1000 Solved Problems in Modern Physics

150 3 Quantum Mechanics – II

3.85 (a) Obtain the angular momentum matrices forj= 1
(b) Hence obtain the matrix forJ^2.

3.86 Two angular momenta withj 1 =1 andj 2 = 1 /2 are vectorially added, obtain
the Clebsch – Gordan coefficients.

3.87 The wave function of a particle in a spherically symmetric potential is

Ψ(x,y,z)=C(xy+yz+zx)e−αr

2

Show that the probability is zero for the angular momentuml=0 andl= 1

and that it is unity forl= 2

3.88 Show that the states specified by the wave-functions

ψ 1 =(x+iy)f(r)

ψ 2 =zf(r)

ψ 3 =(x−iy)f(r)

are eigen states of thez-component of angular momentum and obtain the cor-

responding eigen values.

[Adapted from the University of Manchester 1959]

3.89 The Schrodinger wave function for a stationary state of an atom is

ψ=Af(r)sinθcosθeiφ

where (r,θ,φ) are spherical polar coordinates. Find (a) thezcomponent of the

angular momentum of the atom (b) the square of the total angular momentum

of the atom. (You may use the following transformations from Cartesian to

spherical polar coordinates

x

∂

∂y

−y

∂

∂x

=sinφ

∂

∂θ

+cotθcosφ

∂

∂φ

x

∂

∂z

−z

∂

∂x

=−cosφ

∂

∂θ

+cotθsinφ

∂

∂φ

y

∂

∂x

−x

∂

∂y

=−

∂

∂φ

)

[Adapted from the University of Durham 1963]

3.90 The normalized Schrodinger wavefunctions for one of the stationary states of
the hydrogen atom is given in spherical polar coordinates, by

ψ(r,θ,φ)=

(

1

2 a 0

) 3 / 2

1

√

3

r

a 0

exp

(

−

r

2 a 0

)(

3

8 π

) 1 / 2

sinθexp(−iφ)

(a) Find the value of the component of angular momentum along thezaxis

(θ=0)

(b) What is the parity of this wavefunction?

[a 0 is the radius of the first Bohr orbit]

[Adapted from the University of New Castle 1964]