Building a Localized Single-Particle Wave Packet

We now have a wave function for a free particle with a definite momentum $p$

$$\bgroup\color{black} \psi(x,t)=e^{i(px-Et)/\hbar}=e^{i(kx-\omega t)} \egroup$$

where the wave number $k$ is defined by $p=\hbar k$ and the angular frequency $\omega$ satisfies $E=\hbar\omega$. It is not localized since $P(x,t)=\vert\psi(x,t)\vert^2=1$ everywhere.

We would like a state which is localized and normalized to one particle.

$$\bgroup\color{black} \int\limits_{-\infty}^\infty\psi^*(x,t)\psi(x,t) dx=1 \egroup$$

To make a wave packet which is localized in space, we must add components of different wave number. Recall that we can use a Fourier Series to compose any function $f(x)$ when we limit the range to $-L<x<L$. We do not want to limit our states in $x$, so we will take the limit that $L\rightarrow \infty$. In that limit, every wave number is allowed so the sum turns into an integral. The result is the very closely related Fourier Transform

$$\bgroup\color{black}f(x)={1\over \sqrt{2\pi}}\int\limits_{-\infty}^\infty A(k) e^{ikx} dk\egroup$$

with coefficients which are computable,

$$\bgroup\color{black}A(k)={1\over \sqrt{2\pi}}\int\limits_{-\infty}^\infty f(x) e^{-ikx} dx.\egroup$$

The normalizations of $f(x)$ and $A(k)$ are the same (with this symmetric form) and both can represent probability amplitudes.

$$\bgroup\color{black}\int\limits_{-\infty}^\infty f^*(x)f(x)dx=\int\limits_{-\infty}^\infty A^*(k)A(k)dk\egroup$$

We understand $f(x)$ as a wave packet made up of definite momentum terms $e^{ikx}$. The coefficient of each term is $A(k)$. The probability for a particle to be found in a region $dx$ around some value of $x$ is $\vert f(x)\vert^2dx$. The probability for a particle to have wave number in region $dk$ around some value of $k$ is $\vert A(k)\vert^2dk$. (Remember that $p=\hbar k$ so the momentum distribution is very closely related. We work with $k$ for a while for economy of notation.)