The operator * J*, whose Cartesian components satisfy the commutation
relations is defined as an angular
momentum operator. For such an operator we have [

The index *j* can take on only integral and half integral positive values.
Which
integral and half integral values of *j* are allowed depends on the exact nature of
the physical problem. For a given *j* the index* m* can take on one of *2j+*1
possible values, *m=-j,-j+1,... ,j-1,j*.

We define the operators *J _{+}=J_{x}+iJ_{y}* and

The operators *J±* operating on the basis states {|*k,j,m*>} yield

.

The operator * L*=

(a) What are the possible values for *L _{y}*?

(b) Calculate the probabilities for each of the possible values given in part (a).

- Solution:
(a) Let the vectors {|

*k,1,1*>,|*k,1,0*>,|*k,l1,-1*>} be the eigenvectors of*L*with eigenvalues respectively, which span the subspace E(_{z}*k,l*) with*l*=1. After the measurement of*L*and^{2}*L*the particle is in state |_{z}*k,1,1*>. E(*k,l*) is invariant under. The matrix of**L***L*in the above basis is_{y}.

It can be constructed using and ,

.

The eigenvalues of

*L*are ,_{y}where .

We have .

For the eigenvalue 0 we have

.

For the eigenvalue we have

.

For the eigenvalue we have

.

(b) ,,

.

- Note:
*k*may denote the energy of the free particle. If the particle does not have a well defined energy then the state of the particle after the measurement of*L*and^{2}*L*is_{z},

with

.

- Note:

Let *|y>* be an arbitrary state vector, and let the
result of operating with *L _{x}* on

.

.

.

In coordinate representation the operator *L _{x}* is therefore written as

.

Similarly,

.

It is often more convenient to work in spherical coordinates, *r*,* q, f*;

are the relationships
between Cartesian coordinates and spherical coordinates. Each spherical coordinate is a
function of *x, y*, and *z* and each Cartesian coordinate is a function of *r*,* q, f*.
We can derive the relationships between the partial
derivatives

.

.

Inverting the matrix we find

.

Therefore

Similarly,

.

This yields *L ^{2}=L_{x}^{2}+L_{y}^{2}+L_{z}^{2}*,

.

L_{+}=L_{x}+iL_{y}, L_{-}=L_{x}-iL_{y},

.

In coordinate representation the eigenfunction associated with the eigenvalues and of *L ^{2}* and

,

with* l* integral or half integral and *m=-l,-l+1,...l-1,l*.

The variable* r* does not appear in any differential operator. We may therefore
separate variables and look for solutions of the form

. We will show that the are unique.

To have a normalized wave function we need

.

It is convenient to normalize in the following way

, .

The normalized common eigenfunctions of *L ^{2}* and

- (a) .
Separation of variables is possible. We look for solutions of the form .

Wave functions must be continuous at all points in space. We therefore need

.

In the case of orbital angular momentum the nature of the physical problem dictates that*m*must be an integer, and therefore*l*must be an integer. - (b)

from the general theory..

- .
.

*Y*is uniquely defined up to the arbitrary constant_{ll}*c*._{l}determines

*|c*._{l}|. It is customary to choose the phase of

*c*such that ._{l}is thus

**uniquely determined**. - (c)
allows us to construct all from . The are thus uniquely defined. To each pair

*(l,m)*there corresponds one and only one eigenfunction ..

But we also have

.

Therefore

.

We have

,

.

The may be written as products of the associated Legendre functions and .

,

,

where , and

*P*is the_{l}(u)*l*th order Legendre polynomial.**Note:**The choice of phase for the*P*and therefore the definition of the_{lm}(u)*Y*in terms of the_{lm}*P*is not unique._{lm}

The form a complete set of
functions of *f* and the form a complete set of functions of cos*q*, therefore the form a complete set of functions of angle on the unit sphere.
Orthonormality is expressed through ,
and completeness is expressed through .

We can expand an arbitrary function *f*(*q,f*) in
the series

with .

Most mathematical tables list properties of the spherical harmonics. You need to be aware of these properties, so that you can look them up as needed.

Recursion Relations

.

Complex conjugation

. This can
be derived from the expression of the *Y _{lm}* in terms of the .

.

real positive number.
This is
guarantied by the choice of *c _{l}*.

The addition theorem

, with .

Parity

The parity operator operating on a function of the coordinates replaces in this function the coordinates of any point in space by those of the point which is obtained by reflection through the origin of the coordinate system.

in Cartesian coordinates,

in spherical coordinates.

.

.

**The parity of the spherical harmonics is well defined and
depends only on l.**

Links:

Consider a particle whose state is described by .
We may expand *y* in terms of the
eigenfunctions of *L ^{2}* and

Operating on these eigenfunctions with *L _{±}* yields .

We know that the *y _{klm}(r)* are of the
form .

Therefore operating with *L _{±} *also yields .

Comparison shows that *R _{klm}*

We therefore write .

If [*H, L*]=0, then we can find common eigenfunctions of