Quantum Mechanics Simplified


The idea of writing this post arose during a brief exchange with Llewelyn Morgan – an Oxford classicist whom I follow on Twitter. Llewelyn was responding to an interesting piece in Quanta mag on a classical fluid physics experimental analogy of the famous De Broglie-Bohm interpretation of Quantum Mechanics, and remarked that he found the topic abstruse. I think it is a fair comment, but attributable more to a lack of clarity around the general field of quantum mechanics (in part a reflection of the confusion within the community of physicists themselves) than to some innate difficulty in understanding the basic details of QM itself. So my post is an attempt to explain QM to the lay reader.

Note that this post has literally nothing to do with anything remotely brown or even brownish. So people who read this blog for cultural commentary or South Asian history etc may stop right here.

Also a general note to physicists reading this: in my description of the problem (specifically tangential references to mathematical details) I may introduce some simplification in language. Don’t get unnecessarily triggered by this for you are not the intended audience 🙂

Quantum Mechanics (or wave/matrix* mechanics, as it was known earlier) arose as a field of Modern Physics (as opposed to Classical or Newtonian Physics) in the early part of the 20th century. It was a result of certain observations made as experiments to comprehend and model the basic structure of matter increased in sophistication. Such experiments typically involved firing beams of sub-atomic particles, like alpha rays, on to a target (say a super-thin metal foil) and study the scattering pattern of the particles. These were the proto versions of the modern particle accelerators, like the famous one at CERN.

Experimental physicists realized early on that these experiments behaved identically if electro-magnetic (EM) radiation – which includes the normal visible light by the way – was used instead of beams of particles, e.g. alpha rays are beams of helium nuclei. Until then EM waves were seen not as beams composed of discrete chunks of “stuff” but merely disturbances or waves of ethereal “fields” travelling in space, like waves on the surface of water. It was observed that when the intensity of the EM radiation was tuned down, the energy level of the radiating beam came down continuously only up to a point. Beyond that point, one could see the decrease in discrete jumps, as if the beam was being extinguished one “bit” at a time. That “bit” of the EM radiation is the quantum or photon.

Theoreticians like Max Planck, came up with the mathematics to explain the proportional relationship of the energy content of the quantum with the observed frequency. The constant of proportionality is the famous Planck constant and like Newton’s gravitational constant it forms one of the basic constants of natural law that no mortal messes with. One of the knobs god turned to bring our universe about, or so the religious tell us…

Anyway, so far so good. Light looks like a wavy thing, but is really made up of wee bits (photons or quanta) of energy. It is a little weird to comprehend indivisible bits of energy, but not too much to wrap one’s head around. So what’s the fuss about? Ah, but we’re only getting started. In comes one of the simplest, but also the most famous and controversial experiment ever devised by humanity: the aptly named double-slit experiment. And there’s a Linguistics connection, for the man who first performed the double-slit experiment (in 1801) was the British polymath Thomas Young, the decipherer of the Rosetta Stone. The patron saint of Linguist Physicists. (I sometimes think in my whiggish flights of fancy that it was an absolute pleasure to have been colonized by the Victorians of all people!)

Young’s double-slit experiment with light was to show its wave-like nature. The setup is deceptively simple and anyone can replicate this at home with a laser pointer (or a mobile phone torch light) and a piece of cardboard. All one needs to do is cut two parallel slits in the cardboard so that light can pass through them. Keep it at a suitable distance from the wall (preferably in a darkened room), and shine the light on the cardboard at the other end. What one witnesses on the wall is not a plainly lit area in the shape of two parallel slits, but an interference pattern – a pattern of successive bright and dark patches.

1. The diagram above shows the pattern of interfer

It was obvious to Young as it is to modern physicists that the reason why this pattern emerges is because light travels like a wave, with crests and troughs. And as the wave front passes through the two slits, the resulting fronts interact with each other. The crests of one wave front strengthening the crests of the other wave front but being cancelled by the others’ troughs. The symmetry creates the pattern of alternate waxing and waning of the light wave front we see on the wall.

Again, light + light = dark is a little weirder. But one can still live with that. Maybe light is nothing but a stream of energetic particles moving in tandem, and the double slit causes the motions of some of them to cancel out as they collide head-on (whereas motions of other get reinforced) and we see the result as this pattern. So, maybe the wavy nature of light is an emergent effect of a conglomeration of a stupendously large number of moving particles (like molecules of water) and one could ostensibly reconcile the particulate (or corpuscular) nature of light at very small scales with its gross wavy nature. It seemed sensible for a little while, but physicists are a curious bunch.

