Momentum Space

Abstract

Working in momentum space involves taking the Fourier transform of the eigenfunction ψ(x, t) of the Schrödinger equation. Thus if

$$ \varphi (p,t) \equiv \frac{1} {{\sqrt {2\Pi } }}\int\limits_{ - \infty }^{ + \infty } {e^{\frac{{ - ipx}} {\hbar }} } \psi (x,t)dx $$

((3.1))

it follows from the delta function property of Equation (2.19):

$$ \frac{1} {{\sqrt 2 \Pi }}\int\limits_{ - \infty }^{ + \infty } e ^{\frac{{iy(x - x')}} {\hbar }} \frac{{dy}} {\hbar } = \delta (x - x'), $$

that

$$ \psi (x,t) = \frac{1} {{\sqrt 2 \Pi \hbar }}\int\limits_{ - \infty }^{ + \infty } e ^{\frac{{ipx}} {\hbar }} \varphi (p,t)dp. $$

((3.2))

.