Quantum Mechanics - Angular Momentum

At a given instant of time, a rigid rotator is in the state ψ(θ, ϕ) = √(¾π) sin θ sin ϕ, where θ is the polar angle relative to the z-axis and ϕ is the azimuthal angle. Measurement will find which of the following possible values of the z-component of the angular momentum L_z?

A. 0
B. ħ/2, −ħ/2
C. ħ, −ħ
D. 2ħ, −2ħ
E. ħ, 0, −ħ

(GR9677 #52)

Solution:

The eigenstates of L_z  is L_zY_l^m = mħY_l^m 

Possible values = eigen values of mħ.

L_z = −iħ ∂/∂ϕ → the information about m is contained in the sin ϕ term and it's proportional to e^imϕ:

sin ϕ  = (e^iϕ − e^−iϕ) / 2i

 → m = ±1

Thus, the eigenvalues are ±ħ

Answer: C

Quantum Mechanics - Infinite Potential Well

See Problem 51

The eigenfunctions satisfy the condition

₀∫^aψ_n*(x) ψ_l (x) dx = δ_nl

δ_nl = 1 if n = l, otherwise δ_nl = 0. This is a statement that the eigenfunctions are

A. Solutions to the Schrodinger equation 
B. Orthonormal
C. Bounded 
D. Linearly dependent
E. Symmetric

(GR9277 #52)

Solution:

Orthonormality = orthogonal and normal

₀∫^aψ_n*(x) ψ_l (x) dx = 〈ψ_n|ψ_l〉= δ_nl

Orthogonal,  δ_nl = 0
Normal, δ_nl = 1 if n = l

Answer: B

Electromagnetism - Gauss' Law

A cube has a constant electric potential V on its surface. If there are no charges inside the cube, the potential at the center of the cube is 

A. zero
B. V/8
C. V/6
D. V/2
E. V

(GR8677 #52)

Solution:

Gauss’ Law:  

No charges inside the cube →q_enc = 0 → E = 0

Electric field: E = ∇V
E = 0 → V = constant

Since the potential function has to remain continuous everywhere, at the center of the cube, potential is V.

Answer: E

Condensed Matter - Bravais Lattice

The conventional unit cell of a body-centered cubic Bravais lattice is shown in the figure. The conventional cell has volume a³. What is the volume of the primitive unit cell?

A. a³/8
B. a³/4
C. a³/2
D. a³
E. 2a³

(GR0177 #52)

Solution:

The body-centered cubic Bravais lattice has 2 atoms:
1/8 part of atom at each of its 8 corners + 1 atom in the middle = 2 atoms

Thus, the volume of the primitive unit cell: a³/2

Answer: C