Expansion in terms delta function normalized functions

In many applications it is convenient to express a physical amplitude $\Psi({\bf r})$ not as an expansion of normalized basis functions (as in Eq. (), but in terms of the following functions

$$v_{_{\bf p}}({\bf r}) \equiv {1 \over ({2 \pi \hbar})^{3/2} } \exp(i {\bf p} \cdot {\bf x} ).$$ (1.18)

Note that the integral,

$$\int d^{3}{\bf r} \vert v_{p}({\bf r}) \vert^2 = \infty$$ (1.19)

is undefined, and so these functions do not represent true physical states. Nevertheless they are useful since, according to Fourier theory, any physical amplitude $\Psi({\bf r})$ can be expressed in terms of them, i.e.

$$\Psi({\bf r}) = \int d^{3}{\bf p} \, v_{\bf p}({\bf r}) {\overline\Psi}({\bf p})$$ (1.20)

where

$${\overline\Psi}({\bf p}) = \int d^{3}{\bf r} \, v^{*}_{\bf p}({\bf r}) \Psi({\bf r})$$ (1.21)

is the Fourier transform of $\Psi({\bf r})$. Now according to definition (), ${\overline\Psi}({\bf p}) = (v_{\bf p},\Psi)$ and inserting into () we get

$$\Psi({\bf r}) = \int d^{3}{\bf p} \, v_{\bf p}({\bf r}) (v_{\bf p},\Psi).$$ (1.22)

This relation is similar to that given by Eq. (), indeed if we consider $v_{{ \bf p}}({\bf r})$to be a basis function then the above relation is expansion (), except for the replacement of the discrete index i, and its sum, with that of a continuous index ${\bf p}$ and an integral. (Actually, in this example, I used three continuous indices, ${\bf p} \equiv \{p_{1},p_{2},p_{3} \}$ and three integrals $\int d^{3}{\bf p}$. Using the definition of the delta function we find,

$$\int d^{3}{\bf r} v^{*}_{\bf p}({\bf r}) v_{\bf p'}({\bf r}) = (v_{\bf p}, v_{\bf p}') = \delta^{3}({\bf p} - {\bf p}')$$ (1.23)

a generalization of the normalization conditions exhibited by the basis functions u_i discussed above. The functions $v_{\bf p}$ are said to obey delta function normalization.

We can construct a table, analogous to the one given for the basis functions u_i, that is appropriate for the improper basis functions $v_{\bf p}$. pt

improper basis functions, or vectors	$v_{\bf p}$
delta function normalization	$(v_{ \bf p},v_{ \bf p'})=\delta^{3} ({\bf p-p'})$
State expansion in basis	$\Psi = \int d^{3} {\bf p} \, c_{{\bf p}} \, v_{{\bf p}}$
Expansion coefficients	$c_{{\bf p}}=(v_{{\bf p}},\Psi)$
Closure	$\int d^{3}{\bf p} \, v^{*}_{{\bf p}}({\bf r}') \, v_{{\bf p} }({\bf r})= \delta^{3}({\bf r'}-{\bf r})$