Shankar, R. (1994), *Principles of Quantum Mechanics*, Plenum Press. Chapter 12, Exercise 12.5.14.

[If some equations are too small to read easily, use your browser’s magnifying option (Ctrl + on Chrome, probably something similar on other browsers).]

Here’s another example of using spherical harmonics to study the behaviour of wave functions in 3-d. Under a rotation by about the axis, the coordinates transform using the rotation matrix

This results in the coordinate transformations

Using similar techniques to those for translations, it is found that the wave function transforms into the wave function at the position obtained by rotating by (that is, by rotating by in the opposite direction):

Suppose we have a wave function given by

for some constants and . Under this rotation, using 5 it transforms to

[Note that remains invariant under rotations about the origin, since the distance of a point from the origin is not affected by a rotation. You can verify this directly if you like by working out after the rotation.]

Equation 7 differs from the equation given in Shankar, which is

Curiously, in the errata for Shankar’s book (2006 edition) 7 is listed as the incorrect version, which is ‘corrected’ to 8. In my copy of the book (which doesn’t have a date on the title page), 8 is printed, but I don’t think this is right. In any case, we’ll proceed with the problem.

First, we write 6 in terms of spherical harmonics, using

We have

With the three spherical harmonics , and as the basis, we can write this in vector notation as

A rotation in 3-d for is given by

For , this works out to

We can use this to transform 13 to get

where we used 10 to get the last line. This result agrees with 7 and not with the equation 8 given in Shankar, so (provided I got the signs right) it looks like Shankar’s equation is wrong.

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