1000 Solved Problems in Modern Physics

3.2 Problems 151

3.91 The normalized 2peigen functions of hydrogen atom are

1

√

π

1

(2a 0 )^3 /^2

e−r/^2 a^0

r

2 a 0

sinθeiΦ,

1

√

π

1

(2a 0 )^3 /^2

e−r/^2 a^0

r

2 a 0

cosθ,

1

√

π

1

(2a 0 )^3 /^2

e−r/^2 a^0

r

2 a 0

sinθe−iΦ,form=+ 1 , 0 ,−1 respectively.

Apply the raising operatorL+=Lx+iLyand lowering operator to show that

the states withm=±2 do not exist.

3.92 How can nuclear spin be measured from the rotational spectra of diatomic
molecules?

3.93 An electron is described by the following angular wave function

u(θ,φ)=

1

4

√

15

π

sin^2 θcos 2φ

Re-expressuin terms of spherical harmonics given below. Hence give the

probability that a measurement will yield the eigen value ofL^2 equal to 6^2

You may use the following:

Y 20 (θ,φ)=

√

5

16 π

(

3 cos^2 θ− 1

)

Y 2 ± 1 (θ,φ)=

√

15

8 π

sinθcosθexp(±iφ)

Y 2 ± 2 (θ,φ)=

√

15

32 π

sin^2 θexp(± 2 iφ)

[University College, London]

3.94 Given that the complete wave function of a hydrogen-like atom in a particular

state isψ(r,θ,φ)=Nr^2 exp

(

−Zr 3 a 0

)

sin^2 θe^2 iφdetermine the eigen value of

Lz, the third component of the angular momentum operator.

3.95 Consider an electron in a state described by the wave function

ψ=

1

√

4 π

(cosθ+sinθeiφ)f(r)

where

∫∞

0

|f(r)|^2 r^2 dr= 1

(a) Show that the possible values ofLzare+and zero

(b) Show that the probability for the occurrence of theLzvalues in (a) is 2/3

and 1/3, respectively.

3.96 Show that (a) [Jz,J+]=J+(b)J+|jm>=Cjm+|j,m+ 1 >(c) [Jx,Jy]
=iJz