstates
e2πi/3
1
e−2πi/3
1
1 
1





1
1
|ψa i = √ 
 , |ψb i = √  1  , |ψc i = √ 
,
3 e−2πi/3
3 1
3
2πi/3
e






and corresponding eigenvalues Ea = Ec = −h̄ω and Eb = 2h̄ω.
(a) If the system is initially in the state


1


|ψi i =  0 
0
(1)
and an energy measurement is made, what are the possible outcomes of the measurement and probabilities of those outcomes?
(b) Suppose instead of making an energy measurement the system is evolved in time
for a time t. What is |ψ(t)i?
5
(c) At time t what is the probability of being in the state, |ψi i?
(d) Suppose that instead of part (c) energy measurement is made at time t. What are
the possible outcomes of an energy measurement and the probabilities of those
outcomes?
6
4. Angular Momentum
(a) What is the commutator [Lx Ly , Lz ] expressed as a product of two operators?
(b) What is the matrix element h2, −1|Lx Ly |2, −1i?
7
(c) What is the matrix element h2, 1|Lx Ly |2, −1i?
√
(d) A wavefunction has the form R(r)x2 , where r = x2 + y 2 + z 2 . What are the
possible outcomes with non-zero probabilities of an L2 measurement and of an Lz
measurement?
8
5. Radial Schrodinger Equation
In the following consider the spherically symmetric potential V (r) = Vo > 0 for r1 <
r < r2 and V (r) = ∞ elsewhere, i.e. for 0 < r < r1 and for r > r2 .
(a) Solve the radial Schrodinger equation for u(r) in the region r1 < r < r2 with the
above V (r) and l = 0. Assume that E > Vo .
(b) Because the potential is infinity for r < r1 and r > r2 there are boundary conditions for u(r) just as for the one dimensional infinite well. What are the boundary
conditions?
9
(c) Apply the boundary conditions to find the normalized solutions for u(r).
(d) What are the energies of the eigenstates?
10