Quantum Tunneling

In some respects this logically follows on from a post I wrote in July this year, Quantum Mechanical Philosophy, which is one of the more esoteric essays I’ve written on this blog. Hopefully, this essay will be less so, as the source material is well written and aimed at the uninitiated.

But I need to recount the gist of that post to make the relevant connection: specifically, the enigmatic Bell’s Theorem or Bell Inequality. To summarise, Bell’s Theorem arose from a thought experiment created by Einstein in an attempt to prove Bohr’s interpretation of quantum mechanics (the famous ‘Copenhagen Interpretation’) wrong.

The thought experiment was elaborated upon by Podolsky and Rosen, so it became known as the Einstein-Podolsky-Rosen or EPR experiment. It examines the purported ‘action-at-a-distance’ phenomenon predicted by quantum physics for certain traits of particles or photons, which Einstein described, quite accurately, as ‘spooky’. If you have 2 particles with a common origin (could be photons with opposite polarisation or subatomic particles like electrons with opposite ‘spins’), then separate them over any distance whatsoever, you will not know what the spin or polarity, or whatever quantum mechanical trait you are measuring, is, until you take the actual measurement. The ‘spooky’ bit is that as soon as you make the measurement the ‘twin’ particle will instantaneously become the opposite. Before the measurement or observation is made the particles are in, what’s called, a ‘superposition’ of states – it can be either one or the other.

Einstein realised that this conjecture contradicted his special theory of relativity, which states that no signal or means of communication between particles of any kind can travel faster than the speed of light, which had already been confirmed by experiment. John Bell developed a mathematical equation that analysed correlations of hypothetical results from the thought experiment that would categorically prove either Einstein or Bohr wrong.

Alain Aspect developed a real experiment to test Bell’s Inequality (made the thought experiment actually happen) and proved Einstein wrong (long after Einstein had died, by the way).

As I point out in that previous post, the upshot of this is that either faster-than-light actions are possible (called non-locality) or there is no objective reality. Non-locality is self-explanatory (you can’t communicate faster than the speed of light) but no objective reality means that the thing doesn’t exist until someone measures it or takes an observation. I discuss this in more detail (lots of detail) in my previous post, but that’s effectively the Copenhagen interpretation of quantum mechanics: at the subatomic scale, particles don’t exist until they are measured or observed. A less extreme and more popular interpretation is that they remain in a superposition of states until they interact with something else. If you want to delve deeper, read my previous post, but you may be none the wiser. The philosophical implications of this have never been truly resolved.

My conclusion was to accept non-locality (faster-than-light connections) in order to keep objective reality, and I made specific reference to David Bohm’s unpopular interpretation, known as the ‘hidden variables theory’. Bohm believed that there was a hidden set of parameters that govern the particles which we can’t see or detect.

To quote David Deutsch (who doesn’t agree with Bohm at all): ‘A non-local hidden variable theory means, in ordinary language, a theory in which influences propagate across space and time without passing through the space in between.’

And this leads me to quantum tunneling, because that’s exactly what quantum tunneling does, only it happens over short distances, not the distances used in the EPR experiment, which could theoretically include the other side of the universe.

I’ve just read an excellent book on this subject, Zero Time Space subtitled, How Quantum Tunneling Broke the Light Speed Barrier, authored by Gunter Nimtz and Astrid Haibel. Originally published in German in 2004, it was published in English in 2008. This book could be read by people with only a rudimentary knowledge of physics, as it contains only a few simple equations, among them Planck’s equation: E = hf where E is energy, f is frequency of a ‘wave’ and h is Planck’s constant, 6.6 x 10^-34 Js (Joules seconds). The authors also include Snell’s law of refraction and the universal wave equation of wavelength times frequency equals velocity (I can’t find the symbol, lambda, for wavelength, in my arsenal of fonts). One of the annoyances is that there is a type-setting error in this particular equation (in the book). If someone is going to include equations, especially for people unfamiliar with them, I wish they could at least get them checked during type-setting. The same applies to Richard Feynman’s excellent book on relativity theory, Six Not-So-Easy Pieces where I found 3 type-setting errors amongst the equations scattered throughout the book. In both cases the books are aimed at people who are not familiar with the material, which means they won’t know the errors are there.

Putting that one (some may say petty) criticism aside, it’s a very good book on quantum mechanics for people who know very little about physics. It includes a short history of physics leading up to Einstein’s theories of relativity (with particular reference to the Special Theory) as well as quantum mechanics. They do this because the whole point of the book is to highlight how quantum tunneling breaks Einstein’s special theory of relativity, and therefore reinforces non-locality, as I described in my previous post. So the authors go to some pains to give the reader an overview of both Einstein’s theory and quantum mechanics, in conjunction with the historical context. It’s very well done.

Nimtz and Haibel, by the way, make no reference to Bell’s Theorem, as it would probably confuse readers who are unfamiliar with it – I hope I haven’t put people off by referencing it here. Having said that, they do discuss ‘entanglement’ towards the end of their book, which is the state I described above concerning ‘twin’ particles interacting at a distance. In particular, they raise this phenomenon in their lengthy discussion on causality, as they are at pains to explain that ‘tunneling’ does not affect causality as some people might be led to believe. Even so, it still manages to confound common sense, as I explicate below.

