Day 21 In-Class Activity

Particle on a sphere - probabilities and expectation values

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

\[\ket{\psi} = \dfrac{1}{\sqrt{2}}\ket{1,-1} + \dfrac{1}{\sqrt{3}}\ket{10} + \dfrac{i}{\sqrt{6}}\ket{00}\]

1. Question: What is the probability that a measurement of $L_z$ will yield 2$\hbar$? $-\hbar$? $0\hbar$?
   • Discussion: How does degeneracy feature in this determination, if at all? If it doesn’t feature, how could it?
2. 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?
3. Question: What is the expectation value of $\mathbf{L}^2$ in this state?
   • Discussion: How does degeneracy feature in this determination, if at all? If it doesn’t feature, how could it?
4. 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?
5. 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?