## Saturday, November 14, 2020

### Understanding Quantum Mechanics #8: The Tunnel Effect

[This is a transcript of the video embedded below. Parts of the text will not make sense without the graphics in the video.]

Have you heard that quantum mechanics is impossible to understand? You know what, that’s what I was told, too, when I was a student. But twenty years later, I think the reason so many people believe one cannot understand quantum mechanics is because they are constantly being told they can’t understand it. But if you spend some time with quantum mechanics, it’s not remotely as strange and weird as they say. The strangeness only comes in when you try to interpret what it all means. And there’s no better way to illustrate this than the tunnel effect, which is what we will talk about today.

Before we can talk about tunneling, I want to quickly remind you of some general properties of wave-functions, because otherwise nothing I say will make sense. The key feature of quantum mechanics is that we cannot predict the outcome of a measurement. We can only predict the probability of getting a particular outcome. For this, we describe the system we are observing – for example a particle – by a wave-function, usually denoted by the Greek letter Psi. The wave-function takes on complex values, and probabilities can be calculated from it by taking the absolute square.

But how to calculate probabilities is only part of what it takes to do quantum mechanics. We also need to know how the wave-function changes in time. And we calculate this with the Schrödinger equation. To use the Schrödinger equation, you need to know what kind of particle you want to describe, and what the particle interacts with. This information goes into this thing labeled H here, which physicists call the “Hamiltonian”.

To give you an idea for how this works, let us look at the simplest possible case, that’s a massive particle, without spin, that moves in one dimension, without any interaction. In this case, the Hamiltonian merely has a kinetic part which is just the second derivative in the direction the particle travels, divided by twice the mass of the particle. I have called the direction x and the mass m. If you had a particle without quantum behavior – a “classical” particle, as physicists say – that didn’t interact with anything, it would simply move at constant velocity. What happens for a quantum particle? Suppose that initially you know the position of the particle fairly well, so the probability distribution is peaked. I have plotted here an example. Now if you solve the Schrödinger equation for this initial distribution, what happens is the following.

The peak of the probability distribution is moving at constant velocity, that’s the same as for the classical particle. But the width of the distribution is increasing. It’s smearing out. Why is that?

That’s the uncertainty principle. You initially knew the position of the particle quite well. But because of the uncertainty principle, this means you did not know its momentum very well. So there are parts of this wave-function that have a somewhat larger momentum than the average, and therefore a larger velocity, and they run ahead. And then there are some which have a somewhat lower momentum, and a smaller velocity, and they lag behind. So the distribution runs apart. This behavior is called “dispersion”.

Now, the tunnel effect describes what happens if a quantum particle hits an obstacle. Again, let us first look at what happens with a non-quantum particle. Suppose you shoot a ball in the direction of a wall, at a fixed angle. If the kinetic energy, or the initial velocity, is large enough, it will make it to the other side. But if the kinetic energy is too small, the ball will bounce off and come back. And there is a threshold energy that separates the two possibilities.

What happens if you do the same with a quantum particle? This problem is commonly described by using a “potential wall.” I have to warn you that a potential wall is in general not actually a wall, in the sense that it is not made of bricks or something. It is instead just generally a barrier for which a classical particle would have to have an energy above a certain threshold.

So it’s kind of like in the example I just showed with the classical particle crossing over an actual wall, but that’s really just an analogy that I have used for the purpose of visualization.

Mathematically, a potential wall is just a step function that’s zero everywhere except in a finite interval. You then add this potential wall as a function to the Hamiltonian of the Schrödinger equation. Now that we have the equation in place, let us look at what the quantum particle does when it hits the wall. For this, I have numerically integrated the Schrödinger equation I just showed you.

The following animations are slow-motion compared to the earlier one, which is why you cannot see that the wave-function smears out. It still does, it’s just so little that you have to look very closely to see it. It did this because it makes it easier to see what else is happening. Again, what I have plotted here is the probability distribution for the position of the particle.

We will first look at the case when the energy of the quantum particle is much higher than the potential wall. As you can see, not much happens. The quantum particle goes through the barrier. It just gets a few ripples.

Next we look at the case where the energy barrier of the potential wall is much, much higher than the energy of the particle. As you can see, it bounces off and comes back. This is very similar to the classical case.

The most interesting case is when the energy of the particle is smaller than the potential wall but the potential wall is not extremely much higher. In this case, a classical particle would just bounce back. In the quantum case, what happens is this. As you can see, part of the wave-function makes it through to the other side, even though it’s energetically forbidden. And there is a remaining part that bounces back. Let me show you this again.

