# Potential step and tunneling effect

• I
• Maximilian2
In summary, according to the author, when a plane wave with energy less than the potential crosses a potential step, the wave function decays exponentially and the momentum is imaginary.f

#### Maximilian2

TL;DR Summary
How a particle behave when passing through a potential step when its energy is less than the potential itself
We know that thanks to the tunnel effect, in the case of a finite potential step (V) and considering a stationary state, when a plane wave with energy E < V encounter the step the probabability that the wave-particle coming from -∞ (where potential is V=0) will be ≠ 0, in particular the wave function will be exponential decay. We can also calculate the probability flux (J) through the potential step and the result is J=0. In my book i read that taking into account all these results, the interpretation that we can give is that considering many particles, a certain percentage will cross the step and after a definite amount of time it will turns back before setting out in the direction where it came from, this vision allow us to justify why J=0. Here is my question: once (and if) the wave-particle cross the potential step, shouldn't continue its path without turning back? There is a cause that force it to reverse the direction and that can be explain from an "intuitive" point of view?

We can also calculate the probability flux (J) through the potential step
Would this be for ##x>a## in the Wikipedia casus ? Or only inside the barier ?

In my book
What book is that ?

I'm referring inside the barier, but in the case of a step potential like in the picture:

The book is "Quantum Mechanics" by Claude Choen-Tannoudji, Bernard Diu, and Franck Laloe. (I don't know if it will help but the pages where it's discussed are 65-68, 75-77 and in particular the question i posted is referred to pag. 285) thanks

Ah! Based on the terms 'tunneling'and 'through', I took the barrier example.
For the step function Cohen-Tannoudji chooses appropriate -- careful -- wording.

In the version I can access on p 282 I find Complement BIII Study of probability current in some special cases / 2. Application to potential barrier problems / b. case where E < V0 :

The discussion on 285 is on a two-dimensional case, right ?

But I can't find 'a certain percentage will cross the step' ?

Summary:: How a particle behave when passing through a potential step when its energy is less than the potential itself

In my book i read that taking into account all these results, the interpretation that we can give is that considering many particles, a certain percentage will cross the step and after a definite amount of time it will turns back before setting out in the direction where it came from, this vision allow us to justify why J=0. Here is my question: once (and if) the wave-particle cross the potential step, shouldn't continue its path without turning back? There is a cause that force it to reverse the direction and that can be explain from an "intuitive" point of view?
I think you are referring to

And that suggests a path through region II for the whole wave packet -- the 'particle'. I find it slightly misleading (but correct). Note that the x-momentum in that region is imaginary !

Re 'shouldn't continue its path without turning back?' No. The wave function decays exponentially and the momentum is imaginary. J = 0. Why should it ?

I am visually oriented so I google 'potential barrier animation' and I like this one and this old one

Maximilian2
Thanks that helped a lot to clarify. I have just only one more doubt: I don't know if it's correct but when i think about a poential step I naively think about the classic ball that must overcome the slope of an hill, but once it reaches the top (overcome the potential barrier) i can't see what would be the reason that cause the object to turn back; I mean there are no forces nor interactions.
I read in other threads that in the case of a barrier potential we can consider a "sandwich" of different material (https://www.physicsforums.com/threa...rrier-tunneling-and-potential-barrier.334711/), but I'm not sure if it's a correct vision of the problem.

about a potential step I naively think about the classic ball that must overcome the slope of an hill
Not really a step potential, but nevertheless:

Goes to show that a wave is not a tennis ball

##\mathstrut \ ##

#### Attachments

• 1606252739337.png
37.2 KB · Views: 89
@vanhees71 : What do you think of fig 4 in post #3 ? Is it exaggerating <x> venturing into x>0 ?

I don't know, how to read this figure (I think you mean #4, right?). The modulus-squared wave function tells us the probability distribution for the position at time ##t##. Why Cohen-Tanoudji draws a trajectory I don't know.

you mean #4, right?
Yes, sorry. I have the feeling <x> stays well within x<0 and am curious about the relationship with the Heisenberg uncertainty (but too lazy/not handy enough to work it out in detail)