Editing Free particle (section)

===Wave packet===
{{main|Wave packet}}
Using the [[Fourier inversion theorem]], the free particle wave function may be represented by a superposition of ''momentum'' eigenfunctions, or, ''wave packet'':<ref>{{harvnb|Hall|2013}} Section 4.1</ref>
<math display="block"> \psi(\mathbf{r}, t) =\frac{1}{(\sqrt{2\pi})^3} \int_\mathrm{all \, \mathbf{k} \, space} \hat \psi_0 (\mathbf{k})e^{i(\mathbf{k}\cdot\mathbf{r}-\omega(\mathbf{k}) t)} d^3 \mathbf{k},</math>
where 
<math display="block"> \omega(\mathbf{k}) = \frac{\hbar \mathbf{k}^2}{2m},</math>
and <math>\hat \psi_0 (\mathbf{k})</math> is the [[Fourier transform]] of a "[[Fourier_inversion_theorem#Conditions_on_the_function|sufficiently nice]]" initial wavefunction <math>\psi(\mathbf{r},0)</math>. 

The expectation value of the momentum '''p''' for the complex plane wave is

<math display="block"> \langle\mathbf{p}\rangle=\left\langle \psi \left|-i\hbar\nabla\right|\psi\right\rangle = \hbar\mathbf{k} ,</math>

and for the general wave packet it is

<math display="block"> \langle\mathbf{p}\rangle = \int_\mathrm{all\,space} \psi^*(\mathbf{r},t)(-i\hbar\nabla)\psi(\mathbf{r},t) d^3 \mathbf{r} = \int_\mathrm{all \, \textbf{k} \, space} \hbar \mathbf{k} |\hat\psi_0(\mathbf{k})|^2 d^3 \mathbf{k}. </math>

The expectation value of the energy E is 

<math display="block"> \langle E\rangle=\left\langle \psi \left|- \frac{\hbar^2}{2m} \nabla^2 \right|\psi\right\rangle = \int_\text{all space} \psi^*(\mathbf{r},t)\left(- \frac{\hbar^2}{2m} \nabla^2 \right)\psi(\mathbf{r},t) d^3 \mathbf{r} .</math>