- #1
Shiz
- 9
- 0
Homework Statement
Linear combination is [itex]\hat{A}[/itex] + i[itex]\hat{B}[/itex]. It's given that it is not Hermitian already.
Homework Equations
∫ψi * [itex]\hat{Ω}[/itex] ψj = (∫ψj * [itex]\hat{Ω}[/itex] ψi)*
The Attempt at a Solution
∫ψi * ([itex]\hat{A}[/itex] + i[itex]\hat{B}[/itex]) ψj = (∫ψj * ([itex]\hat{A}[/itex] + i[itex]\hat{B}[/itex]) ψi)*
I chose to work with the right hand side of the equation first.
∫ψi * ([itex]\hat{A}[/itex] + i[itex]\hat{B}[/itex]) ψj = {∫ψj * [itex]\hat{A}[/itex] ψi + i(∫ψj * [itex]\hat{B}[/itex] ψi)}*
So I have to take the complex conjugate of the right hand side (not sure if that's the proper way to say it). What I don't understand is why the operator would become ∫ψj * [itex]\hat{A}[/itex] ψi - i(∫ψj * [itex]\hat{B}[/itex] ψi) and then ∫ψi * ([itex]\hat{A}[/itex] - i[itex]\hat{B}[/itex]) ψj.
What are the mathematical reasons? The complex conjugate of + i[itex]\hat{B}[/itex] is -i[itex]\hat{B}[/itex]. I would just be replacing the operator with its complex conjugate? That would give me the answer, but it doesn't seem that simple. Clarification at this would help! Thank you!