## Orthogonality of Two Eigenvectors

### Orthogonality of Two Eigenvectors

*by: Mariel A. Escobal, MS Physics I, MSU-IIT*

### Prove that two eigenvectors of a Hermitian operator corresponding to two different eigenvalues are orthogonal.

Consider two eigenvectors

|ψ⟩and|φ⟩of the Hermitian operatorÂ,Â|ψ⟩ = λ|ψ⟩ (1)

Â|φ⟩ = μ|φ⟩ (2)

Since

Âis Hermitian, we can write (2) as⟨φ|Â = μ⟨φ| (3)

Multiplying (1) by

⟨φ|on the left and (3) by|ψ⟩on the right, we get⟨φ|Â|ψ⟩ = λ⟨φ|ψ⟩ (4)

⟨φ|Â|ψ⟩ = μ⟨φ|ψ⟩ (5)

Subtracting (4) and (5), we get

(

λ-μ)⟨φ|ψ⟩ = 0Consequently, the fact that

λ≠μ, (λ-μ) ≠ 0which then implies that⟨φ|ψ⟩ = 0.And as we know, this is the condition for two orthogonal eigenvectors.