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In the video Prof. G. Rangarajan is considering the expectation value $$\mathbb{R}~\ni~ \langle \psi |\underbrace{\hat{L}_z}_{\text{self-adj.}} |\psi \rangle ~=~ \int\! d^3r ~\underbrace{\overline{\psi({\bf r})}}_{\text{real}} \underbrace{ (-i\hbar) \frac{\partial}{\partial \varphi}}_{\text{imaginary}} \underbrace{\psi({\bf r})}_{\text{real}} ...


I don't know the term "imaginary operator". I take this to be an antihermitian operator, which eigenvalues are purely imaginary. Then the statement is clearly not true. Take as a counter example any hermitian operator $\hat{A}$ and real wavefunction $\psi$ with $\langle \psi | \hat{A} |\psi\rangle = A_\psi \neq 0$. $A_\psi$ is of course real. Take now ...

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