Editing Probability amplitude (section)

==In the context of the double-slit experiment==
{{main|Double-slit experiment}}
Probability amplitudes have special significance because they act in quantum mechanics as the equivalent of conventional probabilities, with many analogous laws, as described above. For example, in the classic [[double-slit experiment]], electrons are fired randomly at two slits, and the probability distribution of detecting electrons at all parts on a large screen placed behind the slits, is questioned. An intuitive answer is that {{math|1='''P'''(through either slit) = '''P'''(through first slit) + '''P'''(through second slit)}}, where {{math|'''P'''(event)}} is the probability of that event. This is obvious if one assumes that an electron passes through either slit. When no measurement apparatus that determines through which slit the electrons travel is installed, the observed probability distribution on the screen reflects the [[Interference (wave propagation)|interference pattern]] that is common with light waves. If one assumes the above law to be true, then this pattern cannot be explained. The particles cannot be said to go through either slit and the simple explanation does not work. The correct explanation is, however, by the association of probability amplitudes to each event. The complex amplitudes which represent the electron passing each slit ({{math|''&psi;''<sub>first</sub>}} and {{math|''&psi;''<sub>second</sub>}}) follow the law of precisely the form expected: {{math|1=''&psi;''<sub>total</sub> = ''&psi;''<sub>first</sub> + ''&psi;''<sub>second</sub>}}. This is the principle of [[quantum superposition]]. The probability, which is the [[modulus squared]] of the probability amplitude, then, follows the interference pattern under the requirement that amplitudes are complex:
<math display="block">P = \left|\psi_\text{first} + \psi_\text{second}\right|^2 = \left|\psi_\text{first}\right|^2 + \left|\psi_\text{second}\right|^2 + 2 \left|\psi_\text{first}\right| \left|\psi_\text{second}\right| \cos (\varphi_1 - \varphi_2).</math>
Here, <math>\varphi_1</math> and <math>\varphi_2</math>
are the [[Argument (complex analysis)|arguments]] of {{math|''&psi;''<sub>first</sub>}} and {{math|''&psi;''<sub>second</sub>}} respectively. A purely real formulation has too few dimensions to describe the system's state when superposition is taken into account. That is, without the arguments of the amplitudes, we cannot describe the phase-dependent interference. The crucial term <math display="inline"> 2 \left|\psi_\text{first}\right| \left|\psi_\text{second}\right| \cos (\varphi_1 - \varphi_2)</math> is called the "interference term", and this would be missing if we had added the probabilities.

However, one may choose to devise an experiment in which the experimenter observes which slit each electron goes through. Then, due to [[wavefunction collapse]], the interference pattern is not observed on the screen.

One may go further in devising an experiment in which the experimenter gets rid of this "which-path information" by a [[Quantum eraser experiment|"quantum eraser"]]. Then, according to the [[Copenhagen interpretation]], the case A applies again and the interference pattern is restored.<ref>A recent 2013 experiment gives insight regarding the correct physical interpretation of such phenomena. The information can actually be obtained, but then the electron seemingly went through all the possible paths simultaneously. (Certain [[Ensemble interpretation|ensemble-alike]] realistic interpretations of the wavefunction may presume such coexistence in all the points of an orbital.) Cf. {{cite journal |last1=Schmidt|first1=L. Ph. H.| last2=Lower|first2=J.| last3=Jahnke|first3=T. |last4=Schößler|first4=S. |last5=Schöffler|first5=M. S.| last6=Menssen|first6=A. |last7=Lévêque|first7=C. |last8=Sisourat|first8=N. |last9=Taïeb|first9=R. | display-authors=1| year=2013|title=Momentum Transfer to a Free Floating Double Slit: Realization of a Thought Experiment from the Einstein-Bohr Debates |journal=[[Physical Review Letters]]| volume=111|issue=10|pages=103201| doi=10.1103/PhysRevLett.111.103201| first10=H.|last10=Schmidt-Böcking| first11=R.|last11=Dörner| pmid=25166663 |bibcode=2013PhRvL.111j3201S |s2cid=2725093 |url=http://pdfs.semanticscholar.org/e551/f885162ab3b25b16fb7fff48c87dbc1cbd02.pdf | archive-url=https://web.archive.org/web/20190307191633/http://pdfs.semanticscholar.org/e551/f885162ab3b25b16fb7fff48c87dbc1cbd02.pdf |url-status=dead|archive-date=2019-03-07}}</ref>