Now remember that the wave-function tells you what the probability is for something to happen. So what this means is that if you shoot a particle at a wall, then quantum effects allow the particle to sometimes make it to the other side, when this should actually be impossible. The particle “tunnels” through the wall. That’s the tunnel effect.

I hope that these little animations have convinced you that if you actually do the calculation, then tunneling is half as weird as they say it is. It just means that a quantum particle can do some things that a classical particle can’t do. But, wait, I forgot to tell you something...

Here you see the solutions to the Schrödinger equation with and without the potential wall, but for otherwise identical particles with identical energy and momentum. Let us stop this here. If you compare the position of the two peaks, the one that tunneled and the one that never saw a wall, then the peak of the tunneled part of the wave-function has traveled a larger distance in the same time.

If the particle was travelling at or very close by the speed of light, then the peak of the tunneled part of the wave-function seems to have moved faster than the speed of light. Oops.

What is happening? Well, this is where the probabilistic interpretation of quantum mechanics comes to haunt you. If you look at where the faster-than light particles came from in the initial wave-function, then you find that they were the ones which had a head-start at the beginning. Because, remember, the particles did not all start from exactly the same place. They had an uncertainty in the distribution.

Then again, if the wave-function really describes single particles, as most physicists today believe it does, then this explanation makes no sense. Because then only looking at parts of the wave-function is just not an allowed way to define the particle’s time of travel. So then, how do you define the time it takes a particle to travel through a wall? And can the particle really travel faster than the speed of light? That’s a question which physicists still argue about today.

This video was sponsored by Brilliant which is a website that offers interactive courses on a large variety of topics in science and mathematics. I hope this video has given you an idea how quantum mechanics works. But if you really want to understand the tunnel effect, then you have to actively engage with the subject. Brilliant is a great starting point to do exactly this. To get more background on this video’s content, I recommend you look at their courses on quantum objects, differential equations, and probabilities.

You can join the chat on this week’s video here:
• Saturday at 12PM EST / 6PM CET (link)
• Sunday at 2PM EST / 8PM CET (link)

1. It amazes me that this phenomenon was used to create practical device: a scanning tunneling microscope.

2. Dr. Hossenfelder: On the last point, the peak traveling faster than light: I don't understand why there is a controversy; the two waveforms are each still a distribution, I presume with infinite overlap, so who cares where the peaks are? Those are no more definitive of position than the rest of the distribution.

I presume upon measurement in either distribution, the particle will NOT have traveled faster than light, within the bounds of its original positional uncertainty.

So all that has changed with a wall is the positional probability; not the actual position.

The controversy seems (to me) like a misinterpretation of statistical probability. Can you explain why physicists see a controversy?

1. Very interesting. I think they are wrong about one thing, though:

The article says: "Why, though, couldn’t you blast tons of particles at the ultra-thick barrier in the hopes that one will make it through superluminally? Wouldn’t just one particle be enough to convey your message and break physics? Steinberg, who agrees with the statistical view of the situation, argues that a single tunneled particle can’t convey information. A signal requires detail and structure, and any attempt to send a detailed signal will always be faster sent through the air than through an unreliable barrier."

Passing even one bit is a message, yes or no. As the article said earlier, this experiment can be done with multiple kinds of atoms.

Thus in the scenario of "blasting tons of particles", I could theoretically associate different atomic elements with different answers, blast a ton of a particular kind of atom at the barrier, some which traverse it superluminally, and the receiver knows the intended message based upon the kind of atom received.

That message could reasonably consist of 3 or 4 bits of information.

2. While the Quanta article is excellent and a lot of fun to read, I did have one quibble with its presentation of the issue.

For any classically-initiated test of the tunneling velocity of a particle, the front edge of its wave packet will necessarily be well-defined, as opposed to fading out indefinitely. This must be the case because it must begin as a particle that is classically visible to the tester. Thus no matter how one interprets the internal state of the particle -- its phase for example, or some other version of an internal clock -- the detectable front edge of its wave packet will always and only propagate as a quite ordinary Schrodinger wave that cannot travel faster than the speed of light, even for photon wave packets.

This finite velocity of a wave packet with a well-defined leading edge means there is never a possibility that any component of the wave packet can reach a remote location in a way that affects classical causality, regardless of internal phase states that may perplexingly suggest otherwise.

