If you look at the expression and use the explicit form , you may guess that we can use partial integration to get the operator acting on ,
This is the first example of an operator that is clearly not Hermitean, but we see that and are related by “Hermitean conjugation”. We can actually use this to normalise the wave function! Let us look at
If we now use repeatedly until the operator acts on , we find
(9.23) |
Since , we find that
(9.24) |
Question: Show that this agrees with the normalisation proposed in the previous study of the harmonic oscillator!
Question: Show that the states for different are orthogonal, using the techniques sketched above.