Angular Momentum Operators

 

We represent the angular momentum wavefunctions in spherical polar coordinates.

x = r sinq cosf

y = r sinq sinf

z = r cosq

The inverse relations are

r2 = x2 + y2 + z2

cosq = z/r

tanf = y/x

In Cartesian coordinates the angular momentum operators are

These can be transformed into spherical polar coordinates using the nine partial derivatives in which the appropriate Cartesian coordinates are held constant. For example, starting with

r2 = x2 + y2 + z2

we can write

Similar relations are obtained for y and z.

For q we begin with the fact that

Using the fact that

and the quotient rule we find

 

For x we have

For y we have

The derivatives of f are obtained using the generic derivative for arctan

and the quotient rule

In the evaluation of the above partial derivatives we have used the fact that

Finally, we note that

To obtain the angular momentum operators in spherical polar coordinates we substitute into the Cartesian operators above and apply the chain rule.

Zare (Angular Momentum) works out the analogous expression for lx. The three expressions for the angular momenta in spherical polar coordinates are

It follows that