Editing Wave interference (section)

=== Quantum interference ===
{{See also|Double-slit_experiment|l1=Double-slit experiment|Matter wave}}
{{Quantum mechanics|cTopic=Fundamental concepts}}

'''Quantum interference''' – the observed [[matter wave | wave-behavior of matter]]<ref>[[Richard Feynman|Feynman R]], [[Robert B. Leighton|Leighton R]], and [[Matthew Sands|Sands M.]], The Feynman Lectures Website, September 2013.[https://feynmanlectures.caltech.edu/III_toc.html "The Feynman Lectures on Physics, Volume III"] (online edition)</ref> – resembles [[Wave interference#Optical interference|optical interference]]. Let <math>\Psi (x, t)</math> be a [[Wave function|wavefunction]] solution of the [[Schrödinger equation]] for a quantum mechanical object. Then the [[probability amplitude|probability]] of observing the object in the interval <math>[a,b]</math> is <math>P([a,b]) = \int_a^b |\Psi (x, t)|^2 dx = \int_a^b \Psi^* (x, t) \Psi (x, t) dx</math> where * indicates [[complex conjugate|complex conjugation]]. Quantum interference concerns the issue of this probability when the wavefunction is expressed as a sum or [[Quantum superposition|linear superposition]] of two terms <math>\Psi (x, t) = \Psi_A (x, t) + \Psi_B (x, t)</math>:
<math display="block">P([a,b]) = \int_a^b |\Psi (x, t)|^2 = \int_a^b (|\Psi_A (x, t)|^2 +  |\Psi_B (x, t)|^2 + \Psi_A^* (x, t) \Psi_B (x, t) + \Psi_A (x, t) \Psi_B^* (x, t)) dx</math>

Usually, <math>\Psi_A (x, t)</math> and <math>\Psi_B (x, t)</math> correspond to distinct situations A and B. When this is the case, the equation  <math>\Psi (x, t) = \Psi_A (x, t) + \Psi_B (x, t)</math> indicates that the object can be in situation A or situation B. The above equation can then be interpreted as: The probability of finding the object at <math>x</math> is the probability of finding the object at <math>x</math> when it is in situation A plus the probability of finding the object at <math>x</math> when it is in situation B plus an extra term. This extra term, which is called the ''quantum interference term'', is <math>\Psi_A^* (x, t) \Psi_B (x, t) + \Psi_A (x, t) \Psi_B^* (x, t)</math> in the above equation. As in the [[Wave interference#Mechanisms|classical wave case]] above, the quantum interference term can add (constructive interference) or subtract (destructive interference) from <math>|\Psi_A (x, t)|^2 +  |\Psi_B (x, t)|^2</math> in the above equation depending on whether the quantum interference term is positive or negative. If this term is absent for all <math>x</math>, then there is no quantum mechanical interference associated with situations A and B.

The best known example of quantum interference is the [[double-slit experiment]]. In this experiment, [[matter wave|matter waves]] from electrons, atoms or molecules approach a barrier with two slits in it. One slit becomes <math>\Psi_A (x, t)</math> and the other becomes  <math>\Psi_B (x, t)</math>. The interference pattern occurs on the far side, observed by detectors suitable to the particles originating the [[matter wave]].<ref name="Bach Pope Liou Batelaan 2013 p=033018">{{cite journal | last1=Bach | first1=Roger | last2=Pope | first2=Damian | last3=Liou | first3=Sy-Hwang | last4=Batelaan | first4=Herman | title=Controlled double-slit electron diffraction | journal=New Journal of Physics | publisher=IOP Publishing | volume=15 | issue=3 | date=2013-03-13 | issn=1367-2630 | doi=10.1088/1367-2630/15/3/033018 | page=033018 | url=https://iopscience.iop.org/article/10.1088/1367-2630/15/3/033018| arxiv=1210.6243 | s2cid=832961 }}</ref> The pattern matches the optical double slit pattern.