# Free particle problem. Traveling time

1. Mar 9, 2012

### atlantic

I have a particle with a known wave function, and probaility density $|\Psi(x,t)|^{2}$. I also know the expectation value of the position, which is:

$<x> = x_{0} + (l\hbar/m)t$,​

where t is the time, m is the mass of the particle and $x_{0}$ and $l$ are some known constants.

The problem is to determine how long it takes for the particle to travel a distance, y.

I was thinking that the distance y equals the change in $<x>$ from the time $t_{0}$ to $t_{1}$, where ($Δt = t_{1}-t_{0}$). In which case I would have to solve:

$y = x_{0} + (l\hbar/m)t_{1} - (x_{0} + (l\hbar/m)t_{0})$,​

so that the time it takes for the particle to travel the distance y is:
$Δt = (ym)/l\hbar$​

However, I don't have the answer to this question. So would somebody tell me if me solution is correct?