Mathematical Methods for Physics and Engineering : A Comprehensive Guide

19.3 EXERCISES

Now evaluate the expectation value using the eigenvalue properties ofH,namely

H|r〉=Er|r〉, and deduce thesum rule for oscillation strengths,

∑∞

r=0

(Er−E 0 )|〈r|x| 0 〉|^2 =

N
^2

2 m

.

19.9 By considering the function

F(λ)=exp(λA)Bexp(−λA),

whereAandBare linear operators andλis a parameter, and finding its

derivatives with respect toλ, prove that

eABe−A=B+[A, B]+

1

2!

[A,[A, B]]+

1

3!

[A,[A,[A, B]]]+···.

Use this result to express

exp

(

iLxθ

)

Lyexp

(

−iLxθ

)

as a linear combination of the angular momentum operatorsLx,LyandLz.
19.10 For a system containing more than one particle, the total angular momentumJ
and its components are represented byoperators that have completely analogous
commutation relations to those for the operators for a single particle, i.e.J^2 has
eigenvaluej(j+1)
^2 andJzhas eigenvaluemj
for the state|j, mj〉. The usual
orthonormality relationship〈j′,m′j|j, mj〉=δj′jδm′jmjis also valid.
A system consists of two (distinguishable) particlesAandB.ParticleAis in
an= 3 state and can have state functions of the form|A, 3 ,mA〉, whilstBis
in an= 2 state with possible state functions|B, 2 ,mB〉. The range of possible
values forjis| 3 − 2 |≤j≤|3+2|,i.e.1≤j≤5, and the overall state function
canbewrittenas

|j, mj〉=

∑

mA+mB=mj

C

jmj

mAmB|A,^3 ,mA〉|B,^2 ,mB〉.

The numerical coefficientsCjmmAjmBare known asClebsch–Gordoncoefficients.

Assume (as can be shown) that the ladder operatorsU(AB)andD(AB)for

the system can be written asU(A)+U(B)andD(A)+D(B), respectively, and

that they lead to relationships equivalent to (19.34) and (19.35) withreplaced

byjandmbymj.

(a) Apply the operators to the (obvious) relationship

|AB, 5 , 5 〉=|A, 3 , 3 〉|B, 2 , 2 〉

to show that

|AB, 5 , 4 〉=

√

6

10 |A,^3 ,^2 〉|B,^2 ,^2 〉+

√

4

10 |A,^3 ,^3 〉|B,^2 ,^1 〉.

(b) Find, to within an overall sign, the real coefficientscanddin the expansion

|AB, 4 , 4 〉=c|A, 3 , 2 〉|B, 2 , 2 〉+d|A, 3 , 3 〉|B, 2 , 1 〉

by requiring it to be orthogonal to|AB, 5 , 4 〉. Check your answer by considering

U(AB)|AB, 4 , 4 〉.

(c) Find, to within an overall sign, and as efficiently as possible, an expression

for|AB, 4 ,− 3 〉as a sum of products of the form|A, 3 ,mA〉|B, 2 ,mB〉.