After all, just because I can turn off my car clock as I enter the Baltimore Harbor Tunnel and turn it back on as I exit the other end does not mean that I traveled through the tunnel at infinite speed. Similarly, just because the phase of a particle froze as it tunneled through a barrier does not mean that it traveled through that barrier at higher than c velocity.

3. Terry Bollinger: Not a physicist here; but I thought the wave packet of a particle with indefinite position, as shown in the video, was a distribution with infinite tails.

How is a rubidium atom "classically visible", and even if it were, isn't that measuring its position and therefore invalidating the experiment?

4. “Thus in the scenario of "blasting tons of particles", I could theoretically associate different atomic elements with different answers, blast a ton of a particular kind of atom at the barrier, some which traverse it superluminally, and the receiver knows the intended message based upon the kind of atom received.”

“That message could reasonably consist of 3 or 4 bits of information.”

Dr. A. M. Castaldo, that’s a very interesting thought experiment. I was racking my brain, or what’s left of it, being an older person (and brain cells decrease with age), to figure out a loophole in your argument, that would prevent information being transferred superluminally via multiple species of atoms passing through the barrier to provide distinguishable information bits. There might be something quite subtle in the quantum mechanical toolbox that would frustrate efforts to superluminally signal via that strategy. With my knowledge base I’m poorly equipped to come up with a solution. But I love a challenge, and I’m going to keep thinking about it. But, likely, much brainier, and more knowledgeable, people here will figure it out long before I do, if I ever do.

5. Hi Dr A.M., nice to hear from you and I hope you are well!

>… I thought the wave packet of a particle with indefinite position, as shown in the video, was a distribution with infinite tails.

No. To be precise, such wave functions cannot even exist in the real universe. That’s because they would require infinite time and space to form infinite tails via Schrödinger’s equation, and the real universe is finite in both size and time

More importantly for small-scale quantum experiments, and regardless of one’s philosophical stance regarding “wave collapse”, any localization of a wave function that results in an irreversible historical record — actual bits of information, a “detection” in space — unavoidably and completely erases all traces (and thus leading and trailing edges) of that wave function outside of the xyz box in which it was relocalized (“found”). That is what wave collapse is: the complete removal — not just diminution, but erasure — of probability amplitudes outside of the xyz detection box.

The trailing edges then must reform “from scratch”, so to speak, and can only do so via Schrödinger’s equation. In the context of a wave function that has been collapsed into a well-defined initial xyz region, Schrödinger’s equation behaves like an entirely conventional wave function, one that in the case of photons is isomorphic to the electromagnetic wave equation. Since this equation is bound by the speed of light, there is never a leading edge beyond the light cone leading out form the initial xyz box location. The spread of this quantum wave thus is classical in every way except for the final Born interpretation, which is the real spanner in the monkey works (and yes, I couldn’t resist mixing US and UK metaphors there… :)

The idea of a pure sinusoidal particle state is quite nonsensical in this context, for the simple reason that all actual, experimentally meaningful wave functions created naturally or in the lab necessarily have associated with them some classical-origin boundary boxes in xyz. Thus all wave functions are more correctly described as wave packets — wave pulses with sharp edges beyond which their amplitudes are not just very small, but exactly zero. The unreal case of a pure sinusoidal state — meaning a quantum entity with a completely undefined location in the universe — is a state that real wave packets can only approach, and even then only if provided with infinite time and space.

I am genuinely a bit baffled at why so many quantum textbooks are sloppy about this point, since there’s nothing radical about it. The sections about pure states almost always correctly point out that such states can only be approached, but then they start slipping into treating Dirac deltas is if they are real states in the real world, resulting in sloppy equations that assert quantum entities to be in places where they cannot be experimentally.

It’s usually not that bit of a problem, but it can become a big one when folks start talking about things like group velocities in waves, which that can create illusion of faster-than-c propagation. I say “illusion” because faster-than-c group velocities in waves are no more a violation of c than the dot you could get by sweeping a very powerful laser sideways across the surface of the moon at a sufficiently high angular velocity. Such a dot would appear to earth telescopes to move across the moon at faster-than-c velocity, just as some group velocities can appear to create wave that move faster than c. However, in both cases no independent signal is carried from one edge to the other, only the illusion of motion.

When you bound the group velocities by a sharp wave edge, even the illusion disappears as the group velocity waves strike edge of reality and disappear.

6. >… How is a rubidium atom "classically visible", and even if it were, isn't that measuring its position and therefore invalidating the experiment?

