Born rule

The Born rule is a postulate of quantum mechanics that gives the probability that a measurement of a quantum system will yield a given result. In one commonly used application, it states that the probability density for finding a particle at a given position is proportional to the square of the amplitude of the system's wavefunction at that position. It was formulated and published by German physicist Max Born in July 1926.

The Born rule states that an observable, measured in a system with normalized wave function ${\displaystyle |\psi \rangle }$ (see Bra–ket notation), corresponds to a self-adjoint operator ${\displaystyle A}$ whose spectrum is discrete if:

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 (PVM) ${\displaystyle Q}$, the spectral measure of ${\displaystyle A}$. In this case:

For example, a single structureless particle can be described by a wave function ${\displaystyle \psi }$ that depends upon position coordinates ${\displaystyle (x,y,z)}$ and a time coordinate ${\displaystyle t}$. The Born rule implies that the probability density function ${\displaystyle p}$ for the result of a measurement of the particle's position at time ${\displaystyle t_{0}}$ is: $${\displaystyle p(x,y,z,t_{0})=|\psi (x,y,z,t_{0})|^{2}.}$$ The Born rule can also be employed to calculate probabilities (for measurements with discrete sets of outcomes) or probability densities (for continuous-valued measurements) for other observables, like momentum, energy, and angular momentum.

In some applications, this treatment of the Born rule is generalized using positive-operator-valued measures (POVM). A POVM is a measure whose values are positive semi-definite operators on a Hilbert space. POVMs are a generalization of von Neumann measurements and, correspondingly, quantum measurements described by POVMs are a generalization of quantum measurements described by self-adjoint observables. In rough analogy, a POVM is to a PVM what a mixed state is to a pure state. Mixed states are needed to specify the state of a subsystem of a larger system (see purification of quantum state); analogously, POVMs are necessary to describe the effect on a subsystem of a projective measurement performed on a larger system. POVMs are the most general kind of measurement in quantum mechanics and can also be used in quantum field theory. They are extensively used in the field of quantum information.

In the simplest case of a POVM with a finite number of elements acting on a finite-dimensional Hilbert space, a POVM is a set of positive semi-definite matrices ${\displaystyle \{F_{i}\}}$ on a Hilbert space ${\displaystyle {\mathcal {H}}}$ that sum to the identity matrix,: $${\displaystyle \sum _{i=1}^{n}F_{i}=I.}$$

The POVM element ${\displaystyle F_{i}}$ is associated with the measurement outcome ${\displaystyle i}$, such that the probability of obtaining it when making a measurement on the quantum state ${\displaystyle \rho }$ is given by:

$${\displaystyle p(i)=\operatorname {tr} (\rho F_{i}),}$$