Mathematical Methods in the Physical Sciences · Problem 5.24

FUNCTIONS OF A COMPLEX VARIABLE — Mary Boas

Mathematics — Methods · parseval-theorem-quantum-mechanics

Problem

Starting with the symmetrized integrals as in Problem 23, make the substitutions

\alpha = 2\pi p/h

(where

p

is the new variable,

h

is a constant),

f(x) = \psi(x)

g(\alpha) = \sqrt{h/2\pi}\,\phi(p)

; show that then

\begin{aligned} \psi(x) &= \frac{1}{\sqrt{h}} \int_{-\infty}^{\infty} \phi(p) e^{2\pi i p x/h}\, dp,\\ \phi(p) &= \frac{1}{\sqrt{h}} \int_{-\infty}^{\infty} \psi(x) e^{-2\pi i p x/h}\, dx,\\ \int_{-\infty}^{\infty} |\psi(x)|^2\, dx &= \int_{-\infty}^{\infty} |\phi(p)|^2\, dp. \end{aligned}

This notation is often used in quantum mechanics.