Cap 6 - 6.1, 6.2, d, 6.5a, d, d

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11/01/14 Problems
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Problems:
Consider a system of angular momentum j=1, whose state space is spanned by the common
eigenvectors of J2 and Jz ,{ |1>, |0>, |-1> }, with eigenvalues of Jz 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 <Jx
2>, <Jy
2>, and <Jz
2> in terms of the same quantities.
Solution:
(a) In the { |1>, |0>, |-1> } basis the matrices of Jx, Jy, and Jz 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 Jx
2, Jy
2, and Jz
2 are
.
,
.
11/01/14 Problems
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.
 
Consider a system whose four-dimensional state space is spanned by a basis of common
eigenvectors of J2 and Jz , { |1,1> |1,0>, |1,-1>, |0,0> } with j=0 or 1.
(a) Express the common eigenstates of J2 and Jx in terms of the eigenstates of J
2 and Jz.
(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 Jz are measured simultaneously?
 ii) Calculate the mean value of Jz and the probabilities of the various possible results.
 iii) Answer the same questions for J2 and Jx .
 iv) If Jz
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 Jx, Jy,Jz,J+,J-, and J
2 are
,
.
To find the eigenvalues b of Jx we require .
 .
The eigenvalue b=0 is twofold degenerate.
The associated eigenvectors are
11/01/14 Problems
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 J2 Jx
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 Jz
2 are 0 and .
.
For a simple particle moving in space, show that the wave function ylm(r)=x
2+y2-2z2
represents a simultaneous eigenstate of L2 and Lz with eigenvalues l(l+1) and m .
Determine l and m. Find a function with the same eigenvalue for L2 and the maximum
possible eigenvalue for Lz.
Solution:
.
11/01/14 Problems
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Express ylm(r) in terms of the spherical harmonics.
.
ylm(r) is an eigenfunction of Lz with eigenvalue 0 and an eigenfunction of L
2 with
eigenvalue 6 ( l=2 ). For l=2 the maximum possible eigenvalue of Lz is m =2 .
The corresponding eigenfunction is .
 
A system has a wave function , with a real. If Lz and L
2 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 .
.
 
Consider a charged particle on a ring of unit radius with Hamiltonian .
The units are chosen such that =2m=1. Find the complete set of eigenvalues and
11/01/14 Problems
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eigenfunctions.
Solution:
.
Try . .
We need .
.
 
A particle is known to be in an eigenstate of L2 and Lz. Prove that the expectation values
satisfy
.
Solution:
.
.
.
.
The only matrix elements that can be nonzero are the matrix elements of 
.
Therefore .
.