Quantum Mechanics Solution Manual 

© Leon van Dommelen 

2.6.9 Solution hermi
Question:
A complete set of orthonormal eigenfunctions of the operator that are periodic on the interval 0 are the infinite set of functions
Check that these functions are indeed periodic, orthonormal, and that they are eigenfunctions of with the real eigenvalues
Completeness is a much more difficult thing to prove, but they are. The completeness proof in the notes covers this case.
Answer:
Any eigenfunction of the above list can be written in the generic form where is a whole number, in other words where is an integer, one of ..., 3, 2, 1, 0, 1, 2, 3, ... If you show that the stated properties are true for this generic form, it means that they are true for every eigenfunction.
Now periodicity requires that , and the Euler formula verifies this: sines and cosines are the same if the angle changes by a whole multiple of . (For example, , , , etcetera are physically all equivalent to a zero angle.)
The derivative of with respect to is , and multiplying by you get , so is an eigenfunction of with eigenvalue .
To see that is normalized, check that its norm is unity:
To verify that is orthogonal to every other eigenfunction, take the generic other eigenfunction to be with an integer different from . You must then show that the inner product of these two eigenfunctions is zero. Since the normalization constants do not make any difference here, you can just show that is zero. You get
since 1. So different eigenfunctions are orthogonal, their inner product is zero.