# Eigenfunctions of Operators with Continuous Sprectra

I'm self-studying Griffith's Intro to Quantum Mechanics, and on page 100 he makes the claim that the eigenfunctions of operators with continuous spectra are not normalizable. I can't see why this is necessarily true. Hopefully I am not missing something basic.

The problem comes when you try and work out the meaning of "orthogonality". Say x and y are position eigenstates. Then we want
$$\langle x | y\rangle =\delta(x-y)$$
For discrete quantities, the Kronecker delta is either one or zero. The Dirac delta function used here has to be something more like an infinite weight, in the handwaving way we tend to talk about these things. As it's neither a function, nor a number, there's no sense in which we can just divide by $\delta(0)$ to make the "norm" of such a state one.

Hope that helps.

Fredrik
Staff Emeritus
$$\|g\|^2=\int(Ae^{ipx})^*(Ae^{ipx})dx=|A|^2\int dx.$$ There's clearly no choice of A that makes the right-hand side =1, since ##\int dx=\infty##.
If this isn't clear, think about some totally arbitrary continuum of values f, and think about how the identity operator acts on one specific eigenstate with eigenvalue $f_0$:
$$|f_0\rangle=\int df \langle f|f_0\rangle |f\rangle$$