Amplitude

An amplitude is a complex number whose squared modulus gives, or contributes to, a quantum probability after the measurement context has been specified. In a normalized pure state, the amplitude for outcome state $\lvert a\rangle$ is usually written

$$\alpha_a=\langle a\rvert\psi\rangle, \qquad P(a)=\lvert\alpha_a\rvert^2.$$

Amplitudes are not probabilities. Their phases and relative signs matter because amplitudes for indistinguishable alternatives add before probabilities are formed.

Formula Hook

For a transition under a unitary evolution operator,

$$\alpha_{\psi\to\phi}(t) = \langle\phi\rvert U(t,t_0)\lvert\psi\rangle, \qquad P_{\psi\to\phi}(t) = \lvert\alpha_{\psi\to\phi}(t)\rvert^2.$$

If two alternatives contribute amplitudes $\alpha_1$ and $\alpha_2$ to the same outcome, then

$$P = \lvert\alpha_1+\alpha_2\rvert^2,$$

so the cross term encodes interference.

Canonical Home

See Probability Amplitudes for the teaching page, Born Rule for the probability rule, and Transition Probabilities for time-evolution language.

Common Confusions

• A complex amplitude is not directly observed; probabilities and expectation values are.
• An expansion coefficient is an amplitude only relative to a specified basis and measurement context.
• Global phase has no observable effect, but relative phase can change interference.
• A probability density such as $\lvert\psi(x)\rvert^2$ must be integrated over a region to produce a probability.

Related Entries

• Born Rule
• Superposition
• Wavefunction

References

• M. Born, “Zur Quantenmechanik der Stossvorgaenge,” Zeitschrift fuer Physik 37, 863-867, 1926.
• P. A. M. Dirac, The Principles of Quantum Mechanics, 4th ed., Oxford University Press, 1958.
• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020.
• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994.