Speed of Quantum Tunneling

I am no expert in quantum physics, so I might very well say some of this wrong.

Here’s the situation: Some photons or electrons (think tunnel diode) will not cross an impenetrable barrier, but will disappear on one side and reappear on the other. They do not cross the space between.

I realize that’s a simplification, but folks, I can NOT do, or understand the math behind this. So, I ask the Dopers out there, is there any data out there on how fast the wave packets/whatever take to tunnel “under” the barrier? Their original speed? Something greater than the speed of light*? Something in between?

Thanx!

*I do realize that if something disappears in one spot and reappears in another instantaneously without actually crossing the intervening space, you can’t actually say it has a “speed.” I am only talking about…about…uh, equivalents, I guess. Is the time taken greater or less than a photon traveling at c would take to actually traverse the distance?

This is one of those things they really don’t go into in your average Quantum Mechanics class, which is a shame. I vividly recall a series of illustrations in Schiff’s book Quantum Mechanics that shows a computer simulation of a gaussian wavepacket tunneling through a finite rectangular barrier. A couple of decades later and this would’ve been a computer movie on a website (and probably is now), but it had great value as a series iof stills from such a movie. You can see how the packet encounters the barrier, part gets transmitted (even though its energy is smaller than that of the barrier, so it’s like a person transmitting part of themselves through a brick wall), and most of the packet being reflected and interfering with itself, a fascinating effect. The transmitted portion proceeded across the barrier, decreasing exponentially in strength until it encountered the other side, where most was transmitted and appeared on the other side (as if a tiny person appeared on the other side of a brick wall you ran into), but a small portion was reflected back into the barrier, interfering with itself in the process. This packet continued to rattle back and forth, shedding energy to the outside world every time it hit.
In order to make this “movie”, Scgroedinger’s equation had be be cast in a time-dependent form, as it usually isn’t for undergrad presentation. I dug out the original paper that was used for this, but can’t recall the details (and don’t have it or my copy of Schiff here at the moment), but the speed of tunnelling was finite and comparable to the speed of the packet before it hit the barrier.
You can see the same effect with evanescent light waves tunnelling through space. In that case, IIRC, the speed is simply the speed of light in the medium being tunneled through.

There’s nothing mysterious about tunneling with respect to wave phenomena. Both light waves and water waves can display frustrated total internal reflection which can be considered exactly equivalent to the behavior of the quantum mechanical wavefunction.

The problem is that the wave function isn’t real and represents the probability of finding a particle in an energetically forbidden region. This is where the wicket gets sticky and intuition goes out the window. Particles (in the classical sense) shouldn’t be able to do this. And this is an example of why Feynmann said that no one understands Quantum Mechanics.

CalMeacham A nitpick as I’m sure you know this. There’s no shedding of energy. The wave function doesn’t carry energy.

I just realized that I didn’t really answer the question. QM tunneling cannot be instantaneous, and it cannot occur faster than light. It can appear to occur FTL but this is just an artifact of wave packet behavior.

Every post in this thread so far raises more questions than it answers.

-FrL-

The wave behavior associated with tunneling is just common every day stuff. Physicists, even before the development of QM, understood this phenomenon perfectly.

When quantum mechanical particles aren’t being observed they can only be described by a wave.

So the problem arises when the tunneling behavior associated wiht a wave represents the probability of finding a particle.

Since a wave can penetrate a barrier which a particle can’t, we now have a finite probability of finding a particle where it shouldn’t be allowed to be.

This whole mystery is due to the wave / particle nature of reality.

An illustration would help.

-FrL-

Here’s a Java Applet of a gaussian particle tunnelling through a square barrier (I knew there’
d be a Java Applet online. In fact, if you search, you’ll find several):

http://www.physics.brocku.ca/faculty/sternin/teaching/mirrors/qm/packet/wave-map.html

BTW if you just want to understand the bare essentials of this stuff:

Calculus
Differential Equations
Partial Differential Equations
Linear Algebra
Classical Mechanics (With Hamiltonian and Lagrangian formulations)
Introduction to Quantum Mechanics by Griffiths
Principles of Quantum Mechanics by Shankar

Really?

To know whether quantum tunneling happens at the speed of light or at some other speed requires knowledge of all that?

Anyway, I was hoping for an illustration of the wave behavior underlying quantum tunneling which is a straightforward everyday phenomenon. I take it I don’t need to know calculus to understand what you’re referring to in that case. But what are you referring to?

-FrL-

Maybe you should reread posts 4 and 8

I didn’t mean to imply the question in the OP hadn’t already been answered. I was just saying the knowledge you mentioned is not needed in order to understand that answer, at least in its most bare-bones form.

As to post 8, I thought that was an illustration specifically of quantum tunneling, and not an illustration of the everyday, straightforward occurance that happens with waves in any substance. Was I wrong?

By “illustration” I meant, by the way, someone pointing to something I am likely to encounter or be able to imagine encountering in the course of a somewhat normal day. I thought that you were saying the underlying wave phenomenon is of a sort imaginable in this way.

-FrL-

With light, it’s not just exactly equivalent: The light wave is the wavefunction of the photon.

Frylock, the Java applet linked in post 8 is an illustration of a wave tunneling through a behaviour. The same illustration could apply equally well to any wave, be it light, sound, water, or quantum wavefunction.

Well, here is a really good example of something that happens a lot around here. I not only got my basic question answered, but since I said I don’t know much about quantum mechanics, I got some extra, and interesting, knowledge as well. And, in a way I pretty much undestood…at least superficially.

CalMeacham, Ring, Chronos: THANKS!