3.2 Problems
149
3.2.6 Angular Momentum
3.77 Given that L = r × p, show that [L x , L y ] = iL z
3.78 The spin wave function of two electrons is (x ↑ x ↓ –x ↓ x ↑)/
√
2. What is
the eigen value of S 1 .S 2 ? S 1 and S 2 are spin operators of 1 and 2 electrons
3.79 Show that for proton – neutron system
σ p .σ n = −3 for singlet state
= 1 for triplet state
3.80 Write down an expression for the z-component of angular momentum, L z , of a
particle moving in the (x, y) plane in terms of its linear momentum components
p x and p y .
Using the operator correspondence p x = −i
∂
∂ x
etc., show that
L z = −i
x
∂
∂ y
− y
∂
∂ x
Hence show that L z = −i
∂
∂ϕ
, where the coordinates (x,y) and (r, ϕ) are
related in the usual way.
Assuming that the wavefunction for this particle can be written in the form
ψ(r, ϕ) = R(r )Φ(ϕ) show that the z-component of angular momentum is
quantized with eigen value , where m is an integer.
3.81 Show that the operators L x and L y in the spherical polar coordinates are
given by
L x
i
= sin ϕ
∂
∂θ
+ cot θ cos ϕ
∂
∂ϕ
L y
i
= − cos ϕ
∂
∂θ
+ cot θ sin ϕ
∂
∂ϕ
3.82 Using the commutator [L x , L y ] = i L z , and its cyclic variants, prove that
total angular momentum squared and the individual components of angular
momentum commute, i.e [L
2
, L x ] = 0 etc.
3.83 Show that in the spherical polar coordinates
L
2
(i) 2 =
∂
2
∂θ 2 +
1
sin
2
θ
∂
2
∂ϕ 2 + cot θ
∂
∂θ
And show that in the expression for ∇
2 in spherical polar coordinates the
angular terms are proportional to L
2 .
3.84 (a) Obtain the angular momentum matrices for j = 1/2 particles
(b) Hence Obtain the matrix for J
2 .