Introduction to Quantum Mechanics · Problem A.27

Afterword — David J. Griffiths

Physics — Quantum Mechanics · hermitian-transformations

Problem

A hermitian linear transformation must satisfy ⟨α∣T^β⟩=⟨T^α∣β⟩\langle \alpha | \hat{T}\beta \rangle = \langle \hat{T}\alpha | \beta \rangle for all vectors ∣α⟩|\alpha\rangle and ∣β⟩|\beta\rangle. Prove that it is (surprisingly) sufficient that ⟨γ∣T^γ⟩=⟨T^γ∣γ⟩\langle \gamma | \hat{T}\gamma \rangle = \langle \hat{T}\gamma | \gamma \rangle for all vectors ∣γ⟩|\gamma\rangle. Hint: First let ∣γ⟩=∣α⟩+∣β⟩|\gamma\rangle = |\alpha\rangle + |\beta\rangle, and then let ∣γ⟩=∣α⟩+i∣β⟩|\gamma\rangle = |\alpha\rangle + i|\beta\rangle.

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