Infinite Well Hilbert Space

Let us recall what we know from the infinite potential well:

The eigen functions $\phi_n(x)$ of the infinite potential well form the basis of a linear vector space ${\cal H}_{well}$ of functions $f(x)$ defined on the interval $[0,L]$ with $f(0)=f(L)=0$. The $\phi_n(x)$ form an orthonormal basis:

$$\int_0^Ldx \vert\phi_n(x)\vert^2=1,\quad \int_0^Ldx \phi_n^*(x)\phi_m(x)=\delta_{nm}.$$ (249)

(We can omit the $*$ here because the $\phi_n$ 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) ${\cal H}_{well}$ is the main difference to ordinary, finite dimensional vector spaces like the $R^3$. Any function $f(x) \in {\cal H}_{well}$ (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 $\phi_n(x)$:

$$f(x)=\sum_{n=1}^{\infty}c_n\phi_n(x),\quad c_n=\int_0^Ldx f(x)\phi_n(x).$$ (250)

We start to count the eigenstates from $n=1$.