1000 Solved Problems in Modern Physics

3.2 Problems 149

3.2.6 Angular Momentum ................................

3.77 Given thatL=r×p, show that [Lx,Ly]=iLz

3.78 The spin wave function of two electrons is (x↑x↓–x↓x↑)/

√

2. What is
   the eigen value ofS 1 .S 2 ?S 1 andS 2 are spin operators of 1 and 2 electrons

3.79 Show that for proton – neutron system
σp.σn=−3 for singlet state
=1 for triplet state

3.80 Write down an expression for thez-component of angular momentum,Lz,ofa
particle moving in the (x, y) plane in terms of its linear momentum components
pxandpy.
Using the operator correspondencepx=−i

∂

∂x

etc., show that

Lz=−i

(

x

∂

∂y

−y

∂

∂x

)

Hence show thatLz =−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 thez-component of angular momentum is

quantized with eigen value, wheremis an integer.

3.81 Show that the operatorsLxand Ly in the spherical polar coordinates are
given by
Lx
i

=sinφ

∂

∂θ

+cotθcosφ

∂

∂φ

Ly

i

=−cosφ

∂

∂θ

+cotθsinφ

∂

∂φ

3.82 Using the commutator [Lx,Ly]=iLz, and its cyclic variants, prove that
total angular momentum squared and the individual components of angular
momentum commute, i.e [L^2 ,Lx]=0etc.

3.83 Show that in the spherical polar coordinates

L^2

(i)^2

=

∂^2

∂θ^2

+

(

1

sin^2 θ

)

∂^2

∂φ^2

+cotθ

∂

∂θ

And show that in the expression for∇^2 in spherical polar coordinates the

angular terms are proportional toL^2.

3.84 (a) Obtain the angular momentum matrices forj= 1 /2 particles
(b) Hence Obtain the matrix forJ^2.