130_notes.dvi

14 Angular Momentum

14.1 Rotational Symmetry

If the potential only depends on the distance between two particles,

V(~r) =V(r)

the Hamiltonian hasRotational Symmetry. This is true for the coulomb (and gravitational)
potential as well as many others. We know from classical mechanicsthat these are important
problems.

If a the Hamiltonian has rotational symmetry, we can show that theAngular Momentum operators
commute with the Hamiltonian (See section 14.4.1).

[H,Li] = 0

We therefore expect each component ofL~to be conserved.

We will not be able to label our states with the quantum numbers for the three components of
angular momentum. Recall that we are looking fora set of mutually commuting operators
to label our energy eigenstates. We actually want two operators plusHto give us three quantum
numbers for states in three dimensions.

The components of angular momentum do not commute (See section14.4.2) with each other

[Lx,Ly] =i ̄hLz

12.

[Li,Lj] =i ̄hǫijkLk

but thesquare of the angular momentum commuteswith any of the components

[L^2 ,Li] = 0

These commutators lead us to choose themutually commuting set of operatorsto beH,L^2 ,
andLz. We could have chosen any component, however, it is most convenient to chooseLzgiven
the standard definition of spherical coordinates.

The Schr ̈odinger equation can now be rewritten (See section 14.4.3) with only radial derivatives
andL^2.

− ̄h^2

2 μ

∇^2 uE(~r) +V(r)uE(~r) = EuE(~r)

− ̄h^2

2 μ

[

1

r^2

(

r

∂

∂r

) 2

+

1

r

∂

∂r

−

L^2

̄h^2 r^2

]

uE(~r) +V(r)uE(~r) = EuE(~r)