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Mathematics for Physics

118 Expectation Values

A weighted average of an observable in a quantum state that is used to predict the mean outcome of many measurements of that observable, where an observable is a measurable quantity represented by a Hermitian operator.

The Dirac sandwich. The expectation value of an operator \(A\) in a normalized state is the sandwich \(\langle\psi|A|\psi\rangle\). This principle is used to compute the mean of many repeated measurements.

The expectation value is

\[ \langle A\rangle = \langle\psi|A|\psi\rangle \]

where

  • \(A\) is an observable operator.
  • \(|\psi\rangle\) is the normalized state.
  • \(\langle A\rangle\) is the expectation value.

The position mean. The expectation value of position is the first moment of the probability density. This principle is used to compute the mean position from \(\Psi(x)\).

The expectation value of position is

\[ \langle x\rangle = \displaystyle\int x\,\lvert\psi(x)\rvert^{2}\,dx \]

where

  • \(\langle x\rangle\) is the mean position.
  • \(\psi(x)\) is the wavefunction.

The definition of uncertainty. The uncertainty of an observable is the root-mean-square deviation from the expectation value. This principle is used to compute \(\Delta A\) for the uncertainty principle.

The uncertainty is

\[ \Delta A = \sqrt{\langle A^{2}\rangle - \langle A\rangle^{2}} \]

where

  • \(\Delta A\) is the uncertainty of \(A\).
  • \(\langle A\rangle\) is the expectation value of \(A\).

Note: Also called the mean value.

118.1 References

  1. Sakurai, J. J., & Napolitano, J. Modern Quantum Mechanics. Cambridge University Press, 2021. — \(\langle A\rangle=\langle\alpha|A|\alpha\rangle\).
  2. Hall, B. C. Quantum Theory for Mathematicians. Springer, 2013. — \(E(x)=\int x|\psi|^{2}dx\) and \(\langle X\rangle_{\psi}=\langle\psi,X\psi\rangle\).
  3. Shankar, R. Fundamentals of Physics II. Yale University Press, 2020. — \(\langle x\rangle=\int P(x)x\,dx\).