next up previous contents
Next: Harmonic Oscillator: Parity Up: The Hilbert space of Previous: Infinite Well Hilbert Space   Contents

Harmonic Oscillator Hilbert Space

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 $ [0,L]$ but on the infinite interval $ [-\infty,\infty]$.

The eigen functions $ \psi_n(x)$ of the harmonic oscillator form the basis of a linear vector space $ {\cal H}_{osc}$ of functions $ f(x)$ defined on the interval $ [-\infty,\infty]$ with $ f(-\infty)=f(\infty)=0$. The $ \psi_n(x)$ form an orthonormal basis:

$\displaystyle \int_{-\infty}^{\infty}dx \vert\psi_n(x)\vert^2=1,\quad \int_{-\infty}^{\infty}dx \psi_n^*(x)\psi_m(x)=\delta_{nm}.$     (251)

(We can omit the $ *$ here because the $ \psi_n$ are real). Note that the orthonormal basis is of infinite dimension because there are infinitely many $ n$. The infinite dimension of the vector space (function space) $ {\cal H}_{osc}$ is the main difference to ordinary, finite dimensional vector spaces like the $ R^3$. Any function $ f(x) \in {\cal H}_{osc}$ (like any arbitrary vector in, e.g., the vector space $ R^3$) can be expanded into a linear combination of basis `vectors', i.e. eigen functions $ \psi_n(x)$:
$\displaystyle f(x)=\sum_{n=0}^{\infty}c_n\psi_n(x),\quad c_n=\int_{-\infty}^{\infty}dx f(x)\psi_n(x).$     (252)

We start to count the eigenstates from $ n=0$ and not $ n=1$ 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 up previous contents
Next: Harmonic Oscillator: Parity Up: The Hilbert space of Previous: Infinite Well Hilbert Space   Contents
Tobias Brandes 2004-02-04