In the forward to the book, they briefly discuss the ‘myth… about the half-life of knowledge… It suggests that our knowledge is being declared invalid every five years by new knowledge.’ They then go on to dispel the most common representation of that myth: ‘Newton’s theory of gravitation is still valid, even in the light of the theory of relativity… Einstein’s theory has extended theory rather than disproved Newton’s theory.’

I made the same point in my essay on The Laws of Nature (March 08), explaining that Einstein’s equations reduce to Newton’s when certain parameters become negligible. The authors raise this point, because, whilst quantum tunneling appears to contradict Einstein’s special theory of relativity, in their own words: ‘Einstein deals with free space, whereas tunneling is not free space.’ In other words there are constraints on relativity theory in the same way that there are constraints on Newton’s theory, but there are various aspects of nature where one is more significant than the other. It’s one of the reasons that I’m a bit sceptical about a grand unified theory (GUT), a meta-theory of everything. Many people would love to prove me wrong, and a part of me would like to see that, but another part wouldn’t because I don’t believe there will ever be an end to physics.

During this discussion they make another statement, relevant to the stability of scientific knowledge: ‘Mathematical proof has been regarded since Pythagoras and Plato as eternal, metaphysical truth.’ A statement I would agree with. For example, Reimann geometry hasn’t displaced Euclidean geometry, it has just extended our knowledge, both of the mathematical world and the physical world (through Einstein’s theory of General Relativity).

I’ve discussed on other posts, the relationship between mathematics and the natural world (refer The unreasonable effectiveness of mathematics, March 09), but no where is that more significant than in quantum mechanics. QED (Quantum Electrodynamics), for which Richard Feynman, Julian Schwinger and Sin-Itoro Tomonaga jointly won the Nobel Prize, is the most successful theory of all time. Without mathematics, quantum mechanics would be indecipherable, quite literally. Intriguingly, there are imaginary numbers in quantum theory that are completely relevant to quantum tunneling. Without imaginary numbers (created by the square route of -1, called i) quantum mechanics would never have been articulated as a meaningful theory at all.

As Nimtz and Haibel point out, it is the imaginary component of the equation that does the tunneling. When this was first derived, people just assumed that these imaginary components were unnecessary remnants of the mathematics, but that’s not the case. When tunneling occurs there is an interface where part of the signal is reflected and part is transmitted through ‘the tunnel’. The part that is reflected is mathematically ‘real’ and the part that is transmitted is mathematically ‘imaginary’. (I've since been informed this is not correct - refer Addendum 2 below.) A tunnel, by the way, is a barrier, where the particle or wave theoretically can’t travel, because it doesn’t have enough energy. The authors point out that it even occurs in the sun, otherwise the fusion, which gives us sunlight, would never occur. I should add that quantum tunneling is a feature of all transistor devices. In fact, it's the very feature that makes transistors work (called 'tunnel diode' by Nimtz and Haibel).

Both of the authors have performed experiments, to not only detect quantum tunneling, but to also measure the time elapsed. As predicted by Thomas Hartman in 1962, there is a time elapse at the ‘entrance’ to the tunnel, or the ‘interface’, between the medium and ‘the tunnel’, but the actual time spent in the tunnel is zero. This is called the Hartman effect. To quote the authors: ‘So the wave packet spreads in the tunnel in zero time and is everywhere from the entrance to the exit. This non-local phenomenon makes one feel eery.’ An understatement, if I’ve ever read one.

One of the authors, Gunter Nimtz, participated in an experiment that tunneled Mozart’s symphony in g-minor through a waveguide at superluminal speed: 4.7 times the speed of light. The elapsed time occurred at the entrance to the tunnel, as predicted by Hartman, not in the tunnel itself. In an exposition, that I will not try to repeat here, the authors explain how this quirk of nature (the elapsed time at the entrance to the tunnel) allows superluminal communication without impacting causality. The speed in the tunnel is infinite – as the Americans like to say: go figure. The title of the book, Zero Time Space, is therefore entirely appropriate.

They end the book with a brief description of wormholes and hypothetical warp drives, beloved of Sci-Fi writers, like me, that require exotic negative gravity amongst other improbabilities.

Of all the incredible manifestations of the universe, only consciousness is arguably more inexplicable or more mysterious (but no more weird) than quantum phenomena. If we didn’t observe it, no one would believe it. And if we didn’t have the mathematics to describe it, no one would be able to fathom it, even remotely.

Addendum 1: I came across this - it's very entertaining as well as informative.

Addendum 2: I would like to acknowledge Timmo (refer comments thread below) who has valiantly tried to correct all my mistakes. In particular, that the imaginary component of Schrodinger's equation plays no greater role in tunneling than the real component, if I understand Timmo correctly. Also he points out that tunneling and non-locality are independent phenomena, and possibly I misled people on that point.

He also corrects some faux pas I made concerning the Lorenz transformation and Godel's Incompleteness Theorem in response to comments I've made since the post was posted.

I confess I don't know as much as I appear to, and I wish I understood more than I actually do.

And I would like to thank Timmo for reminding me of how much I don't know.