# Quantum Tunneling-Q's I've never seen asked

1. Apr 29, 2004

### MonstersFromTheId

I've seen more explanations than I can count of how Q.T. happens, but all of them involve basically variations of the same specific example of an electron encountering a magnetic field as a barrier.

This leaves me with several questions not answered by this specific example:

1) Are electrons the only kinds of particles known to exhibit this behavior?
Or are there other kinds of particles that could exhibit this behavior as well? For that matter does the type of particle even have anything at all to do with whether or not it could exhibit the behavior of quantum tunneling? Are certain types of particles more or less likely to exhibit this behavior?

2) When a particle tunnels through a barrier, does it (for lack or a better term) "move" from one side of the barrier to the other instantaneously? Or is there some finite time it takes for this transition to occur?

3) Can quantum tunneling, strictly speaking, even be considered "movement" in the every day sense of the term?
The impression I get (and this could be a severely mistaken impression) is that when a particle "tunnels" through a barrier, it isn't like pushing a marble through a wall of clay.
That is to say that the particle, from instant to instant, isn't actually "moving" through the barrier where at a given moment it's a quarter of the way through the barrier, a moment later it's half way through the barrier, a moment later it's three quarters of the way through, and finally it's through the barrier.
Instead, (Geeze this is hard to come up with words to describe), if you were to think of the particle's initial position on one side of the barrier as "Point- A", and its later position on the other side of the barrier as "Point- B", there is no "Point - C" lying between Point - A and Point - B at which you could expect to find the particle. The "movement" is discontinuous and discrete? The particle essentially "jumps" from Point - A to Point - B without ever having actually "traveled" through the intervening space between Point - A & B? Am I getting that essentially correct?

4) The "barrier" used as an example in all the explanations I've seen to date is a magnetic field. What other types of "barriers" could be envisioned?
My impression is that almost any condition representing a level of potential energy in excess of the amount of energy the particle has to cross that barrier would do.
Which leads me to an interesting thought - could a change in position or "effective speed" be considered a "barrier"?
Consider a particle moving at a steady speed of V with no forces acting on it so that S=VT applies.
For a particle moving at a constant speed of Vc, that starts out at an initial position Si, there is a maximum distance S=Sf-Si that it could move for any chosen value of T=Tf-Ti, essentially because it doesn't have the energy required to cover any greater (or lesser) distance in the allowed period of time.
Could that value of "S" also be considered a "barrier" that the particle could also "tunnel" across? That is to say could the phenomenon of quantum tunneling cause a particle to move farther (or even not as far) as it should while moving at a "constant" velocity under conditions where S=VT would normally apply?

"But they forgot one thing. Monsters John. Monsters from the Id!"
Lt. 'Doc' Ostrow in Forbidden Planet; MGM 1956

2. Apr 29, 2004

### chroot

Staff Emeritus
Any particle can tunnel. The exact frequency of tunneling events depends a lot on the specifics of the state of the particles and the width of the barrier.
No, it does not move instantaneously. Keep in mind that particles at the quantum level are not really localised like little billiard balls anyway -- the wave nature of quantum mechanical particles is what enables tunneling to happen in the first place. The wave nature of a particle is effectively a way of describing how a particle is "spread out" in space.
The particle can appear on one side of a barrier during one measurement, and on the other during a subsequent measurement. I suppose it comes down to the semantic definition of "movement," which is not a precise physical term. I would call it movement, however.
Nope. Consider that the particle is really a wave packet -- a little collection of summed waves of different wavenumbers, where the total magnitude at some location represents the probability of finding the particle at that location. When the particle's center-of-probability (the central peak of the wave packet) is near the barrier, some small tail on the end of the wavefunction can actually exist on the other side of the barrier. That means there's a nonzero probability of a measurement finding the particle there. That's all tunneling is, quantum-mechanically.
You can't actually watch a particle moving through the barrier; the best you can do is take a set of closely spaced, repeated measurements. If you do this, you can indeed make measurements where the particle is "inside" the forbidden zone -- it's wavefunction is non-zero there, just like everywhere else.
It's entirely possible that two subsequent measurements will show the particle at point A and point B, with no intervening measurement showing it inside the barrier. That doesn't mean that the particle is moving discontinuously. This is one of the notions that you have to get rid of to fundamentally understand quantum mechanics: what's real is what you can measure, and you can't measure anything continuously.
Well, there are only four fundamental forces, and really electromagnetism is the most convenient for experimentation, but in theory it does not matter at all what creates the potential barrier -- any force will do.
Yes. The notion of position = velocity * time is not in general true in quantum mechanics. The wavepackets themselves obey Newton's laws, though. In other words, the central peak of the wavefunction, the most probable place to find the particle, moves according to Newton's laws. Actual measurements, however, can find the particle anywhere the function is non-zero. Particles can then be said to "move" faster than the wavepacket itself moves -- but that, again, comes down to a semantic definition of movement.

