Day 21 In-Class Activity

Particle on a sphere - probabilities and expectation values

A particle on a sphere is in the following normalized eigenstate:

| ψ ⟩ = \frac{1}{\sqrt{2}} | 1, - 1 ⟩ + \frac{1}{\sqrt{3}} | 10 ⟩ + \frac{i}{\sqrt{6}} | 00 ⟩

Question: What is the probability that a measurement of $L_{z}$ will yield 2 $ℏ$ ? $- ℏ$ ? $0 ℏ$ ?
- Discussion: How does degeneracy feature in this determination, if at all? If it doesn’t feature, how could it?
Question: What is the expectation value of $L_{z}$ in this state?
- Discussion: How does degeneracy feature in this determination, if at all? If it doesn’t feature, how could it?
Question: What is the expectation value of $L^{2}$ in this state?
- Discussion: How does degeneracy feature in this determination, if at all? If it doesn’t feature, how could it?
Question: What is the expectation value of the energy in this state?
- Discussion: How does degeneracy feature in this determination, if at all? If it doesn’t feature, how could it?
Question: What is the expectation value of $L_{y}$ in this state? Hint: think spin-1
- Discussion: How does degeneracy feature in this determination, if at all? If it doesn’t feature, how could it?