It’s classically visible because it’s inside the apparatus. It’s hard to play with rubidium atoms that aren’t there! The game in experiments like this is that you let the location of the rubidium atom wave function get large enough to accommodate some slop in the measured speed of light.

That is all any of this ever is, which is why I’m not personally much impressed by any of this work, even though I fully acknowledge there are some interesting math issues. For example, and as nicely Sabine described, the peak of the tunneled particle looks like it moved faster than light. That sounds impressive until you realize that all that is really going on is that some of the wave components closer to its speed-of-light-bound leading-edge are getting a bigger share of expectation amplitude than they normally would receive.

If that’s breaking the speed of light, then so giving a few dozen folks in a marathon a ride in a car so they can finish closer to but not in front of the actual winner. And since that winner, coincidentally name W. Schrödinger, is never permitted any such boost, the Schrödinger edge-boundary (and classical causality) remain nicely intact, despite the subsequent clump of tunnel-cheaters.

7. Terry Bollinger: Wow, thanks, that is enlightening.

If all that happened is the barrier introduced some skew in a non-infinite distribution and the leading edges are still the same, then I still see no reason for FTL angst. I'd expect the barrier to change the distribution somehow.

I suppose there is some mystery, which might be resolved just by the math, as to why the leading edge of the distribution seems to get a greater than expected through-rate than the rest of the distribution (which would be, to me, the plausible cause of the skewing).

Since the expectations have a not-flat shape, it would be interesting to see what the shape is; e.g. actual probability of tunneling vs. probability of position.

Would you think that a non-zero probability of tunneling requires the leading edge of the wave packet to be on the other side of the barrier? (as opposed to within the barrier).

8. Hi Dr A.M.,

Thanks, I appreciate the feedback! For some reason my own logic always sounds a lot clearer inside my head than it probably does to folks outside said head. (Anyone else ever feel that way?… :)

>... Would you think that a non-zero probability of tunneling requires the leading edge of the wave packet to be on the other side of the barrier? (as opposed to within the barrier).

Good question, one with a nicely straightforward answer: yes.

That is, some non-zero amplitude (a leading edge) must exist on the far side of the barrier for tunneling through the barrier to occur, always. That's because at the Feynman QED level, the leading edge of the wave is really nothing more than the collection of all possible paths by which the particle might get there, subject to the limits of physics and the potentials affecting its motion. Stated that way, it becomes almost a tautology: The wave is the sum of all real paths by which a real particle could get there, not some separate entity. It only looks like a wave because quantum histories interfere and reinforce each other, so that even a single particle looks just like an entire front of electromagnetic waves.

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Tangent 1 of 2: The beauty and predictive power of the Feynman path integral approach is also why I don’t buy into the de Broglie / Bohm pilot wave concept, even though I fully agree that it can be a very powerful and useful analytical tool when used in the right way. My concern is simple: If the best and most precise way to define the ephemeral “pilot wave” is simply to map out every possible path of the real particle, while using phase tracking and mutual interference to keep track of which paths are most energetically favorable… well, then why in the heck do you need a separate, much more magical wave that “just happens” to look like the collection of all those possible particle paths?

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Tangent 2 of 2: The presence of massive redundancy turns out to be a powerful and quite generic analytical heuristic for uncovering logical inconsistencies in theories. For example, massive redundancy also provides a powerful argument against the late-1970s presumption (that’s all it ever was) that gravity “must” me a quantum force, just like the electromagnetic, strong, and weak forces. There’s a bit of a problem with that idea, however, and the problem is very much one of redundancy. Ask yourself this: If gravitons in any way resemble the bosons of the other three forces, then just like those bosons, then they must travel and interact over an xyz-type space, right? So far so good. But now ask yourself the follow-up question: If you then curve the space over which such gravitons are traveling, what do you get in terms of the impact of that curvature on those same particles? Gravity again… the Einstein topological version!

My point is simply that the very concept of boson-based gravity amounts to a sort of theoretical cotton candy, a hypothetical way of constructing a force that kind of looks like gravity in the sense that it pulls together any two objects with “mass charge”. However, on closer examination this mechanism remains flatly unrelated to the actual, topological gravity that Einstein figured out. Smart fellow, that Einstein.

9. I should think you have one or the other, gravitons or curvature; and the relationship between them is analogous to how the equations of fluid dynamics (e.g. Navier-Stokes and others) accurately describe the aggregate behavior of what are ultimately many trillions of discrete atoms; but (as we've seen even with QD) does not capture the behavior of individual particles.

