### Born's Rule

*Not to be confused with the Cauchy–Born rule in crystal mechanics.*

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; the Many Worlds Interpretation for example cannot derive the Born rule.^{[1]} However, within the Quantum Bayesianism interpretation of quantum theory, it has been shown to be an extension of the standard Law of Total Probability, which takes into account the Hilbert space dimension of the physical system involved.^{[2]}

## 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
$\backslash scriptstyle|\backslash psi\backslash rang$
(*see* bra-ket notation), then

- the measured result will be one of the eigenvalues $\backslash lambda$ of $A$, and
- the probability of measuring a given eigenvalue $\backslash lambda\_i$ will equal $\backslash scriptstyle\backslash lang\backslash psi|P\_i|\backslash psi\backslash rang$, where $P\_i$ is the projection onto the eigenspace of $A$ corresponding to $\backslash lambda\_i$.

- (In the case where the eigenspace of $A$ corresponding to $\backslash lambda\_i$ is one-dimensional and spanned by the normalized eigenvector $\backslash scriptstyle|\backslash lambda\_i\backslash rang$, $P\_i$ is equal to $\backslash scriptstyle|\backslash lambda\_i\backslash rang\backslash lang\backslash lambda\_i|$, so the probability $\backslash scriptstyle\backslash lang\backslash psi|P\_i|\backslash psi\backslash rang$ is equal to $\backslash scriptstyle\backslash lang\backslash psi|\backslash lambda\_i\backslash rang\backslash lang\backslash lambda\_i|\backslash psi\backslash rang$. Since the complex number $\backslash scriptstyle\backslash lang\backslash lambda\_i|\backslash psi\backslash rang$ is known as the
*probability amplitude*that the state vector $\backslash scriptstyle|\backslash psi\backslash rang$ assigns to the eigenvector $\backslash scriptstyle|\backslash lambda\_i\backslash rang$, 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 $\backslash scriptstyle|\backslash lang\backslash lambda\_i|\backslash psi\backslash rang|^2$.)

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,

- the probability that the result of the measurement lies in a measurable set $M$ will be given by $\backslash scriptstyle\backslash lang\backslash psi|Q(M)|\backslash psi\backslash rang$.

If we are given a wave function $\backslash scriptstyle\backslash psi$ 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)=$$\backslash scriptstyle|\backslash psi(x,y,z,t\_0)|^2.$

## History

The Born rule was formulated by Born in a 1926 paper.^{[3]}
In this paper, Born solves the Schrödinger equation for a scattering problem and, inspired by Einstein's work on the photoelectric effect,^{[4]} 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.^{[5]} John von Neumann discussed the application of spectral theory to Born's rule in his 1932 book.^{[6]}

## References

## See also

- Gleason's theorem
- Transactional interpretation of quantum mechanics

## External links

- Quantum Mechanics Not in Jeopardy: Physicists Confirm a Decades-Old Key Principle Experimentally ScienceDaily (July 23, 2010)