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