It can be argued that it is the interpretation of the mathematical formalism of quantum mechanics which makes the latter “weird” or “counterintuitive”.
Take theoretical physicist and broadcaster Jim Al-Khalili’s words (in his book Life on the Edge: The Coming of Age of Quantum Biology) on quantum tunnelling. He writes:
“Although it would be wrong to think that quantum tunnelling entails the leaking through barriers of physical waves; rather, it is due to abstract mathematical waves that provide us with the probability of instantaneously finding the quantum particle on the other side of the barrier.”
So, in the above, Al-Khalili explicitly states that
“it would be wrong to think that quantum tunnelling entails the leaking through barriers of physical waves”.
Does that mean that there is no literal tunnelling at all?
Alternatively and at the very least, Al-Khalili is stating that there is nothing equivalent to tunnelling (as it would occur at the “classical” scale) that’s going on at the quantum scale. In other words, there’s no equivalent of, say, tunnelling through a six-foot-thick gold wall with a wooden spoon in quantum mechanics — despite the hype! (Or, less dramatically, there isn’t even an equivalent of simply tunnelling through the ground into your next-door-neighbour’s living room.)
Having written the passage quoted above, however, elsewhere Al-Khalili also writes the following:
“Just as waves can flow around objects… they can flow through objects, like the sound waves that pass through your walls… But if you behave like an atomic nucleus then you would sometimes be able to pass, ghostlike, straight through a solid wall.”
It’s not being said that this second passage from Al-Khalili somehow contradicts the first. However, it does highlight the problems tackled in this piece.
Firstly, “sound waves” passing through bedroom walls may be compared to quantum tunnelling - but this is obviously not the same thing. In addition, “you” can never “behave like an atomic nucleus” because if you did behave that way, then you wouldn’t be you. This is like saying the following: “If x = y, then x would be like y.” Or: “x would be like y, if x = y.” That is, if “you” were a particle/wave, then you would “be able to pass, ghostlike, straight through a solid wall”. But, of course, it makes no sense to say that you could be like a particle/wave passing through a subatomic “barrier”.
Finally, Al-Khalili uses the words “solid wall”. There are no solid walls at the subatomic scale . And even when the words “solid wall” are used as an analogy, this usage may still not work. Or, at the least, the analogy needs to be taken for what it truly is — i.e., simply an analogy. (Who knows, perhaps no one takes it any other way.)
Probabilities
When tackling quantum tunnelling, what we’re mainly dealing with is probability — or what’s called a probability wave. Thus, on this reading at least, the “wave” is literally abstract — i.e., not a wave at all! That is, the word “wave” is simply a colourful way of describing an abstract mathematical function. So, in quantum mechanics, these particles or waves — unlike footballs — can, with a small probability, tunnel to the other side of a barrier.
The important point here is that we’re dealing with quantum waves and particles. Or, at the least, waves and particles as they’re seen in quantum mechanics. And, because we’re dealing with waves and particles (though perhaps not only because), then no probability of a particle/wave tunnelling across a barrier can ever be zero. Another way of putting that is to say that the probability of a given particle/wave being “found” on the opposite side of a given barrier is not zero. In other words, it could happen. In terms of the barrier itself, the wider and higher the barrier is, the lower the probability of a given particle/wave tunnelling “through” it. (It must be noted here that some physicists identify the simple penetration into the barrier — i.e., without it reaching the other side — as an example of tunnelling.)
Quantum Tunnelling is Real
Despite stating all the above, Al-Khalili did also say that there is a
“probability of instantaneously finding the quantum particle on the other side of the barrier”.
And if there’s a probability of x, then it’s possible that x could happen. So, in this case, we could find a quantum particle/wave “on the other side of the barrier”.
So quantum tunnelling most certainly occurs — a hell of a lot! And that’s primarily because a hell of a lot of particles/waves — over various timespans — are involved in this phenomenon.
Now take the following selected examples of quantum tunnelling.
Quantum tunnelling occurs in nuclear fusion. It’s also “used” in the scanning tunneling microscope, in the tunnel diode (when electrons tunnel into solids) and in quantum computing. In addition, quantum tunnelling may be one reason for proton decay. Added to all that is the fact that quantum tunneling has been known about since 1927 (even if the term “quantum tunnelling” was never used in the 1920s or for a long time after).
That said, perhaps Al-Khalili’s own more concrete and up-to-date biological examples of quantum tunnelling are far more interesting. (Interesting in the simple sense that the other examples above have been known about for decades.) For example, Al-Khalili writes:
“[T]unnelling [has] been detected in lots of biological phenomena, from the way plants capture sunlight to the way that all our cells make biomolecules. Even our sense of smell or the genes that we inherit from our parents may depend on the weird quantum world.”
Yet, as in the other examples, these quantum-tunnelling and other cases still occur at the quantum (or micro) scale — even though they all have classical (or macro) effects.
