What is the best estimate for the penetration distance in quantum mechanics?

[sachi]

We have a particle of energy E crossing a potential jump at x=0. for x<=0, V=0, for x>=0 V=V1
We get a wavefunction for x>=0 psi(x) = exp(-iEt/hbar)*exp(-Kx)
where K = (2m(V1-E))^0.5/hbar
N.b E<V1 so classically we get no transmission

we are asked to estimate the penetration distance, and I have found a solution which says let the penetration distance equal 1/K. I can't see physically why we would pick this (it just seems like a random number that means that the wavefunction will decrease by a factor 1/e, but I can't see why this is a sensible estimate).

Thanks

 

[Physics Monkey]

Well, $$1/\kappa$$ is the only relevant length scale in the problem, so the penetration depth has to be proportional to it. The point is that if you were to plot the wavefunction as a function of $$\kappa x$$, it would look the same no matter what the energy or barrier height were. In other words, if $$\kappa x < .1$$ nothing much happens and if $$\kappa x > 10$$ the wavefunction is essentially zero. Clearly, $$\kappa$$ determines the length scale over which the action happens. That being said, you have some freedom in your estimate. Maybe you would like to include a factor of $$\ln{2}$$ so you get the place where the wavefunction is one half its value at the boundary. In some other problem, it might be nice to include a factor of $$\pi$$ for convenience, for example. The convention is basically that anything within a power of ten of $$\kappa$$ is pretty much ok, but this isn't any kind of formal rule and people tend to go with the simple estimate. All you are really doing is identifying the length scale.

 

[Jelfish]

Consider what the probability distribution looks like.

$$\Psi = e^{\frac{-iE}{\hbar}t}e^{-\kappa x}$$
$$P=\Psi^* \Psi = e^{-2\kappa x}$$

Remember that the second part is completely real since V1>E.

As Physics Monkey said, this is a question of length scale. No matter what threshold of probability you choose (say, .1%), you must scale it must be a constant times 1/kappa because kappa can vary depending on what problem you're doing.

Let $$x=\frac{d}{\kappa}$$ be your chosen penetration distance, where d is just a constant.
$$P= e^{-2\kappa x}= e^{-2\kappa \frac{d}{\kappa}} = e^{-2d}$$

Then choose d according to how close you want the probability to be zero. Since this doesn't depend on kappa, it won't matter what E and V1 are in your problem.