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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 the Copenhagen interpretation of quantum mechanics.

The rule

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

• the measured result will be one of the eigenvalues ${\displaystyle \lambda }$ of ${\displaystyle A}$, and
• the probability of measuring a given eigenvalue ${\displaystyle \lambda _{i}}$ will equal ${\displaystyle \scriptstyle \langle \psi |P_{i}|\psi \rangle }$, where ${\displaystyle P_{i}}$ is the projection onto the eigenspace of ${\displaystyle A}$ corresponding to ${\displaystyle \lambda _{i}}$.

(In the case where the eigenspace of ${\displaystyle A}$ corresponding to ${\displaystyle \lambda _{i}}$ is one-dimensional and spanned by the normalized eigenvector ${\displaystyle \scriptstyle |\lambda _{i}\rangle }$, ${\displaystyle P_{i}}$ is equal to ${\displaystyle \scriptstyle |\lambda _{i}\rangle \langle \lambda _{i}|}$, so the probability ${\displaystyle \scriptstyle \langle \psi |P_{i}|\psi \rangle }$ is equal to ${\displaystyle \scriptstyle \langle \psi |\lambda _{i}\rangle \langle \lambda _{i}|\psi \rangle }$. Since the complex number ${\displaystyle \scriptstyle \langle \lambda _{i}|\psi \rangle }$ is known as the probability amplitude that the state vector ${\displaystyle \scriptstyle |\psi \rangle }$ assigns to the eigenvector ${\displaystyle \scriptstyle |\lambda _{i}\rangle }$, 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 ${\displaystyle \scriptstyle |\langle \lambda _{i}|\psi \rangle |^{2}}$.)

In the case where the spectrum of ${\displaystyle A}$ is not wholly discrete, the spectral theorem proves the existence of a certain projection-valued measure ${\displaystyle Q}$, the spectral measure of ${\displaystyle A}$. In this case,

• the probability that the result of the measurement lies in a measurable set ${\displaystyle M}$ will be given by ${\displaystyle \scriptstyle \langle \psi |Q(M)|\psi \rangle }$.

If we are given a wave function ${\displaystyle \scriptstyle \psi (x,y,z,t)}$ for a single structureless particle in position space, this reduces to saying that the probability density function ${\displaystyle p(x,y,z)}$ for a measurement of the position at time ${\displaystyle t_{0}}$ will be given by ${\displaystyle p(x,y,z)=}$${\displaystyle \scriptstyle |\psi (x,y,z,t_{0})|^{2}.}$

History

The Born rule was formulated by Born in a 1926 paper. In this paper, Born solves the Schrödinger equation for a scattering problem and concludes 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. John von Neumann discussed the application of spectral theory to Born's rule in his 1932 book.

References

1. 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.
2. Born's Nobel Lecture on the statistical interpretation of quantum mechanics
3. 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.

• Quantum measurement