Question

A particle of mass $m$ is confined to the surface of a sphere of radius $R$, but is otherwise free. It is put in the normalized state: $\psi(\theta, \phi) = \begin{cases} A, & 0 < \theta < \frac{\pi}{2}, \quad 0 < \phi < \frac{\pi}{2}, \\ 0, & \text{otherwise.} \end{cases}$ (a) Determine A. (b) Find the probability of measuring the square of its angular momentum to be $2\hbar^2$. (c) A small perturbation of the form $\hat{H'} = \eta \cos \theta \sin \phi$ is applied. How does this per- turbation cause the $l = 1$ energy level to split? Give energy corrections to lowest order in perturbation theory, and find the linear combinations of unperturbed states for which the perturbation is diagonal.

          A particle of mass $m$ is confined to the surface of a sphere of
radius $R$, but is otherwise free. It is put in the normalized state:
$\psi(\theta, \phi) = \begin{cases} A, & 0 < \theta < \frac{\pi}{2}, \quad 0 < \phi < \frac{\pi}{2}, \\ 0, & \text{otherwise.} \end{cases}$
(a) Determine A.
(b) Find the probability of measuring the square of its angular momentum to be $2\hbar^2$.
(c) A small perturbation of the form $\hat{H'} = \eta \cos \theta \sin \phi$ is applied. How does this per-
turbation cause the $l = 1$ energy level to split? Give energy corrections to lowest
order in perturbation theory, and find the linear combinations of unperturbed
states for which the perturbation is diagonal.

A particle of mass m is confined to the surface of a sphere of
radius R, but is otherwise free. It is put in the normalized state:
ψ(θ, ϕ) = A, 0 < θ < (π)/(2), 0 < ϕ < (π)/(2),
0, otherwise.
(a) Determine A.
(b) Find the probability of measuring the square of its angular momentum to be 2ħ^2.
(c) A small perturbation of the form Ĥ'̂ = ηcosθsinϕ is applied. How does this per-
turbation cause the l = 1 energy level to split? Give energy corrections to lowest
order in perturbation theory, and find the linear combinations of unperturbed
states for which the perturbation is diagonal.

Added by George W.

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