PHY4604–Introduction to Quantum Mechanics
Fall 2004
Problem Set 8
Oct. 25, 2004
Due: Nov. 8, 2004
Reading: Griffiths Chapter 4
1. Angular momentum operators. Consider a particle that moves in 3 dimensions with wave function ψ. Use operator methods as in sec. 4.3 of Griffiths to
prove that if ψ has total angular momentum quantum number ` = 0, then ψ
satisfies
Lα ψ = 0
for all three components α = x, y, z of the angular momentum operator L.
2. Spherical Harmonics. Use the results of sec. 4.3 to find the spherical harmonics Y11 , Y10 , and Y1−1 .
3. Angular momentum eigenstates of SHO. Let the functions φn (x) be the
1D simple harmonic oscillator eigenfunctions with energy En = h̄ω(n+1/2). We
know from the previous HW that the ground state of the 3D simple harmonic
oscillator is
ψ000 (r) = φ0 (x)φ0 (y)φ0 (z),
while the three degenerate lowest-lying excited states are
ψ100 (r) = φ1 (x)φ0 (y)φ0 (z)
ψ010 (r) = φ0 (x)φ1 (y)φ0 (z)
ψ001 (r) = φ0 (x)φ0 (y)φ1 (z)
2 /x2
0
Remember φ0 is a Gaussian ∝ e−x
2 /x2
0
and φ1 ∝ xe−x
.
Use the results of the previous 2 problems to show that the ground state ψ000
has angular momentum quantum numbers ` = m = 0, that ψ001 has quantum
numbers ` = 1, m = 0, and that ψ100 and ψ010 are linear combinations of
eigenfunctions with ` = 1, m = ±1.
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