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If you look at the expression
f (y)*
g(y) dy and use the explicit form
= 
y -
, you may guess
that we can use partial integration to get the operator acting on f,
f (y)* g(y) dy |
| |
= |
f (y)* y -
g(y) dy |
|
| |
= |
  y +
f (y)*g(y) dy |
|
| |
= |
[ f (y)]*g(y) dy . |
(9.21) |
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
| On |
= |
    e-y2/2![$\displaystyle \left.\vphantom{\left({\hat a}^{\dagger}\right)^{n} e^{-y^2/2}}\right]^{*}_{}$](img326.gif)   e-y2/2 dy |
|
| |
= |
     e-y2/2![$\displaystyle \left.\vphantom{\hat a \left({\hat a}^{\dagger}\right)^{n} e^{-y^2/2}}\right]^{*}_{}$](img328.gif)   e-y2/2 dy |
(9.22) |
If we now use

= 
+
repeatedly until the operator
acts on
u0(y), we find
Since
O0 =
, we find that
Question: Show that this agrees with the normalisation proposed in the
previous study of the harmonic oscillator!
Question: Show that the states un for different n are orthogonal,
using the techniques sketched above.
Next: 10. Time dependent wave
Up: 9. Ladder operators
Previous: 9.3 Eigenfunctions of through