1. Feb 11, 2017

### nashsth

Hello, in my intro to quantum class, we recently started the chapter on quantum tunneling. While I understand the math, I don't understand conceptually what is going on. I do have my thoughts on the matter, but I am not sure that they are correct. I have a few questions:

1) Is the de Broglie wave the same thing as a wavefunction of a particle? Then, would this imply that the de broglie wavelength is the wavelength of the wavefunction?

2) Would it be valid to say that if the de broglie wavelength is large (so large that it is measurable), the particle that the de broglie wavelength "belongs" to has a larger uncertainty in position and that if the de broglie wavelength is small, the particle has a smaller uncertainty in position?

Linking this with the wavefunction concept, I am thinking that if the de broglie wavelength is large, then the wavefunction is more "spread out". Similarly if the de broglie wavelength is small, the wavefunction is more "localized" and so there is a greater probability of finding a particle within a smaller range.

3) Is quantum tunneling a phenomenon where a particle literally burrows through a potential barrier? Or is it just the observation that a particle can be on the other side of the barrier not because it literally burrows through but because the wavefunction of the particle is so "delocalized"/spread out that it spans the length of the potential barrier, hence the probability of observing the particle on the other side is small but non-zero? (In short, is quantum tunneling just an application of the uncertainty principle where there is a large uncertainty in position hence it is possible to find the particle on the other side of the barrier? The de broglie relation looks somewhat similar to the uncertainty principle: λ*p = h vs Δx*Δp ≥ ħ/2 so this is why I thought that there is a connection between the de broglie wave and the wavefunction.)

These questions arose when I attempted to answer the question "can a baseball tunnel through a thin window". My thoughts were that since the de broglie wavelength of a massive object like a baseball is very tiny (beyond detectable), it would mean that the wavefunction of the baseball is highly localized. Because of this localization, the wavefunction of the baseball doesn't span the length of the window hence the probability that the baseball tunnels through the window is practically 0.

-Nash

2. Feb 11, 2017

### houlahound

If I understand yr question, the particle is either inside or outside the barrier, it is not spread between the two.

3. Feb 11, 2017

### ZapperZ

Staff Emeritus

Zz.

4. Feb 11, 2017

### nashsth

Hi Zapper, it seems that you and I both agree on the wavefunction "leaking" through the barrier, as you wrote in this question:

So based on what you wrote there, the particle literally burrows through the barrier because its wavefunction "spans" the length of the barrier. Is this correct?

Also, could you provide insights as to the relationship between the de broglie wave and the wavefunction?

-Nash

5. Feb 11, 2017

### houlahound

The dB wavelength is not correlated to the wavefunction (WF).
the dB wavelength can exceed the wave function in fact, at normal energies it is much shorter than the WF.

6. Feb 11, 2017

### nashsth

I see... so there's no relationship whatsoever between the dB wave and the wavefunction?

7. Feb 11, 2017

### houlahound

Not in the sense that one relies on and varies with the other in a mathematical equation.

The external environment can affect both tho.

8. Feb 12, 2017

### A. Neumaier

The particle moves continuously from a position on one side to one on the other side of the barrier. But it does not move ''through the barrier'' since the barrier is not a wall but a metaphorical object - a bulge in the potential energy. On this metaphorical level it moves over the barrier, not through it. (For in the latter case, the tunneling probability should be independent of the height of a rectangular barrier. But it isn't. In particular, an infinitely high barrier cannot be penetrated, no matter how thin it is.) This complements the observation made by ZapperZ:

9. Feb 12, 2017

### ZapperZ

Staff Emeritus
I actually don't see how this compliments what I mentioned. I also do not know what it means that that particle moves over the barrier, and not through it. If we take this literally and not "metaphorically", then it somehow means that the particle has a higher energy than the barrier height, and simply moves over it rather than through it. This gives a different physics, i.e. it doesn't give a decaying wavefunction. So physic-wise, I have no idea what going "over the barrier" means.

I wrote elsewhere, and gave references to two papers that dealt with the tunneling matrix element explicitly:

In these two papers, one gets the group velocity of the particle in the barrier itself! Bardeen did an extensive, detailed study of the earlier Harrison paper to reveal this.

Considering that (i) these are two important papers in the physics of tunneling and (ii) that these are classic papers from many, many years ago, I am puzzled why this question of whether the particle goes through and inside the barrier is an issue now.

Zz.

10. Feb 12, 2017

### Karolus

The answer to this question can simply be given by solving the equation schrodinger in the case of a potential well. Mathematically there are solutions of the wave outside the barrier function. This function is beyond that of the barrier corresponds to a finite probability of finding the particle beyond the barrier. That the particle actually passes through, physically the barrier is experimentally demonstrated in the application of the scanning tunneling microscope. And that is all. More questions, merely deduced from the principles of QM, as the uncertainty principle, and non-locality. Of course you can set the schrodinger equation with a tennis ball, but as you have observed the wavelength of de broglie associated with a tennis ball is so small that it makes no sense to apply the equation schroringer a ball tennis. The MQ has the sense of scale of the Planck constant.

11. Feb 12, 2017

### Staff: Mentor

This is not an answer to the OP question, but an interesting side point.

Professor Leonard Susskind in a lecture about QM commented on a student's question about tunneling. He said that to "catch" the tunneling particle in the middle of the barrier you would at the minimum have to hit it with a photon of sufficiently high energy to localize its position to within the barrier's width. When you calculate the minimum energy of such a photon, it comes out to be the same as the energy needed to lift the particle over the barrier in the first place. That would add a new question -- did the particle tunnel or get lifted over? Susskind said that was an example of the kinds of problems you run into when you try to think of clever ways to peek under the covers of QM principles. There is always a got-ya catch that defeats your experiment.

12. Feb 12, 2017

### Karolus

Yes. since the wave function, the schrodinger equation solution exists even inside the barrier, as well as outside. It is not a magician who makes appear and disappear rabbits in his hat

13. Feb 12, 2017

### A. Neumaier

I didn't argue against this!

Tunneling is the quantum analogue of the motion of a classical particle with a slightly random kinetic energy (a Brownian particle) moving in a potential with a barrier. The particle has a small probability of being kicked (by the surrounding bath) over the barrier and ending up outside the well it was in originally. The higher the barrier the lower the probability of landing on the other side, with an exponential relationship. This is the case both in classical stochastic motion and in quantum stochastic motion, although one conventionally only refers to the quantum setting as tunneling.

In a typical experiment the barrier has no geometric height; the height is in energy space, hence a figurative concept. Thus in $x$-space, the particle simply moves through the barrier.

But one may picture the particle as a little ball moving close to the surface of a multimodal bowl, with an autonomous moving part inside that randomly supplies or removes some of the kinetic energy energy of the ball. Then the potential is realized as a gravitational potential and the height of the particle corresponds to its potential energy. This analogy is what I referred to as the figurative view. There you think of the particle as moving figuratively on a potential energy surface in $x-E$ space with $E$ being the height. Now there is no tunneling anymore but only climbing over a barrier.

Because of the above, I conclude that the particle climbed the barrier!

Last edited: Feb 12, 2017
14. Feb 12, 2017

### Karolus

intersting observation of prof Leonard Susskind. But , if I understand, I cannot get an answer