Meaning, Einstein's curvature formulation might be just a very accurate approximation of how trillions of gravitons behave, but might not accurately capture the behavior of just a few gravitons.

3. By statistics you can imagine classically that if the particle is measured as tunneled, it (its info) was actually travelled at front in the wave. Logically you can also imagine that the tunneling particle is actually from the barrier matter structure via energetic interactions.

Still, the human imagination does not produce new physics but only helps avoid obstacles for it. The same other way: no equally varying interpretation is physics but a new prediction really is if it's more accurate than earlier.

4. As a young nuclear engineer, I accepted radioactive behavior without explanation. I genuinely appreciate having a better understanding - however slight - of radioactive decay as a manifestation of tunneling.

5. This is an excellent video, Sabine! I liked in particular your strategy of starting with how particles smear out, so that folks can see how the idea of a single particle with a well-defined location breaks down in quantum mechanics.

One thought that occurred to me as you showed a ball bouncing near the top of the wall is that there is also an even closer analogy possible: A very slow-moving neutron heading toward the top of a quite literal wall of dense metal. Viewed as a point, the neutron would either bounce off of (or more likely, be absorbed by) the wall, or it would sail over it, exactly like a classical wall. But because the neutron wave function smears out in exactly the fashion you described at the start of your video, there will instead be a part of the neutron wave function that, if it is very close to the top of the metal wall, will diffract over the wall, allowing some neutrons to cross over the wall even though the same neutrons as points would not be able to do so.

6. Tunneling in part is manifestated by the occurrence of a wave function in a region that is forbidden classically. Classically if the energy of the particle is smaller than the potential the particle bounces off. The generator of the unitary motion for a particle in a potential barrier V is ε = √[2m/ħ(E – V)]. For V > E this is complex valued and the complex valued e^{-iEt/ħ} becomes real valued. It is also exponential decaying. One result is that we can write εt/ħ with ε = √[2m/ħ(E – V)] as ε = i√|2m/ħ(E – V)| → iε and associate the i = √-1 with time so it = τ. Then for this we have ετ/ħ = ε/kT for some pseudo-temperature T = ħτ/k.

I have thought this pseudo-temperature might be interpreted as a temperature of nonlocal hidden variables, maybe local if we cast realism aside, that in a statistical sense define this pseudo-temperature. This is also the imaginary Euclidean time above. This would the correspond to the time of tunneling.

Quantum tunneling is a big cornerstone of a lot of technology, in particular quantum electronics.

7. One of the cornerstones of Quantum Mechanics the Quantum Tunneling that appears to be well understood, being controllable with many applications in solid state physics e.g. semiconductors, it is actually non-understood at all. The wave function and the uncertainty principle serve as the veil of Quantum weirdness that ignore the cause behind the effect pointing to Quantum Mechanics incompleteness.

Below are listed the two inconsistencies being saved by the probabilistic observational justification:
a) A quantum particle overcomes a potential barrier having less energy would normally lead to violation of energy conservation if the particle was classical
b) A quantum particle overcomes a potential barrier in the name of the Wave function (not an intrinsic property of the particle) and the uncertainty principle corresponding the first to the probabilistic (pure mathematics unrelated to physics) outcome and the later a justification that has nothing to do with the observation but to the determination (uncertainty) of the position (Δx) and momentum (Δp) or the energy (ΔE) and time (Δt).

Obviously, due to (a) and (b) no one understands why the particle was actually able to tunnel through the barrier, because the Schrodinger equation has nothing to say about it except to calculate the probability of seeing something on the other side (beyond the barrier).

The question remains: If one tells you the probability to experience/measure an outcome that under classical conditions defies the energy conservation, it cannot be justified just by using the definition of "quantum particle" (see Schrodinger equation).

The missing part in the puzzle is according to my opinion, a shielding effect that leads to the reduction of quantum particle effective charge. All those mentioned in the video clip make really sense when the cause e.g. shielding effect is integrated in the Schrodinger equation, otherwise the Quantum Tunneling tends to be classified as an illusion than a genuine effect.

1. It is interesting to consider that any type of wave can tunnel even sound waves. Both Yang and Robertson have shown that acoustic waves can tunnel They found that inside the stop band the acoustic group delay was relatively insensitive to the length of the structure, a verification of the Hartman effect. Furthermore the group velocity increased with length and was greater than the speed of sound, a phenomenon they refer to as "breaking the sound barrier".

2. John,

The potential of the barrier is an average one. You do not actually know what was the potential at the exact time and place the particle was there. It might be the case it was low enough so that no violation of energy conservation is required.

