Construct operator such that division of expectation values is equal to expectation value of the operator – physics.stackexchange.com

Is is possible to construct an operator $\hat{C}$ out of $\hat{A}$ and $\hat{B}$ such that: $$\frac{\langle \psi|\hat{A}|\psi\rangle}{\langle\psi|\hat{B}|\psi\rangle} = \langle \psi|\hat{C}|\psi\...

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