Shankar, R. (1994), Principles of Quantum Mechanics, Plenum Press. Chapter 12, Exercise 12.5.12.
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The parity operator in 3-d reflects every point directly through the origin, so that a position vector . In rectangular coordinates this means replacing each coordinate by its negative. In spherical coordinates, the angular coordinates change according to
If this isn’t obvious, picture reflecting a vector through the origin. If the original vector makes an angle with the (vertical) axis, then the reflected vector makes an angle with the axis, which is equivalent to an angle of with the axis. The azimuthal angle just gets rotated by to lie on the other side of the axis.
Using this, we can see that the parity operator commutes with both and , as follows. Since neither of these operators involves the radial coordinate, we can consider their effect on a function . Under parity, we have
Thus the derivatives transform under parity according to
The angular momentum operators are
Thus the combined operation gives
If we apply to , we have
where in the first line we used .
Since involves only a derivative with respect to which doesn’t change under parity, we have
Since commutes with both and it is possible to find a set of functions that are simultaneous eigenfunctions of all three operators. These functions turn out to be the same spherical harmonics that we’ve been using all along. We can show this by starting with the top spherical harmonic
where we’ve included the to be consistent with Shankar’s equation 12.5.32. Under parity, this transforms as
where we used in the second line. Thus is an eigenfunction of with eigenvalue .
To show that the other spherical harmonics are also eigenfunctions, we can use the lowering operator . In spherical coordinates, we have
The operator can be expressed as
Under parity, we can transform 22 using and , so that . We therefore have
Thus is unchanged by parity, which means that from 21, has the same parity as . Starting with and using the lowering operator successively to reduce the superscript index, we have therefore
Thus all spherical harmonics are also eigenfunctions of parity.