If any operator is commute with Hamilton then they are labelled such a way that the energy eigenstate are equal and we also know it is a constant of motion. I don't related constant of motion with above theory. please help me


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Let $A$ be any self-adjoint operator on a Hilbert space and $H$ the Hamiltonian of a quantum system: one can show (Ehrenfest theorem) that $$ \frac{d}{dt}\langle A \rangle = \frac{1}{i\hbar}\langle [A, H] \rangle + \langle \frac{\partial A}{\partial t} \rangle $$ where $\langle A \rangle$ denotes the expectation value onto a generic state $|\psi\rangle$. Therefore unless the operator explicitly depends on time, it is a constant of motion as long as it commutes with the Hamiltonian.

  • $\begingroup$ Can you explain the explicitly dependence by an example. $\endgroup$ – SUMANTA SAHOO Apr 2 at 9:21
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    $\begingroup$ What in particular don't you understand? $\endgroup$ – gented Apr 2 at 9:25
  • $\begingroup$ The explicitly dependence of operator on time $\endgroup$ – SUMANTA SAHOO Apr 2 at 11:28
  • $\begingroup$ Well, take any function explicitly depending on time (this has nothing to do with quantum mechanics: if the problems are what a partial derivate is then it is another question). $\endgroup$ – gented Apr 2 at 11:49

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