Why we dont have "direct" velocity operator just as $p$? ( as use $p$ space not $v$? ) in quantum mechanics?

why there is no direct velocity operator on quantum mechanic while there is for mumentum ( $p_{x}=d/dx$ ) Also why use mumentum space not velocity?

Answer

There is a speed operator in quantum mechanics, as there's a time derivative operator for all operators, using the Heisenberg equation :

$$\begin{equation} \frac{d}{dt}A(t)=\frac{i}{\hbar}[H,A(t)]+ \frac{\partial A}{\partial t} \end{equation}$$

For speed, this will be

$$\begin{equation} v = \frac{d}{dt}x=\frac{i}{\hbar}[H,x] \end{equation}$$

A simple Hamiltonian is $H = \frac{p^2}{2m} + V(x)$. $x$ will commute with the potential, leaving

$$\begin{equation} v = \frac{i}{\hbar2m}[p^2,x] = \frac{p}{m} \end{equation}$$

which is the same relation as in classical mechanics, except with operators. A similar relation exists for $F = ma$, which is $\frac{dp}{dt} = \frac{i}{\hbar}[H,p] = -\nabla V(x)$.