- #1
EquationOfMotion
- 22
- 2
So say our inner product is defined as ##\int_a^b f^*(x)g(x) dx##, which is pretty standard. For some operator ##\hat A##, do we then have ## \langle \hat A ψ | \hat A ψ \rangle = \langle ψ | \hat A ^* \hat A | ψ \rangle = \int_a^b ψ^*(x) \hat A ^* \hat A ψ(x) dx##? This seems counter-intuitive. Say our operator is ##\frac{d}{dx}## and ##ψ(x)=x##. Then, evidently ## \langle \hat A ψ | \hat A ψ \rangle = b-a## and ## \langle ψ | \hat A ^* \hat A | ψ \rangle = 0##, which is obviously incorrect. What am I missing?