# Quantum mechanics on operator [closed]

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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## 1 Answer

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.

• Can you explain the explicitly dependence by an example. – SUMANTA SAHOO Apr 2 at 9:21
• What in particular don't you understand? – gented Apr 2 at 9:25
• The explicitly dependence of operator on time – SUMANTA SAHOO Apr 2 at 11:28
• 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). – gented Apr 2 at 11:49