- #1
friendbobbiny
- 49
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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?