Introduction to Quantum Mechanics · Problem 3.3

Formalism — David J. Griffiths

Physics — Quantum Mechanics · hermitian_operators

Problem

Show that if

\langle h|\hat{Q}h\rangle = \langle \hat{Q}h|h\rangle

for all

h

(in Hilbert space), then

\langle f|\hat{Q}g\rangle = \langle \hat{Q}f|g\rangle

for all

f

and

g

(i.e. the two definitions of "hermitian"—Equations 3.16 and 3.17—are equivalent). Hint: First let

h = f + g

, and then let

h = f + ig