A student offers this argument that is Hermitian: "

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

A student offers this argument that

\hat Q = d/dx

is Hermitian: "

\int f^*(\hat Qg)\,dx = \int f^*g'\,dx

. Integrating by parts,

= \big[f^*g\big]_{-\infty}^{\infty} - \int(f^*)'g\,dx

. The boundary term vanishes for normalisable functions, and

(f^*)' = (f')^*

, so

\int f^*(\hat Qg)\,dx = -\int(f')^*g\,dx = -\int(\hat Qf)^*g\,dx

. Since this differs from

+\int(\hat Qf)^*g\,dx

only by a minus sign, and a minus sign is just a convention,

\hat Q

is Hermitian." Find the error in this argument.