How can we define a velocity for quantum objects? I have a question about quantum mechanics: I know that velocity is defined as the change of position with time, $v = \frac{\mathrm{d}x}{\mathrm{d}t}$.
In quantum mechanics, the position of a particle is not certain, but of a statistical nature. How can we define a velocity for quantum objects?
 A: In the Heisenberg picture of quantum mechanics, the position operator is itself time-dependent, and you may just define the "velocity operator" $\dot{x}$ as in classical mechanics. However, the Heisenberg equation of motion says
$$ \dot{x} = \mathrm{i}[H,x]$$
and e.g. for a free particle with $H= \frac{p^2}{2m}$, we have $[H,x] \propto p$, so this velocity operator is just proportional to the momentum operator (as one would classical expect). In particular, it does not commute with the position operator, you cannot know the position and the velocity of a particle simultaneously to arbitrary precision.
A: QM usually operates with momentum. Momentum operator is giver by $$ \hat p =-i\hbar{\frac{\partial }{\partial x}}$$
You could say that velocity is momentum divided by mass of the particle.
A: Defining velocity of a quantum  object- say a particle-
In quantum  representation the physical variables are operators in the vector space mapped by eigen vectors .
 so a position of a particle , its  momentum ,Energy , spin ...are all represented by an operator - 
instead of velocity the momentum operator(P) can be measured therefore the expected value of measurement of P(operator) can give us idea about value of its velocity;
i.e.  one can write   = m  or
 result of velocity measurement 
                               = /m
If one has set up the  Hamiltonian(op) of the physical system;  H=Top +Vop
where T is kinetic energy operator and V is Potential energy operator; Thus the Hamiltonian is Total Energy  operator of the system.
then Rate of Change of  x = (i/h bar) [ Hop. , x] where the bracketed term is quantum mechanical commutator (analogous to Poisson Bracket in classical mechanics).
dxop/dt = (i/h bar) [ T + V , x]=(i/h bar){ [T,x]  + [ v,x]}
 be substituted and using commutation relations  of P and x  the rate of change of x can be calculated.
as T(op) = (Pop)^2/2m  
