Eigenstate

Definition

An eigenstate for an operator $\hat{A}$ is a state $| a ⟩$ such that $A | a ⟩ = a | a ⟩$ .

A state vector $| ψ ⟩$ is an eigenvector of $\hat{A}$ iff $\hat{A} | ψ ⟩ = α | ψ ⟩$ . $α$ is the eigenvalue of the operator.

The eigenvalues of a Hermitian operator are real and eigenstates of $\hat{A}$ correspond to different eigenvalues are orthogonal.

Proof.

$A | φ_{n} ⟩ = a_{n} | φ_{n} ⟩$ .

(1) $⟨ φ_{m} | A | φ_{n} ⟩ = a_{n} ⟨ φ_{m} | φ_{n} ⟩$ .

$⟨ φ_{m} | A^{†} = a_{m}^{*} ⟨ φ_{m} |$ .

(2) $⟨ φ_{m} | A^{†} | φ_{n} ⟩ = a_{m}^{*} ⟨ φ_{m} | φ_{n} ⟩$ .

(1) - (2): $a_{n} ⟨ φ_{m} | φ_{n} ⟩ - a_{m}^{*} ⟨ φ_{m} | φ_{n} ⟩ = (a_{n} - a_{m}^{*}) ⟨ φ_{m} | φ_{n} ⟩ = 0$

For $m = n$ , $a_{n} - a_{m}^{*} = 0$ , hence $a_{n} = a_{n}^{*}$ . Therefore, the eigenvalues are real.

For $a_{m} \neq a_{n}$ (different eigenvalues), $⟨ φ_{m} | φ_{n} ⟩ = 0$ , thus they are orthogonal.

Measuring $A$ . $ψ (\vec{r}, t) = \sum_{n} c_{n} | φ_{n} ⟩$ , $a_{n} \to | c_{n} |^{2}$ , $c_{n} = ⟨ φ_{n} | ψ ⟩$ .