>>10898843>How do the quantum phenomena emerge into classic physics phenomena?they emerge in the classical limit.
The time derivative of a quantum operator , according to ehrenfest's theorem, is
where the angle brackets have their usual meaning as the expectation value of an operator given a state , i.e.
consider the time derivative of the position and momentum operators, and dispensing with the angle brackets since I don't want to write \langle and \rangle again, just understand that everything is an expectation value of a quantum state.
If we have the hamiltonian , then
the partial derivative vanishes since the momentum operator has no explicit time dependence.
Now it is easy to show by writing as a taylor series and using the canonical commutation relation that
, thus we have that
i.e. Newton's second law. Note that this is not exactly the same as Newton's second law in classical mechanics, since this relates expectation values of the operators. However, in the classical limit, the probability distribution of the operators approaches a delta function, and thus we have
where the lower case denotes that they are classical variables rather than quantum operators.
