UPSC Combined Geo-Scientist, Mains · 2020 · Mains Paper II

Q1(k). Hermitian operators

Question

State the condition for an operator to be Hermitian and prove its eigenvalues are real.

Â=Â^ on the appropriate domain; every eigenvalue of a normalisable eigenfunction is real.

Hermiticity means ⟨ f|Âg⟩=⟨Âf|g⟩ for allowed states, with boundary conditions that remove any integration-by-parts boundary terms.
If Âψ=aψ, then ⟨ψ|Âψ⟩=a⟨ψ|ψ⟩. By Hermiticity the same quantity equals ⟨Âψ|ψ⟩=a*⟨ψ|ψ⟩.
Therefore (a-a*)⟨ψ|ψ⟩=0. A nonzero physical state has positive norm, so a=a* and the eigenvalue is real.

The full model answer

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