An operator is called anti-Hermitian if for all normalisable
PHYSICS · Subjective · Fermi University Physics · Quantum Mechanics · The formalism of quantum mechanics · The formalism of quantum mechanics
Quantum Mechanics: From the Photon to the Hydrogen Atom, Ch Observables, commutators and the uncertainty relations
Problem
An operator R^ is called anti-Hermitian if ∫f∗(R^g)dx=−∫(R^f)∗gdx for all normalisable f,g, the opposite sign from the Hermitian condition. Using the same sandwich argument this chapter used for Hermitian operators, show that every eigenvalue of an anti-Hermitian operator is purely imaginary (or zero). Then confirm that R^=d/dx, shown in this chapter not to be Hermitian, is anti-Hermitian, and find its eigenvalue for the eigenfunction eikx.