- Continuous Spectra: Normalization and Completeness in Vector Spaces
Normalization in Continuous Spectra
In a vector space with continuous spectra, we can express states as follows:
$$|\psi\rangle = \lim_{\Delta x \to 0} \sum_{n=-\infty}^{\infty} \Delta x \ \psi(n\Delta x)|n\Delta x\rangle$$
$$|\phi\rangle = \lim_{\Delta x \to 0} \sum_{m=-\infty}^{\infty} \Delta x \ \phi(m\Delta x)|m\Delta x\rangle$$
The inner product between these states can be written as:
$$\langle\phi|\psi\rangle = \lim_{\Delta x \to 0} \sum_{m=-\infty}^{\infty}\sum_{n=-\infty}^{\infty} $$
$$\left (\ \Delta x \ \phi^*(m\Delta x) \ \psi(n\Delta x) \ \Delta x \ \langle m\Delta x|n\Delta x\rangle\ \right )$$
Since
$$\langle m\Delta x|n\Delta x\rangle = 0\tag{1}$$
when $m \neq n$ $\tag{1}$, the expression simplifies to:
$$\begin{align} \langle\phi|\psi\rangle = \lim_{\Delta x \to 0} \sum_{n=-\infty}^{\infty} &\Delta x \ \phi^*(n\Delta x) \ \psi(n\Delta x)\\ &\quad \left ( \Delta x \ \langle n\Delta x|n\Delta x\rangle \right ) \end{align}$$If we normalize such that
$$\Delta x \ \langle n\Delta x|n\Delta x\rangle = 1,\tag{2}$$
then:
$$\begin{align} \langle\phi|\psi\rangle &= \lim_{\Delta x \to 0} \sum_{n=-\infty}^{\infty} \Delta x \ \phi^*(n\Delta x) \ \psi(n\Delta x) \\ &= \int_{-\infty}^{\infty} dx \ \phi^*(x') \ \psi(x) \ \delta(x'-x) \\ &= \int_{-\infty}^{\infty} dx \ \phi^*(x) \ \psi(x) \end{align}$$$$\tag{3}$$
where $\lim_{\Delta x \to 0} m\Delta x = x’$, $\lim_{\Delta x \to 0} n\Delta x = x$, and from equations (1) and (2), we have
$$\lim_{\Delta x \to 0}\ \langle m\Delta x|n\Delta x\rangle = \langle x’|x\rangle = \delta(x’-x)$$
$$\tag{4}$$
Completeness Relation
Since $\phi^*(x) = \langle\phi|x\rangle$ and $\psi(x) = \langle x|\psi\rangle$, we can rewrite equation (3) as:
$$\begin{align} \langle\phi|\psi\rangle &= \int_{-\infty}^{\infty} dx \ \langle\phi|x\rangle \langle x|\psi\rangle \\ &= \langle\phi|\left(\int_{-\infty}^{\infty} dx \ |x\rangle \langle x|\right)|\psi\rangle \end{align}$$Therefore, the completeness relation is:
$$1 = \int_{-\infty}^{\infty} dx \ |x\rangle \langle x|\tag{5}$$