Q1(l). Eigenvalues of a Hermitian operator are real

Answer in brief

For Âψ = aψ, the Hermitian property gives (a − a*)∫|ψ|² dτ = 0; since ∫|ψ|² dτ > 0, a = a*, so a is real.

Key points

• Hermitian definition: ∫f*Âg dτ = ∫g(Âf)* dτ.
• Multiply Âψ = aψ by ψ* and integrate: ∫ψ*Âψ dτ = a∫ψ*ψ dτ (1).
• The complex conjugate (Âψ)* = a*ψ* and the Hermitian property with f = g = ψ give ∫ψ*Âψ dτ = a*∫ψψ* dτ (2).
• (1) − (2): (a − a*)∫|ψ|² dτ = 0; the integral is positive for a non-zero ψ, so a = a* and a is real.
• Every observable is a Hermitian operator because a measured value must be real; p̂x = −iℏ d/dx is Hermitian, d/dx alone is not.