Eigenstate

An eigenstate is a state represented by a nonzero vector that is unchanged up to scale by an operator. For an operator $A$,

$$A\lvert a\rangle = a\lvert a\rangle.$$

If $A$ is an observable and $\lvert a\rangle$ is normalized, an ideal projective measurement of $A$ gives the eigenvalue $a$ with probability one.

Formula Hook

For a nondegenerate discrete observable,

$$A = \sum_n a_n \lvert a_n\rangle\langle a_n\rvert,$$

and an arbitrary normalized state expanded in that basis has amplitudes

$$\lvert\psi\rangle = \sum_n c_n\lvert a_n\rangle, \qquad c_n=\langle a_n\rvert\psi\rangle.$$

The probability of outcome $a_n$ is $\lvert c_n\rvert^2$.

Canonical Home

See Eigenvalues and Eigenstates for the Core discussion, Spectral Decomposition for the operator-level statement, and Born Rule for Discrete Spectra for probabilities in an eigenbasis.

Common Confusions

• The zero vector is never an eigenstate.
• A degenerate eigenvalue corresponds to an eigenspace, not one preferred vector.
• Continuous-spectrum “eigenstates” are usually generalized states, not normalizable Hilbert-space vectors.
• Being an eigenstate of one observable does not make a state an eigenstate of every observable.

Related Entries

• Observable
• Born Rule
• Hilbert Space

References

• D. J. Griffiths and D. F. Schroeter, Introduction to Quantum Mechanics, 3rd ed., Cambridge University Press, 2018.
• 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.