150
3 Quantum Mechanics – II
3.85 (a) Obtain the angular momentum matrices for j = 1
(b) Hence obtain the matrix for J
2 .
3.86 Two angular momenta with j 1 = 1 and j 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 (x y + yz + zx)e
−αr
2
Show that the probability is zero for the angular momentum l = 0 and l = 1
and that it is unity for l = 2
3.88 Show that the states specified by the wave-functions
ψ 1 = (x + iy) f (r )
ψ 2 = z f (r )
ψ 3 = (x − iy) f (r )
are eigen states of the z-component of angular momentum and obtain the corresponding eigen values.
[Adapted from the University of Manchester 1959]
3.89 The Schrodinger wave function for a stationary state of an atom is
ψ = A f (r ) sin θ cos θe
iϕ
where (r, θ, ϕ) are spherical polar coordinates. Find (a) the z component 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
= sin ϕ
∂
∂θ
+ cot θ cos ϕ
∂
∂ϕ
x
∂
∂z
− z
∂
∂ x
= − cos ϕ
∂
∂θ
+ cot θ sin ϕ
∂
∂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
2a 0
3/2 1
√
3
r
a 0
exp
−
r
2a 0
3
8π
1/2
sin θ exp (−iϕ)
(a) Find the value of the component of angular momentum along the z axis
(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]