Editing Matter wave (section)

=== Single-particle matter waves ===
The more general description of matter waves corresponding to a single particle type (e.g. a single electron or neutron only) would have a form similar to
<math display="block">\psi (\mathbf{r}) = u(\mathbf{r},\mathbf{k})\exp(i\mathbf{k}\cdot \mathbf{r} - iE(\mathbf{k})t/\hbar)</math>
where now there is an additional spatial term <math>u(\mathbf{r},\mathbf{k})</math> in the front, and the energy has been written more generally as a function of the wave vector. The various terms given before still apply, although the energy is no longer always proportional to the wave vector squared. A common approach is to define an [[Effective mass (solid-state physics)|effective mass]] which in general is a tensor <math>m_{ij}^*</math> given by
<math display="block"> {m_{ij}^*}^{-1}  = \frac{1}{\hbar^2} \frac{\partial^2 E}{\partial k_i \partial k_j}</math>
so that in the simple case where all directions are the same the form is similar to that of a free wave above.<math display="block">E(\mathbf k) = \frac{\hbar^2 \mathbf k^2}{2 m^*}</math>In general the group velocity would be replaced by the [[probability current]]<ref name=Schiff>{{Cite book |last=Schiff |first=Leonard I. |title=Quantum mechanics |date=1987 |publisher=McGraw-Hill |isbn=978-0-07-085643-1 |edition=3. ed., 24. print |series=International series in pure and applied physics |location=New York}}</ref>
<math display="block">\mathbf{j}(\mathbf{r}) = \frac{\hbar}{2mi} \left( \psi^*(\mathbf{r}) \mathbf \nabla \psi(\mathbf{r}) - \psi(\mathbf{r}) \mathbf \nabla \psi^{*}(\mathbf{r}) \right) </math>
where <math>\nabla</math> is the [[del]] or [[gradient]] [[operator (mathematics)|operator]]. The momentum would then be described using the [[momentum operator|kinetic momentum operator]],<ref name="Schiff" />
<math display="block">\mathbf{p} = -i\hbar\nabla</math>
The wavelength is still described as the inverse of the modulus of the wavevector, although measurement is more complex. There are many cases where this approach is used to describe single-particle matter waves:
* [[Bloch wave]], which form the basis of much of [[band structure]] as described in [[Ashcroft and Mermin]], and are also used to describe the [[Electron diffraction|diffraction]] of high-energy electrons by solids.<ref>{{Cite book |last=Metherell |first=A. J. |title=Electron Microscopy in Materials Science |publisher=Commission of the European Communities |year=1972 |pages=397–552}}</ref><ref name="Peng"/>
* Waves with [[angular momentum]] such as [[electron vortex beam]]s.<ref>{{Cite journal |last1=Verbeeck |first1=J. |last2=Tian |first2=H. |last3=Schattschneider |first3=P. |date=2010 |title=Production and application of electron vortex beams |url=https://www.nature.com/articles/nature09366 |journal=Nature |language=en |volume=467 |issue=7313 |pages=301–304 |doi=10.1038/nature09366 |pmid=20844532 |bibcode=2010Natur.467..301V |s2cid=2970408 |issn=1476-4687|url-access=subscription }}</ref>
* [[Evanescent field|Evanescent waves]], where the component of the wavevector in one direction is complex. These are common when matter waves are being reflected, particularly for [[Grazing incidence diffraction|grazing-incidence diffraction]].