Calculate the rate of probability density's movement?

In summary, the conversation discusses calculating the rate at which the right-going wave brings probability density up to a barrier in a finite potential well. The solution involves using the average probability density and velocity to find the rate. The discussion also touches on the presence of left-going waves and the concept of particles attempting to cross the barrier. Ultimately, it is determined that the left-going waves have a negligible effect on the calculation.
  • #1
friendbobbiny
49
2
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1. Homework Statement

Given the following diagram of a finite potential well, calculate the rate at which the right-going wave is bringing probability density up to the barrier. (Ignore interference with the left-going wave. ) (Hint: you can get the velocity from the energy, and the average probability density from assuming that the integral over the well must give 1, when both left and right-going parts are included.). You can think of this rate as the rate at which the particle ‘attempts’ to cross the barrier.

Homework Equations



[tex]\frac{dP}{dt} = \frac{dP}{dx} \frac{dx}{dt} [/tex]

The Attempt at a Solution


Using the concession given in the question -- that we can use the average probability density to calculate the answer, [tex]\frac{dP}{dx} = \frac{1}{W+L}[/tex].

Speed is given by solving for v in [tex]E = 0.5mv^2[/tex]

Thus, we should have [tex] \frac{\sqrt(2E/m)}{(W+L)} [/tex]

The actual answer is [tex] \frac{\sqrt(2E/m)}{(2W)} [/tex]

For this to be true, average probability would have to be estimated as [tex]\frac{dP}{dx} = \frac{1}{2W}[/tex]. Why?
 
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  • #2
friendbobbiny said:
You can think of this rate as the rate at which the particle ‘attempts’ to cross the barrier.
In this statement, you may ignore [itex] L [/itex] and since there are two ways of waves, left and right, you just divide it as 2.
 
  • #3
Daeho Ro said:
since there are two ways of waves, left and right, you just divide it as 2.

I understand why waves would travel rightwards. Why would they travel leftwards? To travel leftwards, a wave would have to first tunnel through the barrier and then tunnel back. A wave wouldn't tunnel back, because once it tunnels rightwards (ie. exits the barrier), it is unenclosed and will now move rightwards.
 
  • #4
If the time is passed, then it can be correct but in this problem, I guess the time is fixed.

Addendum : The probability density what you get is averaged. So, for each point, the probability density is given by
[tex] \dfrac{dP}{dx} = \dfrac{\sqrt{2E/m}}{W}. [/tex]
However, there always exist left and right moving particles for each point and as the statement says, the particle "attempts" to cross the barrier can only be the right moving particles at the boundary. The left moving particle can be thought as the bounced particles at the wall.

When the time passes, the particles only escape to the right side but the probability density will not change. Some particles can bounce again and again.
 
Last edited:

1. What is the formula for calculating the rate of probability density's movement?

The formula for calculating the rate of probability density's movement is the change in probability density divided by the change in time.

2. How do you measure the change in probability density?

The change in probability density can be measured by taking the difference between the current probability density and the previous probability density at a specific time interval.

3. What units are used to express the rate of probability density's movement?

The rate of probability density's movement is typically expressed in units of probability per unit of time, such as per second or per day.

4. What factors can affect the rate of probability density's movement?

The rate of probability density's movement can be affected by various factors such as changes in the underlying variables, external influences, and the accuracy of the data used to calculate the probability density.

5. How can the rate of probability density's movement be used in scientific research?

The rate of probability density's movement is a valuable tool in scientific research as it can provide insights into the changes and trends of a particular phenomenon over time. It can also help in predicting future outcomes and identifying potential patterns in data.

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