125 Measurement

A process that yields an eigenvalue of an observable and updates the state that is used to connect a quantum state to a definite laboratory outcome, where an eigenvalue is an allowed measured value of that observable.

The eigenvalue-outcome rule. A measurement of an observable \(A\) yields one of the eigenvalues of \(A\). This principle is used to list the possible laboratory outcomes.

Collapse to an eigenstate. If the result is the eigenvalue \(a'\), the state immediately afterward is the corresponding eigenstate. This principle is used to update the state after a measurement.

The collapse to an eigenstate is

\[ \hat{A}|\psi'\rangle = a'|\psi'\rangle \]

where

\(\hat{A}\) is the operator for the observable.
\(a'\) is the measured eigenvalue.
\(|\psi'\rangle\) is the state immediately after the measurement.

The Born probability rule. If the state before measurement is \(\sum c_{n}|a_{n}\rangle\), the probability of result \(a_{n}\) is \(|c_{n}|^{2}\). This principle is used to compute the statistics of repeated measurements.

The Born rule for a discrete spectrum is

\[ P(a_{n}) = \lvert\langle a_{n}|\psi\rangle\rvert^{2} \]

where

\(P(a_{n})\) is the probability of eigenvalue \(a_{n}\).
\(|\psi\rangle\) is the state before measurement.
\(|a_{n}\rangle\) is the corresponding eigenket.

Note: Also called the collapse of the wave function when the post-measurement state is emphasized.

125.1 References

Hall, B. C. Quantum Theory for Mathematicians. Springer, 2013. — Axiom 4: measurement result \(\lambda\) and collapse to an eigenstate of \(\hat{f}\).
Sakurai, J. J., & Napolitano, J. Modern Quantum Mechanics. Cambridge University Press, 2021. — measurement yields an eigenvalue of the observable.

124 Magnetic Moment

126 Momentum Operator