Editing Matter wave (section)

== General matter waves ==
The preceding sections refer specifically to [[free particle]]s for which the wavefunctions are plane waves. There are significant numbers of other matter waves, which can be broadly split into three classes: single-particle matter waves, collective matter waves and standing waves.

=== 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]].

=== Collective matter waves ===
{{See also|List of quasiparticles}}
Other classes of matter waves involve more than one particle, so are called collective waves and are often [[quasiparticle]]s. Many of these occur in solids – see [[Ashcroft and Mermin]]. Examples include:
* In solids, an '''electron quasiparticle''' is an [[electron]] where interactions with other electrons in the solid have been included. An electron quasiparticle has the same [[electric charge|charge]] and [[Spin (physics)|spin]] as a "normal" ([[elementary particle]]) electron and, like a normal electron, it is a [[fermion]]. However, its [[Effective mass (solid-state physics)|effective mass]] can differ substantially from that of a normal electron.<ref name="Kaxiras">{{cite book|author=Efthimios Kaxiras|title=Atomic and Electronic Structure of Solids|url=https://books.google.com/books?id=WTL_vgbWpHEC&pg=PA65|date=9 January 2003|publisher=Cambridge University Press|isbn=978-0-521-52339-4|pages=65–69}}</ref> Its electric field is also modified, as a result of [[electric field screening]].
* A '''[[electron hole|hole]]''' is a quasiparticle which can be thought of as a [[Vacancy defect|vacancy]] of an electron in a state; it is most commonly used in the context of empty states in the [[valence band]] of a [[semiconductor]].<ref name="Kaxiras" /> A hole has the opposite charge of an electron.
* A '''[[polaron]]''' is a quasiparticle where an electron interacts with the [[polarization density|polarization]] of nearby atoms.
* An '''[[exciton]]''' is an electron and hole pair which are bound together.
* A [[Cooper pair]] is two electrons bound together so they behave as a single matter wave.

=== Standing matter waves ===
{{See also|Standing wave}}
[[File:InfiniteSquareWellAnimation.gif|thumb|200px|right|Some trajectories of a particle in a box according to [[Newton's laws]] of [[classical mechanics]] (A), and matter waves (B–F). In (B–F), the horizontal axis is position, and the vertical axis is the real part (blue) and imaginary part (red) of the [[wavefunction]]. The states (B,C,D) are [[energy eigenstate]]s, but (E,F) are not.]]
The third class are matter waves which have a wavevector, a wavelength and vary with time, but have a zero [[group velocity]] or [[probability flux]]. The simplest of these, similar to the notation above would be
<math display="block">\cos(\mathbf{k}\cdot\mathbf{r} - \omega t)</math>
These occur as part of the [[particle in a box]], and other cases such as in a [[Particle in a ring|ring]]. This can, and arguably should be, extended to many other cases. For instance, in early work de Broglie used the concept that an electron matter wave must be continuous in a ring to connect to the [[Bohr–Sommerfeld quantization|Bohr–Sommerfeld condition]] in the early approaches to quantum mechanics.<ref>{{Cite book |last=Jammer |first=Max |title=The conceptual development of quantum mechanics |date=1989 |publisher=Thomas publishers |isbn=978-0-88318-617-6 |edition=2nd |series=The history of modern physics |location=Los Angeles (Calif.)}}</ref> In that sense [[atomic orbital]]s around atoms, and also [[molecular orbital]]s are electron matter waves.<ref>{{Cite journal |last=Mulliken |first=Robert S. |date=1932 |title=Electronic Structures of Polyatomic Molecules and Valence. II. General Considerations |url=https://link.aps.org/doi/10.1103/PhysRev.41.49 |journal=Physical Review |volume=41 |issue=1 |pages=49–71 |doi=10.1103/PhysRev.41.49 |bibcode=1932PhRv...41...49M |url-access=subscription }}</ref><ref>{{Cite book |last=Griffiths |first=David J. |title=Introduction to quantum mechanics |date=1995 |publisher=Prentice Hall |isbn=978-0-13-124405-4 |location=Englewood Cliffs, NJ}}</ref><ref>{{Cite book |last=Levine |first=Ira N. |title=Quantum chemistry |date=2000 |publisher=Prentice Hall |isbn=978-0-13-685512-5 |edition=5th |location=Upper Saddle River, NJ}}</ref>