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