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If we write Schrodinger equation in imaginary time $\tau = it$, then one can easily show that the energy $E(\tau) = \langle \psi(\tau)| \hat{H} |\psi(\tau)\rangle$ is a diminishing quantity, i.e. $$\partial_{\tau} E(\tau) \le 0$$

Imagine that we have a state that depends on variational parameters denoted collectively by $\{R_{k} \}$. Can one prove in general that energy is a diminishing quantity if we impose some constraints on the state evolution, e.g. constant normalization ?

Equation of motion for $\{ R_{k}\}$ is derived from stationary action principle: $$ S[R_{k}] = \int d\tau \langle R_{k}(\tau)| \partial_{\tau} + \hat{H} | R_{k}(\tau)\rangle - \nu(\tau) \langle R_{k}(\tau)| R_{k}(\tau) \rangle$$

It holds for a trivial case: $$|C_{k}(\tau)\rangle = \sum\limits_{k}C_{k}(\tau) |k\rangle$$ where $|k\rangle$ are eigenstates of the Hamiltonian.

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