Quantum Mechanics - Spherical Harmonics

A diatomic molecules is initially in the state Ψ(Θ, Φ) = (5Y₁¹ + 3Y₅¹ + 2Y₅⁻¹) / (38)^½ where Y_l^m is a spherical harmonics. If measurements are made of the total angular momentum quantum number l and of azimuthal angular momentum quantum number m, what is the probability of obtaining the results l = 5?

A. 36/1444
B. 9/38
C. 13/38
D. 5/(38)^½
E. 34/38

(GR9677 #33)

Solution:

l = 5 → Y₅¹ and Y₅⁻¹

Probability, P = ∑_i |c_i|² 

P = |3/√38|² + |2/√38|² =  9/38 + 4/38 =  13/38

Answer: C

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 - Probability

A system is know to be in the normalized state described by the wave function



Where  are the spherical harmonics. The probability of finding the system in a state with azimuthal orbital quantum number m = 3 is

A. 0
B. 1/15
C. 1/6
D. 1/3
E. 13/15

(GR9277 #28)

Solution:

For the state  ,  probability is  

States with  m = 3 → and  with  c₁ = 5/√30 and c₂ = 1/√30

Therefore,



Answer: E