Softened potential well / potential step

[hilbert2]

TL;DR Summary

Approximating a finite potential step with a continuous function

Do any of you know of an article or book chapter that discusses the difference between a discontinuous potential well of length $2L$

$V(x)=\left\{\begin{array}{cc}0, & |x-x_0 |<L\\V_0 & |x-x_0 |\geq L\end{array}\right.$

and a differentiable one

$\displaystyle V(x) = V_0 \exp\left[-\left(\frac{|x-x_0 |}{L}\right)^n\right]$

Edit: the correct equation above should have been

$\displaystyle V(x) = V_0 - V_0 \exp\left[-\left(\frac{|x-x_0 |}{L}\right)^n\right]$with $n$ some large but finite exponent (limit $n\rightarrow\infty$ is equivalent to the discontinuous version) ?

There was one article discussing the non-classical reflection of a wave packet at a downward potential step that considered the difference between these: https://aapt.scitation.org/doi/abs/10.1119/1.3636408

It should be possible to handle the difference between these two as a small perturbation, if the energy eigenstates and eigenvalues of the discontinuous basic potential well are already known (at least if $V_0$ isn't a very large or infinite number). Doing this for a 2D or 3D well of some shape would be more involved.

 

[PeterDonis]

Shouldn't the sign inside the exponential be positive? It looks to me like the negative sign, in the limit $n \to \infty$, gives a "downward step", but the potential well you define is an "upward" step.

 

[BvU]

It becomes a well if $V_0<0$

 

[PeterDonis]

It becomes a well if $V_0<0$

The exponential does, but the discontinuous version doesn't; it's a "downward step" if $V_0 < 0$. So they're still not the same.

 

[BvU]

I didn't look at the article (no access) only at the two potentials in post #1.

Don't think it very likely an exponential yields an analytical solution ...

 

[hilbert2]

Oh, sorry, it was supposed to be like

$\displaystyle V(x) = V_0 - V_0 \exp\left[-\left(\frac{|x-x_0 |}{L}\right)^n\right]$.

The reflection from downward potential step was just an example of an article that discusses the effects of making an initially sharp potential barrier continuous and differentiable. I'm mostly interested on the effect of this on the energy eigenvalues of a finite-depth potential well.

Edit: the problem has been studied in these two articles I found:

https://www.scielo.br/pdf/rbef/v42/1806-9126-RBEF-42-e20190222.pdf

https://iopscience.iop.org/article/10.1088/0953-4075/45/10/105102

The first is about the general problem and the second one specifically about modelling the confinement of an atom inside a fullerene cage.