Classical Pictures and Analogies
On this essay’s interpretation of Jim Al-Khalili’s words, there’s a stress on the mathematical formalism alongside literal physical interpretations of that formalism. Yet even though such physical events as quantum tunnelling occur, it may still be inadvisable to compare them to anything which occurs at the classical level. Thus not only may the words “wave” and “particle” create complications (at least for laypersons), so too may such words as “tunnelling” and “barrier”.
Again, obviously no one will question the fact that these physical things occur at the quantum scale. The question is whether the “classical” words or pictures accurately capture what actually happens. Indeed, as Werner Heisenberg (1901–1976) argued some nine decades ago (if not about exactly the same thing), perhaps no classical words, descriptions or visualisations will ever accurately capture these quantum phenomena. Indeed, in stronger terms, this is what Heisenberg once wrote (in a 1926 letter) to Wolfgang Pauli:
“The more I think of the physical part of the Schrödinger theory, the more detestable I find it. What Schrödinger writes about visualization makes scarcely any sense, in other words I think it is shit.”
Yet, of course, as Niels Bohr (1885–1962) also argued that we have no choice but to use classical terms (or classical descriptions) in our interpretations of quantum mechanics. Bohr once wrote the following:
“It is decisive to recognize that, however far the phenomena transcend the scope of classical physical explanation, the account of all evidence must be expressed in classical terms. The argument is simply that by the word ‘experiment’ we refer to a situation where we can tell to others what we have done and what we have learned and that, therefore, the account of the experimental arrangement and of the results of the observations must be expressed in unambiguous language with suitable application of the terminology of classical physics.”
The above seems to be a simple account of Bohr’s more technical correspondence principle; which states (broadly speaking) that the behavior of systems described by quantum mechanics reproduce classical physics when it comes to large (what’s called) quantum numbers.
So perhaps even trained physicists (i.e., those who’re adept at the mathematical formalism/s of quantum mechanics) also need such “classical pictures” in order to understand what’s going on. And that may be because in physics it’s almost impossible to rely entirely on the mathematical formalism/s. That said, many commentators have claimed that, for example, Paul Dirac (1902–1984) didn’t require any additional “classical pictures” or interpretations (see here). Indeed Dirac himself (more or less) said the same about his own stance on these matters. ( For example, he once wrote: “The interpretation of quantum mechanics has been dealt with by many authors, and I do not want to discuss it here. I want to deal with more fundamental things.”) Yet Dirac did class himself as “mathematical physicist”, not as a “mathematician”. (Incidentally, Dirac didn’t even class himself as a “theoretical physicist” — though most other physicists always saw things differently.)
Jim Al-Khalili himself tells us that he
“tr[ies]… to provide intuitive analogies wherever possible to explain quantum phenomena”.
So that surely means that the words “wave”, “barrier”, “tunnelling” — and especially “particle” — are analogical too.
But what are these words analogies of?
Basically, they must be analogies of (to use Philip Ball’s words) “what the maths tells us”.
But what does the maths tell us?
Perhaps the maths tells us nothing beyond itself.
Yet surely that can’t be the case!
Anyway, Al-Khalili also says that
“the reality is that quantum mechanics is utterly counterintuitive and there is a danger of oversimplifying for the purpose of clarity”.
Perhaps it’s the case that the “analogies” Al-Khalili refers to are almost entirely responsible for the “counterintuitive” or “weird” nature of quantum mechanics. That is, QM becomes weird the moment it’s interpreted using analogies or classical descriptions.
The other thing is that such analogies don’t actually (as Al-Khalili put it) “oversimplify[]” at all — they do the opposite. Perhaps the analogies are actually creating the complications (alongside the weirdness). That is, these analogies are effectively attempting to place round shapes in square holes. And, because of that, rather than analogies (or classical pictures) simplifying things, what we get are complications… And we most certainly get weirdness or counterintuitive scenarios.
So on the one hard we have the seeming clarity which everyday analogies or pictures provide (i.e., those analogies/pictures which help the laypersons who don’t know the mathematics of quantum mechanics). Yet, on the other hand, these analogies or pictures actually add to — or even complicate — the mathematics. More strongly, the analogies or pictures simply don’t work in that they don’t — and even in principle can’t — describe or express the mathematics.
All this should be obvious really.
Think about what we have here.
Particles, waves or events at the quantum scale are at a level that is fantastically smaller than, say, a football. In terms of detail, a sheet of paper is about six orders of magnitude thicker than an atom. Now think about how much “thicker” a football is to a sheet of paper. (100,000,000 atoms would stretch along the width of your fingernail.)
More relevantly to this piece, take the case of the barrier through which the particle/wave tunnels. Usually, this tunneling only happens when it comes to barriers of thickness around 1–3 nm (i.e., nanometres) or smaller. A nanometre is equal to one billionth of a metre.
A Final Word
So when we use the kind of words we use to describe the kicking of a ball to describe what happens at the quantum scale, no wonder things become weird. Doing so is roughly like using poetry to describe the internal workings of a cell phone. Or, alternatively, it’s like expecting a virus or lice to behave (entirely) like a cat or a human being.
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