8. The Hartman effect is the tunnelling effect through a barrier where the tunnelling time tends to a constant for thick enough barriers. This was first described by Thomas E. Hartman in 1962. This leads to the conclusion that the wave does not travel through the barrier but just appears at the other side of the barrier in the same constant time no matter how long the barrier happens to be.

But the probability of transmission through such a thick barrier becomes vanishingly small, since the probability density inside the barrier is an exponentially decreasing function of barrier length.

For waves traveling at the speed of light, this effect implies that for a thick barrier, the wave may go superluminal.

1. I am familiar with this. My ancient memory on this was jogged a while back with news of experimental data on this. I will say that I do not think this determines the speed information crosses a barrier. The Hartman effect is the time for phase velocity to cross or the phase time.

I suggested above this time is related to a pseudo-temperature determined by statistics on hidden variables. Either this HV is nonlocal, or if local one abandons realism as with Wigner's friend and related developments of late. This would then have no sensitivity to the scale of the potential barrier.

9. In a previous universe, a photon could have tunnels through a seemingly insurmountable barrier. Since this event has a non-zero probability, given enough time, such an event will happen.

With its passage through the barrier, this tunneling event would have greatly amplified the energy of the photon to the point of big bang causation and subsequent superluminal inflation. This process is referred to as Quantum Creation.

10. The particle can be anywhere and everywhere within distributions both before tunneling and after. The distribution peak after tunneling appears ahead of the peak without tunneling so lets behave ourselves and bound the distribution after tunneling by x=tc. Steinberg says the probability of tunneling is extraordinarily low but he didn't say to bound the distribution by 'c'. I suppose this model looses fidelity when the barrier is the strong force and we examine nuclei collisions on the Sun?

11. There must be a typo error in the Quanta magazine article that Sabine referenced. It states: “The researchers reported that the rubidium atoms spent, on average, 0.61 milliseconds inside the barrier, in line with Larmor clock times theoretically predicted in the 1980s. That’s less time than the atoms would have taken to travel through free space.” In .61 milliseconds a light beam, in free space, traverses 183 kilometers. So, that’s a rather big laboratory for this experiment. It must have been pico-seconds, which would correspond to .183 millimeters for the width of the laser beam. Or, maybe, it was a really wide laser beam at 183 millimeters, or about 7.2 inches. At least those would fit into a terrestrial laboratory.

1. The atoms don't move with the speed of light. The next sentence explains it.

2. Sabine, thanks, I didn't make that connection.

12. lagunastreets: Statistically speaking, if you bound an infinite tail of a distribution at some point X, the volume of probabilities in the excluded region (beyond X) must be somehow incorporated back into the acceptable, realizable portion of the distribution.

Which may not be much, admittedly, but would still suggest the Schrödinger equation is not accurately capturing the probabilities; but consistently falling short.

Not being a physicist, I would have guessed such a discrepancy between theoretical and realized probabilities would have been noticed by now.

1. Mathematically, Schrödinger's equation is akin to a diffusion equation, and inaccurate in the context discussed here. A relativistic equation (Klein-Gordon or Dirac) should be used. The solutions could then actually have sharp edges instead of long tail precursors.

Incidentally, the tunnel effect in the case of photons is adequately described by classical optics ("evanescent waves"). The distinction between "classical" and "quantum" effects is not Nature's, but of how we choose to describe them.

13. As someone with an understanding of physics limited by not having the math involved, I still found this to be a very clarifying description of tunneling. Thanks, Dr. Hossenfelder!

14. Referring to the last part of your video, I noticed how the 2007 conjectures of Günter Nimtz - the superluminal tunneling of information and the violation of Special Relativity - can be debunked with a single stroke. kudos!

15. The tunnel effect can also be understood by the particle model of Louis de Broglie; which means in a more classical way which is helpful for imagination and avoids much of the weirdness of QM.

A potential wall in this context is realized by a field which repels the charges of the particle trying to pass. These field forces build the potential wall which the particle has to pass. This is the field of the particles which are the cause of the wall. The oscillations which are present in all particles cause the internal charges to build an external oscillating field. If there are several particles, which is the normal case, then the field is built by a random superposition of the individual fields. So the repelling force is in a permanent change. If this constellation is momentarily such that the resulting field is smaller than the average, then the particle in view can pass even if its energy is too small for passing at the average potential. The probability that the superposition of the single fields cause a considerably reduced one is statistically a rare case. In such rare case the arriving particle may pass at a comparatively small energy.

