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Mathematics for Physics

111 Eigenvector

A nonzero state that an operator maps to a scalar multiple of itself that is used to identify the state left unchanged in direction by that operator.

The definition of an eigenvector. If \(A|\psi\rangle=a|\psi\rangle\) with \(|\psi\rangle\neq 0\), then \(|\psi\rangle\) is an eigenvector of \(A\) belonging to \(a\). This principle is used to name the post-measurement state when \(a\) is obtained.

The eigenvector equation is

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

where

  • \(A\) is an operator.
  • \(|\psi\rangle\) is the eigenvector.
  • \(a\) is the corresponding eigenvalue.

Orthogonality for Hermitian operators. Eigenvectors of a Hermitian operator belonging to different eigenvalues are orthogonal. This principle is used to expand a general state in an orthonormal eigenbasis.

The post-measurement eigenstate. After a measurement that yields \(a\), the system is in the corresponding eigenvector. This principle is used to update the state.

Note: Also called an eigenket when the state is written as a ket. Also called an eigenfunction when the state is a wavefunction.

111.1 References

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