__
__

The state of the system is |

(a) Calculate the mean value <* J*> of the
angular momentum in terms of

(b) Give the expressions for <*J _{x}^{2}*>, <

- Solution:
(a) In the { |1>, |0>, |-1> } basis the matrices of

*J*, and_{x}, J_{y}*J*are_{z}.

The vector |

*y*> is represented by the column matrix .Assume |

*y*> is normalized and*|a|*1.^{2}+|b|^{2}+|c|^{2}=.

Similarly,

.

(b) The matrices for*J*,_{x}^{2}*J*, and_{y}^{2}*J*are_{z}^{2}.

,

.

.

Consider a system whose four-dimensional state space is spanned
by a basis of common eigenvectors of* J ^{2}* and

(a) Express the
common eigenstates of *J ^{2}* and

(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 *J ^{2}* and

ii) Calculate the mean value of *J _{z} *and the
probabilities of the various possible results.

iii) Answer the same questions for *J ^{2}* and

iv) If *J _{z}^{2}* 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*and_{x}, J_{y},J_{z},J_{+},J_{-},*J*are^{2},

.

To find the eigenvalues

*b*of*J*we require ._{x}.

The eigenvalue

*b=0*is twofold degenerate.The associated eigenvectors are

*J*^{2}*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|*1.^{2}+|b|^{2}+|c|^{2}+|d|^{2}=

- (i) .
- (ii) .
- (iii)
.

as on the previous line.

.

.

- iv) The possible results of measuring
*J*are 0 and ._{z}^{2}.

**Problem:**

For a simple particle moving in space, show that the wave function ψ_{lm}(**r**) = x^{2} + y^{2}
- 2z^{2} represents a simultaneous eigenstate of L^{2} and L_{z}
with eigenvalues l(l + 1)ħ^{2}
and mħ. Determine l and m. Find a
function with the same eigenvalue for L^{2} 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 L^{2}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**) = x^{2}+ y^{2}- 2z^{2}=

r^{2}sin^{2}θcos^{2}φ + r^{2}sin^{2}θsin^{2}φ - 2r^{2}cos^{2}θ = r^{2}(sin^{2}θ - 2cos^{2}θ)

in spherical coordinates.

Express ψ_{lm}(**r**) in terms of the spherical harmonics.

ψ_{lm}(**r**) = r^{2}(1 - cos^{2}θ - 2cos^{2}θ) = r^{2}(1 - 3cos^{2}θ)

= -r^{2}(16π/5)^{½}Y_{20}(θ,φ).

[Formulas for some of the spherical harmonics:

Y_{00}= (4π)^{-½}, Y_{1±1}= ∓(3/8π)^{½}sinθ exp(±iφ), Y_{10}= (3/4π)^{½}cosθ,

Y_{2±2}= (15/32π)^{½}sin^{2}θ exp(±i2φ), Y_{2±1}= ∓(15/8π)^{½}sinθ cosθ exp(±iφ),

Y_{20}= (5/16π)^{½}(3cos^{2}θ - 1).]

ψ_{lm}(**r**) is an eigenfunction of L_{z}with eigenvalue 0 and an eigenfunction of L^{2}with eigenvalue 6ħ^{2}(l = 2). For l = 2 the maximum possible eigenvalue of L_{z}is mħ = 2ħ. The corresponding eigenfunction is

Cr^{2}Y_{22}(θ,φ) = Cr^{2}(15/32π)^{½}sin^{2}θ exp(±i2φ).

**Problem:**

- 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*..

The units are chosen such that =*2m*=1.
Find the complete set of
eigenvalues and eigenfunctions.

- Solution:
.

Try . .

We need .

.

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- Solution:
.

.

.

.

The only matrix elements that can be nonzero are the matrix elements of .

Therefore .

.