- Warren

3. Apr 30, 2004

### MonstersFromTheId

Very helpful, another Q

These wave packets, as they fade from most probable location, to less, to much less, never actually reach zero, they approach zero asymptotically.
Potentially any particle could essentially appear to "jump" several miles, or even several light-years from one measurement to the next, essentially traveling at speeds much greater than the speed of light.
But the probabilities of that happening are SO incredibly low that you'd have to be waaaaaaaaaaaay beyond lucky to ever be able to catch it happening. And we're talking REALLY lucky here. You could take millions of measurements over millions of years and STILL be very unlikely to witness such an event.
But it can happen.
If fact for all we know it just did. A single particle from my behind may have just left my chair and now be in orbit of Mu Leporis. But it may take hundreds of millions of trillions of years for it to happen again and I missed it.
DAMN.
So it's not that "nothing can travel faster than light", so much as the chances of ever being in exactly the right place, at exactly the right time, to catch the one single solitary particle that did it in the last several hundred trillion years, is so remote that it's a lot more practical to say "things don't travel faster than light".
And that's just talking about a single particle. The chances any two particles, let alone the trillions of them there are in my body, doing this at the same time, and winding up having moved in the same direction, for the same distance, is so vanishingly small that the terms improbable and impossible, for all practical intents and purposes, take on the same meaning.

Ergo; I really don't have to worry about the rest of my behind suddenly winding up in orbit of Mu Leporis as well.
At least not this week.

But things CAN, and at least in theory DO, on exceedingly rare occasions move "faster than light" in this manner?

Monsters

4. Apr 30, 2004

### chroot

Staff Emeritus
As I said, the wavepacket itself cannot move faster than light, but the particle can seem to if two repeated measurements just happen to work out that way. This is another way of expressing the uncertainty principle: when you measure its position precisely, you lose all information about its momentum. When you know exactly where it is at time 0, you cannot say at all where it will be at time 1.

Quantum mechanics deals with the evolution of the wavefunction itself, though, and it is the wavefunction that obeys "normal (classical) physics."

- Warren

5. Apr 30, 2004

### MonstersFromTheId

Tx Warren, VERY helpful, seriously @

123123123123123 .

6. Apr 30, 2004

### jcsd

Chroot, surely though the wavefunction ios non-zero,he you can't detect it in the classically forbidden zone as it would require the partcile to have negative kinetic energy thus making it undetectable.

7. Apr 30, 2004

### chroot

Staff Emeritus
jcsd,

You're right, of course -- good point.

- Warren

8. Apr 30, 2004

### turin

For all intents and purposes, experimentally, there is no way to determine whether the electron that was in your butt is the same one that you find in orbit, and theoretically, there is not even meaning in the issue.

9. May 5, 2004

### alpha_wolf

Just wanted to add that alpha radiation (and I think other types too) are associated with tunneling. The alpha particle essentially tunnels out of the nucleus during a decay. The potential energy involved is formed through the strong and/or weak nuclear force(s) (not sure which of the two). So there's one example of a non-electron tunneling through a non EM barrier. Another example that I've heard of is a C60 molecule tunneling through some barrier, although I'm not familiar with the details of the experiment.

10. May 6, 2004

### gnome

I'm puzzled by some of the statements here.

I've just been studying the chapters in the Serway Modern Physics text about square wells, quantum oscillators, and tunnelling. It definitely says that there is a non-zero probability of finding the particle in the forbidden zone in the case of tunnelling, just as there is a probability of finding it in the forbidden zone of a finite square well, and beyond the classical turning points of an oscillator.

The place(s) where the particle is never found are those points within a well where $$\psi(x)$$ for an excited state is transitioning from negative to positive or vice versa: at these points $$|\psi|^2$$ is zero.

jcsd: what is this about "negative kinetic energy"? I can't find any mention of that in the Serway or the Tipler textbook. Please explain that.

