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Momentum wave function for a free particle

A non-relativistic free particle of mass $m$ moving in the positive $x$-direction with speed $v_{x}$ has kinetic energy

$E=\frac{1}{2} m v^{2}_{x}$

and momentum

$p_{x}=mv_{x}$

The energy and momentum are associated with a wave of wavelength $\lambda$ and frequency $\nu$ given by

$\lambda = \frac{h}{p_{x}}$

and

$\nu=\frac{E}{h}$

The propagation constant $k_{x}$ of the wave is

$k_{x}= \frac{2\pi}{\lambda}=\frac{2\pi}{\left(\frac{h}{p_{x}} \right)}=\frac{p_{x}}{\left(\frac{h}{2\pi} \right)}=\frac{p_{x}}{\hbar}$

and the angular frequency $\omega$ is

$\omega = 2\pi \nu = \frac{2\pi E}{\hbar}=\frac{E}{\hbar}$

A plane wave traveling along the $x$ axis in the positive direction may be represented by

$\psi(x,t)=A e^{-i\left(k_{x} \: x - \omega t\right)}$

Now subtitute the value of $\omega$ and $k_{x}$ in above equation then we get

$\psi(x,t)=A e^{i\left( \frac{p_{x}}{\hbar} \: x - \frac{E}{\hbar} \: t\right)}$

$\psi(x,t)=A e^{\frac{i}{\hbar}\left( p_{x} \: x - E \: t\right)}$

The superposition of a number of such waves of propagation number slightly different from an average value traveling simultaneously along the same line in the positive $x$- direction forms a wave packet of small extension. By Fourier's theorem the eave packet may be expressed by

$\psi(x,t) = \frac{1}{\sqrt{2 \pi \hbar}} \int_{-\infty}^{+\infty} A (p_{x}) e^{\frac{i}{\hbar}\left( p_{x} \: x - E \: t \right)} \: \: dp_{x} \qquad(1)$

The function $\psi(x,t)$ is called the momentum wave function for the motion of the free particle in one dimension.

The amplitude $A(p_{x})$ of the $x$-component of the momentum is given by the Fourier tranform

$A(p)=\frac{1}{\sqrt{2 \pi \hbar}} \int_{-\infty}^{+\infty} \psi (x,t) e^{-\frac{i}{\hbar}\left( p_{x} \: x - E \: t \right)} \: \: dx \qquad(2)$

In three dimension the wave function is represented by

$\psi(\overrightarrow{r},t) = \frac{1}{(2 \pi \hbar)^{3/2}} \int_{-\infty}^{+\infty} A (\overrightarrow{p}) e^{\frac{i}{\hbar}\left( \overrightarrow{p} . \overrightarrow{r} - E \: t \right)} \: \: d^{3}\overrightarrow{p} \qquad(3)$

Where $d^{3}\overrightarrow{p}=dp_{x} \: dp_{y} \: dp_{z}$ is the volume element in the momentum space. In equation $(1)$, equation $(2)$ and equation $(3)$ $\frac{1}{\sqrt{2 \pi \hbar}}$ and $\frac{1}{(2 \pi \hbar)^{3/2}}$ are normalization constants.

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