The experimentalists of the 20th century were not quite content with this view, and took Young’s experiment to a place Young couldn’t even have dreamt about. They tested how the light-dark interference pattern changed as the intensity of the light was tweaked lower and lower until only one quantum of energy was fired at the slits at a time. Just one teeny weeny photon hitting the slits. What is the “common sense” expectation? It is the same thing one would expect if one were to throw a cricket ball through two large, parallel slits to the wall behind them. Some throws will hit the card-board and the ball bounces right back without having reached the wall. Some throws will send the ball through the right slit and hit the wall, and the rest through the left. These are all mutually exclusive possibilities and the resulting pattern of marks is merely two areas (with some overlap) where the ball hits the wall.

Yet when the experimentalists tried it with a photon (in place of a classical cricket ball) that is not what happens. Sending through a single photon at a time through the slits and collating the results of many such hits still results in the same interference pattern. Stop reading and take a moment to mull over how supremely weird this is. How does a single photon “know” that the experiment will be repeated later with other photons separately, and it lands in a manner so as to accommodate the final interference pattern? A single photon thrown at the slits is just one indivisible entity. It does not have any observable counterpart – another photon moving in tandem or cancelling motion – to guide its trajectory. Yet that is what seems to happen with every single one of them. The plot thickens.

Something’s got to give. You may say that maybe the indivisible quantum really isn’t indivisible after all. Maybe it is smeared out in space – some fraction at one slit and some fraction at the other, and these fractional photons then interfere in the same way individual molecules or photons do. There is a serious problem with that picture, namely we never see it. Remember we only ever register a single photon hit the wall at the back – so this smeared out sub-quantum cloud, if it exists, must magically coalesce as it impinges the wall. Ok, our credulity is at breaking point now and we may just about manage to cope with this weirdity. But what about the slits? There are 2 of them and if this smeared out cloud is real, we ought to measure fractional quanta at both slits. As you may have guessed, we do not. Every time we measure the photon, from the point it is fired to the point it passes through the slits to the point it lands on the wall we always see just one, indivisible photon, either at the left slit or the right but never both. We see no magical smeared-out clouds of fractional photons or magical invisible waves buffetting our single photon or one photon in two places at once. Just one guy moving as if it were in a crowd of other guys – a mysterious effect technically known as the superposition.

There are various theories, more correctly ontological interpretations, of this experiment. Don’t let the word “ontology” scare you – it is simply fancy short-hand for stuff that is really happening out there. One interpretation, originally floated by the French Nobel laureate Louis de Broglie and later fine-tuned by the American David Bohm is that quantum particles are buffeted by an invisible but real wave that guides/pilots it to its final landing place. And this happens with every single quantum, such that the final result is the interference pattern. In other words, de Broglie-Bohm get out of the problem by stressing the ontology of the pilot wave – the wavy essence separate from and not an emergent effect of the particles or quanta. Of course we cannot see such a wave, nor do we have any way to measure its existence other than the mysterious effect it seems to have on our quantum particles. So is it truly real, or an artefact of our theory? Besides it seems to affect only the very small, but not large objects like cricket balls. But why not?

This brings me to the Quanta piece that Llewelyn commented on, which I referred to at the start of this blog post. The article is about experimentation done on the fluid physics analogue of modelling the pilot wave. Note that we cannot directly observe this quantum mechanical pilot wave (assuming it exists), but we can observe classical waves in fluids and the equation for those waves are mathematically analogous to that for the pilot wave. So, if we can setup a experiment showing how fluid waves, going through the double-slit, move particles so as to produce an interference pattern akin to QM, it is an important (albeit indirect) confirmation of pilot waves doing the same at microscopic scale. The study shows that such an interference pattern was not found as originally reported, and therefore represents a set-back for the de Broglie-Bohm pilot wave theory.

Another interpretation, the least satisfying but also the most widespread, is the so-called Copenhagen interpretation (CI). The word “Copenhagen” here is really a euphemism for the redoubtable Danish physicist, Niels Bohr, who was instrumental in creating it but didn’t want to make it obvious by naming it after himself 😉 It has remained physics dogma ever since – the stuff for the textbooks. The basic idea of CI is that the world of the very, very small (technically speaking, of the order of Planck scale) is fundamentally different from the world of the macroscopic. So the concept of a trajectory of a photon does not make the same sense as trajectory of, say, a cricket ball. Asking questions of the ontology of where a photon is and how it reached there are silly. Physics students shouldn’t be asking them. They should shut the hell up and calculate.

In other words, Bohr favoured an instrumentalist – as opposed to a Popperian or realist – view of fundamental reality. Don’t care about what really is, but care about what you can measure. Left to its own devices a photon does not exist as me and you do, namely in one place and time coherently, but as pregnant with possibilities of where it could be. It is only in the act of measurement of a photon that we realize one of those possibilities and it pops into classical existence – unitary and indivisible, just how we like our quanta to be. The mysterious state of suspended animation, of possibilities of what could be, is the superposition and its mathematical definition is called the wave-function. The act of measurement is akin to the collapse of the wave-function to yield a single classical state. The quantum wave-function cannot tell us where the photon is, but it can tell us where it is more probable to occur and various meta-rules and theorems (cf. Born rule) have been devised to do an accounting of these probabilities.

