Born rule

Details

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

• 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 ${\displaystyle ⟨\psi |P_i|\psi ⟩}$, 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 ${\displaystyle |{\lambda }_i⟩}$, ${\displaystyle P_i}$ is equal to ${\displaystyle |{\lambda }_i⟩⟨{\lambda }_i|}$, so the probability ${\displaystyle ⟨\psi |P_i|\psi ⟩}$ is equal to ${\displaystyle ⟨\psi |{\lambda }_i⟩⟨{\lambda }_i|\psi ⟩}$. Since the complex number ${\displaystyle ⟨{\lambda }_i|\psi ⟩}$ is known as the probability amplitude that the state vector ${\displaystyle |\psi ⟩}$ assigns to the eigenvector ${\displaystyle |{\lambda }_i⟩}$, it is common to describe the Born rule as saying that probability is equal to the amplitude-squared (really the amplitude times its own complex conjugate). Equivalently, the probability can be written as ${\displaystyle |⟨{\lambda }_i|\psi ⟩{|}^2}$.)

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

• the probability that the result of the measurement lies in a measurable set ${\displaystyle M}$ is given by ${\displaystyle ⟨\psi |Q(M)|\psi ⟩}$.

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:

$${\displaystyle p(i)=\mathrm{tr}(\rho F_i),}$$

where ${\displaystyle \mathrm{tr}}$ is the trace operator. This is the POVM version of the Born rule. When the quantum state being measured is a pure state ${\displaystyle |\psi ⟩}$ this formula reduces to: $${\displaystyle p(i)=\mathrm{tr}(|\psi ⟩⟨\psi |F_i)=⟨\psi |F_i|\psi ⟩.}$$

The Born rule, together with the unitarity of the time evolution operator ${\displaystyle e^{-i\hat{H}t}}$ (or, equivalently, the Hamiltonian ${\displaystyle \hat{H}}$ being Hermitian), implies the unitarity of the theory: a wave function that is time-evolved by a unitary operator will remain properly normalized. (In the more general case where one considers the time evolution of a density matrix, proper normalization is ensured by requiring that the time evolution is a trace-preserving, completely positive map.)