16  Completeness of Polynomials

We can rephrase all the cases of orthogonal polynomials above, including Fourier series, more directly into Dirac notation. The underlying principle is the following.

Let \(\left\{ |f_{n}\rangle\right\}\) be orthogonal on the interval \(\left[a,b\right]\) with respect to the weight function \(w\left(x\right)\):

\[\langle f_{n}|f_{m}\rangle=\int^{b}_{a}f_{n}\left(x\right)^{*}f_{m}\left(x\right)w\left(x\right)\text{d}x\propto\delta_{nm}.\]

NB the existence (non-infiniteness) of this norm defines these spaces to be \(L^{2}\left(\left[a,b\right],w\left(x\right)\text{d}x\right)\), and therefore Hilbert spaces:

\[\langle f_{n}|f_{m}\rangle=\int^{b}_{a}f_{n}\left(x\right)^{*}f_{m}\left(x\right)w\left(x\right)\text{d}x<\infty.\]

Then we can always decompose any function \(c\left(x\right)\in L^{2}\left(\left[a,b\right],w\left(x\right)\text{d}x\right)\) into the \(\left\{ |f_{n}\rangle\right\}\) basis:

\[\begin{aligned} c\left(x\right) & =\sum_{n}c_{n}f_{n}\left(x\right) \end{aligned}\]

with

\[c_{n}=\frac{\langle f_{n}|c\rangle}{\langle f_{n}|f_{n}\rangle}=\frac{\int^{b}_{a}f^{*}_{n}\left(x\right)c\left(x\right)w\left(x\right)\text{d}x}{\int^{b}_{a}f^{*}_{n}\left(x\right)f_{n}\left(x\right)w\left(x\right)\text{d}x}.\]

This is the main use of orthogonality – it allows a formal extension of vector decomposition to the case of functions.

The statement of completeness is equivalent to a statement of the existence of a resolution of the identity, as follows:

\[\begin{aligned} |c\rangle & =\sum_{n}c_{n}|f_{n}\rangle\\ & =\sum_{n}\left(\frac{\langle f_{n}|c\rangle}{\langle f_{n}|f_{n}\rangle}\right)|f_{n}\rangle\\ & =\sum_{n}\frac{|f_{n}\rangle\langle f_{n}|}{\langle f_{n}|f_{n}\rangle}|c\rangle \end{aligned}\]

and therefore

\[\mathbb{I}=\sum_{n}\frac{|f_{n}\rangle\langle f_{n}|}{\langle f_{n}|f_{n}\rangle}.\]

Assuming the basis is normalised (always possible), \(\langle f_{n}|f_{n}\rangle=1\), and

\[\mathbb{I}=\sum_{n}|f_{n}\rangle\langle f_{n}|.\]

The basis \(\left\{ |f_{n}\rangle\right\}\) is said to be complete if any of the equivalent statements is true:

• \(\hat{\mathbb{I}}=\sum_{n}|f_{n}\rangle\langle f_{n}|\) (the identity can be decomposed into the basis)

• All functions \(f_{n}\left(x\right)\in L^{2}\left(\left[a,b\right],w\left(x\right)\text{d}x\right)\) are orthogonal to the zero function

• The zero function is the only function orthogonal to all \(f_{n}\left(x\right)\in L^{2}\left(\left[a,b\right],w\left(x\right)\text{d}x\right).\)

In general, proving completeness is difficult and subtle. Recall that QM3, the third axiom of quantum mechanics, has to postulate completeness!