Problems

Problems:

Consider a system of angular momentum j=1, whose state space is spanned by the common eigenvectors of J² and J_z ,{ |1>, |0>, |-1> }, with eigenvalues of J_z of

, 0, and -

respectively.
The state of the system is |y>=a|1> + b|0> + c|-1>.

(a) Calculate the mean value <J> of the angular momentum in terms of a, b, and c.

(b) Give the expressions for <J_x²>, <J_y²>, and <J_z²> in terms of the same quantities.

Solution:
(a) In the { |1>, |0>, |-1> } basis the matrices of J_x, J_y, and J_z are
.

The vector |y> is represented by the column matrix .

Assume |y> is normalized and |a|²+|b|²+|c|²=1.

.

Similarly,

.
(b) The matrices for J_x², J_y², and J_z² are
.

,

.

.

Consider a system whose four-dimensional state space is spanned by a basis of common eigenvectors of J² and J_z , { |1,1> |1,0>, |1,-1>, |0,0> } with j=0 or 1.

(a) Express the common eigenstates of J² and J_x in terms of the eigenstates of J² and J_z.

(b) Consider a system in the normalized state |y>=a|1,1> + b|1,0> + c|1,-1> + d|0,0>.

i) What is the probability of finding 2 and if J² and J_z are measured simultaneously?

ii) Calculate the mean value of J_z and the probabilities of the various possible results.

iii) Answer the same questions for J² and J_x .

iv) If J_z² is measured, what are the possible results, their probabilities and their mean value?

Solution:

(a) The basis vectors are { |1,1> |1,0>, |1,-1>, |0,0> }. In this basis the matrices for J_x, J_y,J_z,J₊,J_-, and J² are

To find the eigenvalues b of J_x we require .

The eigenvalue b=0 is twofold degenerate.

The associated eigenvectors are

	J²	J_x
b=0: \|0,0>	0	0
	2	0
	2
	2	-

(b) |y>=a|1,1> + b|1,0> + c|1,-1> + d|0,0> = .
|a|²+|b|²+|c|²+|d|²=1.

(i) .
(ii) .
(iii)

.

as on the previous line.

.

.
iv) The possible results of measuring J_z² are 0 and .
.

Problem:

For a simple particle moving in space, show that the wave function ψ_lm(r) = x² + y² - 2z² represents a simultaneous eigenstate of L² and L_z with eigenvalues l(l + 1)ħ² and mħ. Determine l and m. Find a function with the same eigenvalue for L² and the maximum possible eigenvalue for L_z.

Solution:

Concepts:
The eigenfunctions of the orbital angular momentum operator, the spherical harmonics
Reasoning:
The common eigenfunctions of L² and L_z are the spherical harmonics. We have to write the given wave functions in terms of the spherical harmonics.
Details of the calculation:
ψ_lm(r) = x² + y² - 2z² =
r²sin²θcos²φ + r²sin²θsin²φ - 2r²cos²θ = r²(sin²θ - 2cos²θ)
in spherical coordinates.
Express ψ_lm(r) in terms of the spherical harmonics.
ψ_lm(r) = r²(1 - cos²θ - 2cos²θ) = r²(1 - 3cos²θ)
= -r²(16π/5)^½Y₂₀(θ,φ).
[Formulas for some of the spherical harmonics:
Y₀₀ = (4π)^-½, Y_1±1 = ∓(3/8π)^½sinθ exp(±iφ), Y₁₀ = (3/4π)^½cosθ,
Y_2±2 = (15/32π)^½sin²θ exp(±i2φ), Y_2±1 = ∓(15/8π)^½sinθ cosθ exp(±iφ),
Y₂₀ = (5/16π)^½(3cos²θ - 1).]
ψ_lm(r) is an eigenfunction of L_z with eigenvalue 0 and an eigenfunction of L² with eigenvalue 6ħ² (l = 2). For l = 2 the maximum possible eigenvalue of L_z is mħ = 2ħ. The corresponding eigenfunction is
Cr²Y₂₂(θ,φ) = Cr²(15/32π)^½sin²θ exp(±i2φ).

Problem:

A system has a wave function

, with a real. If L_z and L² are measured, what are the probabilities of finding 0 and 2

Solution:
Express the wave function in terms of spherical harmonics.
.

.

.

.

for properly chosen N.

.

Let P(l,m) denote the probability of finding the eigenvalues l(l+1) and m .

.