By the way, another example of tunnelling is the oscillation of the nitrogen atom in an ammonia molecule, NH3. It's arranged with the hydrogen atoms situated at the corners of a triangle and the nitrogen atom forming the apex of a pyramid. The nitrogen atom sees a potential wall at the plane of the hydrogen triangle but repeatedly flip-flops through the wall, re-forming the pyramid on the opposite side of the plane of the hydrogen triangle, oscillating at about 2.4x10^10 Hz.

11. May 6, 2004

### ZapperZ

Staff Emeritus
I knew this would happen sooner or later! :)

There is certainly nothing to prevent one from calculating the probability of finding the particle in the barrier. After all, it is just a matter of applying $$\psi$$ with the right operator and over the right boundaries of integration. However, when one makes a measurement, one then is making a CLASSICAL observation (position). Keep in mind that within the barrier is the classically forbidden region. So we're forcing the particle to become a classical particle in a classically forbidden region. So now what? :)

It is interesting to note that in the tunneling phenomena between two superconductors separated by a normal insulator, the physical description requires that the superfluid "suspends" it's superconducting property while it goes through the barrier, and then reforms the superfluid state when it reenters the superconductor on the other side of the barrier.[1,2] This means that while it is in the barrier, the electrons are normal, typical, run-of-the-mill conduction electrons.

As far as the "negative KE" thing, I can see how that might come about since we typically define the total energy E as E=PE + KE. Since in the barrier, E is less than PE, one would tend to think that KE is negative here. However, again, this may not be the right picture. Remember that all we care about is the wave vector "k" in the wavefunction. For a free particle in 1D, it is just exp(ikx). k here is defined as being proportional to sqrt(E-PE). Depending on the sign of E-PE, one and re-adjust k to be either real or imaginary, resulting in either a propagating wave, or a decaying wave. This is all that matters. It makes it an added complication to try and equate this to "KE", which in itself is not well-defined in the barrier (remember, v or p is again an observable that we don't know of in the barrier).

Zz.

[1] W. A. Harrison, Phys. Rev. v.123, p.85 (1960).
[2] J. Bardeen (this is the "B" in BCS theory), PRL v.6, p.57 (1961).

12. May 6, 2004

### jcsd

I'm not 100% on this, I can only say what I was taught:

It's pretty clear that the probabilty of finding the particle in the barrier is in theory non-zero, this can be ascertained by a fairly rudimentary caculation or even just looking at the graph of the wavefunction. Now what I was taught is that the problem comes in detecting it, infact it seems that it is impossible to detect a partcile in the barrier, as the very act of inserting a detector (though I think if you had a theoretical device that could detect negative KE this wouldn't be a problem, but nonsuch device exists)will create a hole in the barrier menaing the particle will no longer be in a classically forbidden zone when detected.

13. May 6, 2004

### gnome

Thanks for those explanations. I guess there's a certain ring of familiarity in that. It seems similar to the description we were given of two-slit electron scattering in which the act of trying to detect the electrons passing through one of the slits forces all of the electrons to pass through the other.

And I guess that what jcsd is describing as "negative kinetic energy" is the same phenomenon as what my professor referred to as "borrowing" kinetic energy (although it's not clear where it's being borrowed from). That would be the difference between the potential of the barrier and the total energy of the particle, correct?

14. May 6, 2004

### turin

Here's a thought about observability in the barrier. If we are trying to determine whether the particle is ever found in the barrier, then we must use a position measurement, no? That said, then the measurement should collapse the particle to a position eigenstate. If this is so, then the p and KE wouldn't be well defined in this determination in the first place, regardless of whether the particle is found inside or outside of the barrier. It seems that the only reason why negative KE would prevent measurement is if one is measuring p. In this case, it is only axiomatic in my mind, but I've been taught that only real eigenvalues are observable. Therefore, p must be real valued, and thus only KE ~ p2 that are positive semidefinite can be measured. But, if one is measuring p, then one cannot make a determination of where the particle is, because the p eigenstate obscures the position. So, one can demonstrate that, when measuring the energy, only E > V are found, but this can never rule out any position, because this must incorporate an integration of the wavefunction in the position basis over all of space.

In one picture, it's being borrowed from time, like a magincian borrows from time (slight of hand) the knowledge of the correct card out of the deck of 52. The longer the time, the harder it is to borrow energy. Energy and time go hand in hand.