Quantum Mechanics Solution Manual |
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© Leon van Dommelen |
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2.6.9 Solution herm-i
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.