Born rule

The Born rule (also called the Born law, Born's rule, or Born's law) is a law of quantum mechanics which gives the probability that a measurement on a quantum system will yield a given result.It is named after its originator, the physicist Max Born.

The rule

The Born rule states that if an observable corresponding to a Hermitian operator $A$ withdiscrete spectrum is measured in asystem with wave function $|psi>$, then
* the measured result will be one of the eigenvalues $lambda$ of $A$, and
* the probability of measuring any given eigenvalue $lambda\_i$ will equal , where $P\_i$ is the projection onto the eigenspace of $A$ corresponding to $lambda\_i$.

In the case where the spectrum of $A$ is not wholly discrete, the spectral theorem proves the existence of a certain projection-valued measure $Q$, the spectral measure of $A$. In this case,
* the probability that the result of the measurement lies in a measurable set $M$ will be given by . If we are given a wave function $psi(x,y,z,t)$for a single structureless particle in position space, this reduces to saying that the probability density function$p(x,y,z)$ for a measurement of the position at time $t\_0$ will be given by $p(x,y,z)=|psi(x,y,z,t\_0)|^2.$

History

The Born rule was formulated by Born in a 1926 paper. ["Zur Quantenmechanik der Stoßvorgänge", Max Born, Zeitschrift für Physik, "37", #12 (Dec. 1926), pp. 863–867 (German); English translation in "Quantum theory and measurement", section I.2, J. A. Wheeler and W. H. Zurek, eds., Princeton, NJ: Princeton University Press, 1983.] In this paper, Born solves the Schrödinger equation for a scattering problem and concludes that the Born rule givesthe only possible interpretation of the solution. In 1954, together with Walter Bothe, Born was awarded the Nobel Prize in Physics for this and other work. [ [http://nobelprize.org/physics/laureates/1954/born-lecture.pdf Born's Nobel Lecture on the statistical interpretation of quantum mechanics] ] John von Neumann discussed the application of spectral theory to Born's rule in his 1932 book. ["Mathematische grundlagen der quantenmechanik", John von Neumann, Berlin: Springer, 1932 (German); English translation "Mathematical foundations of quantum mechanics", transl. Robert T. Beyer, Princeton, NJ: Princeton University Press, 1955.]

References