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 Born rule is one of the key principles of quantum mechanics. There have been many attempts to derive the Born rule from the other assumptions of quantum mechanics, with inconclusive results.[1]

Contents

The rule

The Born rule states that if an observable corresponding to a Hermitian operator A with discrete spectrum is measured in a system with normalized wave function \scriptstyle|\psi\rang (see Bra-ket notation), then

(In the case where the eigenspace of A corresponding to \lambda_i is one-dimensional and spanned by the normalized eigenvector \scriptstyle|\lambda_i\rang, P_i is equal to \scriptstyle|\lambda_i\rang\lang\lambda_i|, so the probability \scriptstyle\lang\psi|P_i|\psi\rang is equal to \scriptstyle\lang\psi|\lambda_i\rang\lang\lambda_i|\psi\rang. Since the complex number \scriptstyle\lang\lambda_i|\psi\rang is known as the probability amplitude that the state vector \scriptstyle|\psi\rang assigns to the eigenvector \scriptstyle|\lambda_i\rang, it is common to describe the Born rule as telling us that probability is equal to the amplitude-squared (really the amplitude times its own complex conjugate). Equivalently, the probability can be written as \scriptstyle|\lang\lambda_i|\psi\rang|^2.)

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,

If we are given a wave function \scriptstyle\psi 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)=\scriptstyle|\psi(x,y,z,t_0)|^2.

History

The Born rule was formulated by Born in a 1926 paper.[2] In this paper, Born solves the Schrödinger equation for a scattering problem and, inspired by Einstein's work on the photoelectric effect,[3] concluded, in a footnote, that the Born rule gives the 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.[4] John von Neumann discussed the application of spectral theory to Born's rule in his 1932 book.[5] A 2011 article by Armando V.D.B. Assis claims to show that Born's Rule can be derived within a game-theoretical framework.[6]:

References

  1. ^ N.P. Landsman, "The conclusion seems to be that no generally accepted derivation of the Born rule has been given to date, but this does not imply that such a derivation is impossible in principle.", in Compendium of Quantum Physics (eds.) F.Weinert, K. Hentschel, D.Greenberger and B. Falkenburg (Springer, 2008), ISBN 3540706224
  2. ^ Zur Quantenmechanik der Stoßvorgänge, Max Born, Zeitschrift für Physik, 37, #12 (Dec. 1926), pp. 863–867 (German); English translation, On the quantum mechanics of collisions, in Quantum theory and measurement, section I.2, J. A. Wheeler and W. H. Zurek, eds., Princeton, NJ: Princeton University Press, 1983, ISBN 0691083169.
  3. ^ "Again an idea of Einstein’s gave me the lead. He had tried to make the duality of particles - light quanta or photons - and waves comprehensible by interpreting the square of the optical wave amplitudes as probability density for the occurrence of photons. This concept could at once be carried over to the psi-function: |psi|2 ought to represent the probability density for electrons (or other particles)." from Born's Nobel Lecture on the statistical interpretation of quantum mechanics
  4. ^ Born's Nobel Lecture on the statistical interpretation of quantum mechanics
  5. ^ 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.
  6. ^ Armando V.D.B. Assis (2011). "Assis, Armando V.D.B. On the nature of \scriptstyle a^{*}_{k}a_{k} and the emergence of the Born rule. Annalen der Physik, 2011.". Annalen der Physik (Berlin). doi:10.1002/andp.201100062. http://onlinelibrary.wiley.com/doi/10.1002/andp.201100062/abstract. 

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