This probability is described by the Schrödinger equation.

And I think that the apparent case of a superluminal velocity can be explained by the difficulty to determine the position of a particle, if only the surrounding wave is detectable.

16. I have to wonder what happens to the portion of the wavefuction that was associated with the tunneled particle, but is reflected at the barrier and doesn’t continue with the particle? This, I imagine, impinges on the old debate whether the wavefunction is something real, or is just knowledge of the observer. My understanding is that the Copenhagen interpretation would fit in the latter category. Hidden variable theories like the de Broglie-Bohm model by imputing reality to the wavefunction (as an ensemble of pilot waves associated with the particle, I think?), would (presumably) treat this reflected portion of the wavefunction as objectively real and able to exist on its own.

But, I’m guessing that this depiction of the reflected portion of the wavefunction, originally associated with the particle, in both the Quanta magazine article, Sabine’s excellent video, and textbooks in general, is just for illustration and not intended to represent reality. In the de Broglie-Bohm pilot wave model perhaps the reflected part of the wavefunction (as an ensemble of pilot waves), in a real world situation like a radioactive nucleus, just continues its existence inside the potential barrier of the nucleus and either mixes with other pilot waves or dissipates eventually. Don’t have time this AM to research this in more detail, with a pending appointment, but will check it out later today.

17. Commonly tunnel effects are noticeable through diminishing hustle and bustle.

18. To Engineers, i.e. people who know PDEs, the Fourier theory, and a bit about numerical solutions. I write this for you. (Sorry, laymen! I just don't know how to explain it *briefly* enough---and I have RSI.):

See if this helps. (Sabine, could you please to chime in if I am describing it wrong somewhere.)

---

A neat applet quite similar to what Sabine has shown, is by Prof. Dr. Daniel V. Schroeder. (Google on his name + "applet" + "BarrierScattering.html".)

I now assume you've played a bit with this applet.

This applet has thankfully been written in HTML5, and so I could easily take a peek at the code (right click, View Page Source). Very brief comments follow. (I assume that the scheme for Sabine's simulation goes similarly.):

1. There is a finite box with infinitely large potential walls on its two extreme ends.

2. Assume no PE (i.e. V =0) i.e. no middle barrier. In Schroeder's applet, you can't make V zero, but you can make it ("Barrier energy") very small; smallest possible is 0.001.

3. The simulation puts a Gaussian wave-packet in this box, as the initial condition. Its initial "width" is small.

(Technically, the Gaussian *always* fills all the space of any domain---finite or infinite. The "width" for the Gaussian is therefore defined differently. (Recall/refer to the normal distribution.))

4. The peak of the initial Gaussian is put into the left half of the box. Though the Gaussian's tail is not zero anywhere in the box, it's almost zero in the right hand-side half.

5. With time, due to the Schrodinger evolution, the Gaussian wave-packet spreads. (It "diffuses", though this is not quite the exact word.)

Schroeder implements the time-marching through the leap-frog, not FFT, but yes, you can always think about the time evolution using the Fourier theoretical terms.

6. The interaction of the Gaussian wave-packet with the boundaries (and the potential barrier, if any), together with the Schrodinger evolution, makes the peak go to the right.

This rightward motion of the peak is not a primitive; it is a result of a process. The motion does not exactly occur at a constant speed, but I guess it might be OK to describe it thus in a pop-sci video.

7. Next, you increase V ("Barrier energy") to some significant value, and repeat.

8. With V = 0, there is only one peak. With V = nonzero value suitable for tunnelling, there are two peaks. Yes, it's a bit complicated, but basically it's directly a result of the Fourier theory (and the Schroedinger equation).

9. In both cases, the peak is a result of superpositions of sinusoidals of *all* frequencies (f = upto half the number of cells used in the simulation; Shannon's theorem). Remember, Gaussian in theory carries a continuum of pure frequencies.

10. The motion of the peak (e.g. its speed) thus is a result of superpositions of plane waves (sinusoidals).

11. Different frequencies "react" differently to the potential wall of the specified width and height.

Think of it this way: each sinusoidal has to maintain continuity. To appreciate this point, check out the PhET simulation on quantum tunnelling. (QM *theory* requires only C0 continuity, but the IC is Gaussian, so C1 also would get satisfied.)

12. Now, feel free to interpret what(ever) it all means.

Best,
--Ajit
PS: Guess it would be a good idea to turn this reply into a blog post at my blog too!

