1000 Solved Problems in Modern Physics

(Tina Meador) #1

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

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