Review

The operator J, whose Cartesian components satisfy the commutation relations is defined as an angular momentum operator.  For such an operator we have [Ji,J2]=0, i.e. the operator J2=Jx2+Jy2+Jz2 commutes with each Cartesian component of J.  We can therefore find an orthonormal basis of eigenfunctions common to J2 and Jz.  We denote this basis by {|k,j,m>}. 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+=Jx+iJy  and J-=Jx-iJy.  We then have and .

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

.

Orbital angular momentum

The operator L=R´P satisfies the commutation relations and is called the orbital angular momentum operator.  We denote the common eigenstates of L2 and Lz by {|k,l,m>}.

Problem:

A measurement of L2 and Lz for a free particle yields the values l=1 and m=1.  Later a measurement of Ly is made.

(a) What are the possible values for Ly?
(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 Lz with eigenvalues respectively, which span the subspace E(k,l) with l=1. After the measurement of L2 and Lz the particle is in state |k,1,1>. E(k,l) is invariant under L. The matrix of Ly in the above basis is

.

It can be constructed using  and  ,

.

The eigenvalues of Ly are ,

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 L2 and Lz is

,

with

.

Let |y> be an arbitrary state vector, and let the result of operating with Lx on |y> be |f>, i.e. Lx |y>=|f>.  In coordinate representation we write

.

.

.

In coordinate representation the operator Lx 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 L2=Lx2+Ly2+Lz2,

.

L+=Lx+iLy, L-=Lx-iLy,

.

In coordinate representation the eigenfunction associated with the eigenvalues and of L2 and Lz respectively are therefore solutions to the partial differential equations

,

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 L2 and Lz are called the spherical harmonics.

Properties of the spherical harmonics

• (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.

.

• .

.

Yll is uniquely defined up to the arbitrary constant cl.

determines |cl|.

.  It is customary to choose the phase of cl such that .

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 Pl(u) is the lth order Legendre polynomial.

Note: The choice of phase for the Plm(u) and therefore the definition of the Ylm in terms of the Plm is not unique.

The form a complete set of functions of f and the form a complete set of functions of cosq, 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 Ylm in terms of the .

.

real positive number.  This is guarantied by the choice of cl.

, 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.

Consider a particle whose state is described by .  We may expand y in terms of the eigenfunctions of L2 and Lz.

Operating on these eigenfunctions with L± yields .

We know that the yklm(r) are of the form .

Therefore operating with L± also yields  .

Comparison shows that Rklm±1(r)=Rklm(r), i.e. that Rklm(r) is independent of m.

We therefore write .

If [H,L]=0, then we can find common eigenfunctions of H, L2, and Lz.  The eigenvalues Ekl of H and the radial function Rkl(r) are then independent of m.