Free particle problem. Traveling time

[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?

 

[mark726]

Hello, thank you for your question. Your solution is correct. The time it takes for the particle to travel a distance y is indeed given by Δt = (ym)/l\hbar. This can be seen by rearranging the equation you provided:

Δt = t_{1}-t_{0} = (y/m)/l\hbar

This shows that the time taken is dependent on the distance y and the mass of the particle, but is independent of the initial position x_{0} and the constants l and \hbar. This makes intuitive sense, as the particle's position and wave function are constantly evolving over time, but the distance traveled is solely determined by the initial conditions and the properties of the particle.