Orthogonality of Two Eigenvectors | Quantum Science Philippines
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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 operator Â,

Â|ψ⟩ = λ|ψ⟩     (1)

Â|φ⟩ = μ|φ⟩     (2)

Since Â is 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⟩ = 0

Consequently, the fact that λ≠μ, (λ-μ) ≠ 0 which then implies that ⟨φ|ψ⟩ = 0.

And as we know, this is the condition for two orthogonal eigenvectors.



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