1. Hi Ajit,

Thanks for pointing out, I hadn't seen this app! I have really just dumbly forward integrated the Schr eq. Anything more sophisticated seemed to me an overkill given that I'd only need a few seconds of it. I did not put infinite walls on the side, and the Gaussian you see in my sim actually isn't exactly a Gaussian because it goes to zero at a point you can't see (which produces some artifacts). Oh, and the color in my animation doesn't show the phase, I found that too be too confusing and also superfluous somehow.

2. Hi Sabine,

1. Sure! Also see the PhET simulation. You can try plane-waves in it.

2. For pop-sci videos, the forward Euler is quite great! I vaguely recall that Chad Orzel used it for showing how precession (with nutation) arises, and Rhett Allain for demo'ing 3-body simulation, in their blog posts.

(BTW, I always check out *others'* code first, before implementing anything. After all, I am, ahem, a *professional* programmer.)

3. What approximate Gaussian with smoothly approached zero-ends did you use? Any systematic means to construct such functions for numerical work? In any case, looks like it gives great results for this case. Would be a handy tool to have. (For my work, I was actually thinking of just "pulling down" the whole Gaussian so that its support becomes finite!)

4. Phases *are* confusing. I don't "get" much anything using them. I have to see the separate Re and Im plots, really speaking. But for pop-sci, prob. density is best, I think. No confusions!

5. I caught a mistake I made in my comment above:

QM theory requires only square-normalization, not even the C0 continuity. E.g., in theory, you can have a wave-function whose Re and Im parts *separately* go like ideal square waves (with the Re and Im squares not even matching). Neither Re nor Im part would be, technically, even C0 continuous.

Best,
--Ajit

3. Sabine,

Sorry to bother you again, but I've another question:

If you didn't put infinite walls on the sides, then how come the peak ends up travelling in one direction (to the right)?

...On second viewing of the video, looks like your wavefunction isn't symmetrical around the peak---it has a sharper drop on the right hand side. That can explain it.

(May be, if you don't mind, could you please share the code? But of course, a hint would be good enough too.)

Best,
--Ajit

4. My "code" isn't a code really, and it's in no condition to share. I'm not sure I understand your question, I just used the initial condition for a wave-packet with a non-vanishing momentum, that's why it travels to the right. I don't know why you think it's not symmetric. It should be. Maybe that's from the shading? The shading may not be symmetric. (It's actually just a blurred gradient that I didn't do myself, so god knows what that is.) The issue with the boundary condition is that the program I used expected the boundary condition to be constant with t. So I set it to zero at some initial time, but then if the wave-packet gets close to the boundary, that produces nonsense. I didn't have the patience to deal with that, so I just put the boundary very far away from the center of the Gaussian, so it's to excellent precision zero and all is fine. Hope that clarifies it.

5. Sabine,

1. OK, I got it! I mean the travel to the right. (Stupid me! Somehow, had got stuck in the special case of k_0 = 0 in my imagination, once I *began* with that case).

2. Well, the profile in the video does seem to become asymmetric as it moves. Could be a numerical artifact, but guess it's best to leave the matter at that, because the overall picture *is* quite clear now. Thanks!

Best,
--Ajit

6. Hi Ajit,

I am still not sure what asymmetry you mean. Could you let me know what time in the video you are referring to so I can look into this?

7. Hi Sabine,

Check out at 02.49 in the video. The peak is under your left palm. I took a screenshot and verified. While the absolute number of pixels would depend on the size of the window of the video when the screenshot was taken, it is clear that the x-axis extents of the left- and right halves of the profile, as measured at the bottom of the profile, are roughly in the 0.44 : 0.56 proportion. Not 0.5 : 0.5.

Best,
--Ajit

8. Oops. The left half is longer (~0.56), and the right half is shorter (~0.44). --Ajit

9. Hi Ajit,

Ok, thanks, I see now what you mean. I thought you were talking about the tunneling part. Yes, I suspect that this comes from the boundary condition. You see, the boundary condition will try to push the value of the Gaussian down to zero somewhere off to the right, though actually it isn't zero. I guess I should have put the boundary farther away, but that'd have brought up the computation time. Sorry about that & thanks for pointing out.

19. Dr. Hossenfelder:

Is there some mass limit (or complexity limit) to tunneling? For example, can a water molecule tunnel through a physical barrier? Or is it impossible to make a physical barrier that thin?

(I have long thought pancake syrup tunnels; I have no other explanation for how it can get everywhere.)