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:

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

$$f(x)=\sum_{n=0}^{\infty}c_n\psi_n(x),\quad c_n=\int_{-\infty}^{\infty}dx f(x)\psi_n(x).$$ (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.