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For the harmonic oscillator, the situation is completely analogous. The difference now
is that the potential is no longer the infinite well but a harmonic potential.
The wave functions are not defined on the finite interval but on the infinite
interval
.
The eigen functions of the harmonic oscillator form the basis
of a linear vector space
of functions defined on the
interval
with
.
The form an orthonormal basis:
|
|
|
(251) |
(We can omit the here because the are real).
Note that the orthonormal basis is of infinite dimension because there are infinitely many
. The infinite dimension of the vector space (function space)
is the main difference to ordinary, finite dimensional vector spaces
like the .
Any function
(like any arbitrary vector in, e.g., the vector space ) can be expanded into
a linear combination of basis `vectors', i.e. eigen functions :
|
|
|
(252) |
We start to count the eigenstates from and not as for the infinite well.
To prove the orthogonality (4.27) is a bit more difficult for the
harmonic oscillator then for the infinite potential well. It can be done by using the properties of the
Hermite polynomials.
Next: Harmonic Oscillator: Parity
Up: The Hilbert space of
Previous: Infinite Well Hilbert Space
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Tobias Brandes
2004-02-04