Next: 9.3 Eigenfunctions of through
Up: 9. Ladder operators
Previous: 9.1 Harmonic oscillators
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
,
Since
(yf (y)) = y f (y) + f (y), |
(9.5) |
or in operator notation
(the last term is usually written as just 1)
we find
If we define the commutator
we have
Now we see that we can replace the eigenvalue problem for the scaled
Hamiltonian by either of
 
+  u(y) |
= |
u(y), |
|
 
-  u(y) |
= |
u(y). |
(9.10) |
By multiplying the first of these equations by
we get
If we just rearrange some brackets, we find
If we now use
we see that
 
+   u(y) = (
- 1) u(y). |
(9.14) |
Question: Show that
 
+   u(y) = (
+ 1) u(y). |
(9.15) |
We thus conclude that (we use the notation un(y) for the
eigenfunction corresponding to the eigenvalue
)
un(y) |
 |
un - 1(y), |
|
un(y) |
 |
un + 1(y). |
(9.16) |
So using
we can go down in eigenvalues, using
a
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!
Next: 9.3 Eigenfunctions of through
Up: 9. Ladder operators
Previous: 9.1 Harmonic oscillators