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Mathematics for Physics

110 Eigenvalue

A number for which a nonzero state is scaled by an operator that is used to identify an allowed measured value of an observable.

The definition of an eigenvalue. If \(A|\psi\rangle=a|\psi\rangle\) with \(|\psi\rangle\neq 0\), then \(a\) is an eigenvalue of \(A\). This principle is used to list the possible outcomes of a measurement of \(A\).

The eigenvalue equation is

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

where

  • \(A\) is an operator.
  • \(|\psi\rangle\) is a nonzero eigenket.
  • \(a\) is the eigenvalue.

Reality for Hermitian operators. A Hermitian operator has real eigenvalues. This principle is used to guarantee that measured values are real numbers.

Energy eigenvalues of \(\hat{H}\). The eigenvalues of the Hamiltonian are the allowed energies. This principle is used to obtain the discrete spectrum of a bound system.

The energy eigenvalue equation is

\[ \hat{H}\psi = E\psi \]

where

  • \(\hat{H}\) is the Hamiltonian.
  • \(\psi\) is an energy eigenfunction.
  • \(E\) is the energy eigenvalue.

110.1 References

  1. Sakurai, J. J., & Napolitano, J. Modern Quantum Mechanics. Cambridge University Press, 2021. — eigenvalues of Hermitian observables.
  2. Hall, B. C. Quantum Theory for Mathematicians. Springer, 2013. — self-adjoint operators and real spectra.