In a previous chapter I have discussed a solution by a power series expansion. Here I shall look at a different technique, and define two operators and ,
(9.4) |
Since
(9.5) |
or in operator notation
(9.6) |
(the last term is usually written as just 1) we find
If we define the commutator
(9.8) |
we have
(9.9) |
Now we see that we can replace the eigenvalue problem for the scaled Hamiltonian by either of
By multiplying the first of these equations by we get
(9.11) |
If we just rearrange some brackets, we find
(9.12) |
If we now use
(9.13) |
we see that
(9.14) |
Question: Show that
(9.15) |
We thus conclude that (we use the notation for the eigenfunction corresponding to the eigenvalue )
So using we can go down in eigenvalues, using we can go up. This leads to the name lowering and raising operators (guess which is which?).
We also see from ( 9.15 ) that the eigenvalues differ by integers only!