Free particle problem. Traveling time

In summary, the time it takes for a particle to travel a distance y can be calculated using the equation Δt = (ym)/l\hbar, which accounts for the mass and the distance traveled, but is independent of the initial position and other constants.
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I have a particle with a known wave function, and probaility density [itex]|\Psi(x,t)|^{2}[/itex]. I also know the expectation value of the position, which is:

[itex]<x> = x_{0} + (l\hbar/m)t[/itex],​

where t is the time, m is the mass of the particle and [itex]x_{0}[/itex] and [itex]l[/itex] 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 [itex]<x>[/itex] from the time [itex]t_{0}[/itex] to [itex]t_{1}[/itex], where ([itex]Δt = t_{1}-t_{0}[/itex]). In which case I would have to solve:

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

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

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

1. What is a free particle problem?

A free particle problem is a concept in physics that describes the motion of a particle with no external forces acting upon it. In other words, the particle is not subject to any forces such as gravity or friction, and therefore, its motion is determined solely by its initial conditions.

2. How is the traveling time of a free particle calculated?

The traveling time of a free particle is calculated using the equation t = d/v, where t is the time, d is the distance traveled, and v is the velocity of the particle. This equation assumes that the particle is traveling at a constant velocity.

3. Can a free particle have a negative traveling time?

No, a free particle cannot have a negative traveling time. Time is a scalar quantity and cannot have a negative value. However, the displacement of a particle can be negative if it moves in the opposite direction of its initial position.

4. What factors can affect the traveling time of a free particle?

The traveling time of a free particle can be affected by its initial velocity, the distance it travels, and any external forces that may act upon it. Other factors such as air resistance or a change in direction can also affect the traveling time.

5. How is the traveling time of a free particle related to its kinetic energy?

The traveling time of a free particle is not directly related to its kinetic energy. However, the kinetic energy of a particle can affect its velocity, which in turn can affect its traveling time. A particle with a higher kinetic energy will have a higher velocity and therefore, a shorter traveling time compared to a particle with a lower kinetic energy.

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