# Why is quantum mechanical momentum the derivative of the wave function with respect to the position? [duplicate]

In classical mechanics the momentum is defined as mass times the time-derivative of position.

In quantum mechanics, however, the time-derivative of the wave function is the hamiltonian, while the momentum is defined as $i\hbar \frac{\partial}{\partial x} \Psi(x)$, which is a space-derivative and not a time-derivative.

Note that I understand why momentum is an operator on the wave function (it's a measurable quantity, so it's an operator as per a postulate of QM). I understand the derivation from spatial translation, but I don't understand why it's an equivalent of the classical momentum as it's a space derivative and not a time derivative.