The fact that probabilities are an embedded part of nature was an anathema to not just positivists (Ernst Mach and his ilk) but also realists like Einstein. One of the fundamental (and often unsaid) assumptions of the Scientific Method is that the set of what exists is identical to the set of what can be measured-calculated-comprehended. Yet how can we truly comprehend something that which we cannot exactly repeat in principle? And the nature of quantum randomness of being baked into reality is precisely the sort of thing that offends repeatability and therefore comprehension. It is not the same as the lack of repeatability of the kind seen in normal (classical) experiments prone to human error or noise. Those factors can be reduced to arbitrarily low levels. However, quantum uncertainty is a law of nature and therefore irreducible. This ostensibly unscalable wall of ignorance implicit in natural law is what Einstein deeply militated against – whence God does not play dice!

Bohr’s instrumentalist shut-up-and-calculate solution was to cleave measurability from existence – a price too great for Einstein and many students of physics since (including yours truly). Yet Bohr, Heisenberg and others managed to gang-up on Uncle Albert (at the famous Solvay conference of 1927) and were successful in foisting their deeply dissatisfying instrumentalist dogma on generations of physicists.

https://youtu.be/nbdf-_4TWYw

My own frustration with CI started fairly early in secondary school, when the basics of orbital configurations of electrons in base elements are introduced in the Chemistry curriculum. Orbitals are nothing but wave-functions of electrons (as opposed to photons) inside an atom that show where electrons are most likely to be found. QM showed that our simplistic picture of an atom comprising electrons revolving around a central nucleus is woefully inadequate. Actually electrons inside atoms come in different spatial configurations, but one of the most common is the so-called p-orbital. It is a dumbbell shaped region where the two spheroid zones are separated by a nodal plane that passes through the nucleus.

Image result for p-orbital

Consider Boron, an element found in much abundance on Earth (mainly Turkey and US). Each Boron atom has 5 electrons, of which one is in a p-orbital. In simple terms, this p-electron has 50% chance of being found in the one spheroid and 50% chance of being found in the other spheroid. Ok, but we’ve already figured this shit is weird, so what? The trouble is the electron cannot exist at all in the nodal plane according to QM. So if we took snapshots of this little guy and found it in spheroid 1 of the orbital in the first, and then in spheroid 2 in the second – how did it go from non-zero chance of existence to nothingness to non-zero chance of existence? The dogmatic CI answer –  don’t ask about “how did it go” – always seemed like a patronizing unscientific cop-out to me. In the meanwhile, Boron deposits continued to exist and nobody saw them spontaneously turn to Boron ions (with 4 electrons) and back. Heck! there’s even a mining town called Boron in California and they’d spot it by now, right? I’m not sure Americans were bothered about their boron supply, but goodness knows I was tormented by it – right through to my first year of PhD when I finally managed to cure myself of it. Read on to find out how I managed this…

Dogmas have short shelf-lives in Physics. Physicists aren’t quite the sort who’d toe the line just because one of them (or anyone else for that matter) with a redoubtable reputation says so. So CI dissidents (like Einstein, de Broglie, Schroedinger, Wheeler etc) continued their work and major schools of interpretational theories emerged, of which de Broglie-Bohm discussed above was one. Yet the best was yet to come. It took another 30 years after Solvay, for an American Phd student at Princeton (advised by John Wheeler) called Hugh Everett to posit the Relative State formulation. It is now popularly known as the Many Worlds Interpretation (MWI) – a phrase coined by the American physicist Bryce DeWitt who built on Everett’s work.

The basic idea of Everett was to resolve the question of what’s really happening by taking the wavefunction seriously – not as an artefact of our theories or a container of probabilistic information – but a real feature of the universe (like length and mass). Taking it seriously and following through the logic of measurement carefully we begin to unravel what is truly happening. Let’s understand it by following through the most famous gedanken-experiment** in all of Physics, of Schroedinger’s cat:

The gedanken (thought) experiment is quite simple. Erwin’s poor cat is locked in a box which contains a poison gas vial, connected to a mechanism for opening it triggered by a quantum experiment, say, Young’s double slit-experiment. If the photon passes through the right slit, the mechanism stays put and the vial is not opened and the cat lives. Whereas if it passes the left slit, the mechanism triggers the vial open and the cat dies. QM says that the photon is in a weird superposition between the left and right slits until it is measured. Obviously, therefore, if the box is unopened the superposition stays and the cat is both dead and alive at the same time – whatever that means!

Note that the equations of QM have no scale dependence built into them and macroscopic objects, like cats, can actually be in a superposition state if the experiment is setup carefully. There is no cut-off boundary between quantum and classical behaviour and the oft-repeated phrase that QM is the theory of the very small is actually, manifestly, wrong! People making the statement confound our current technological inability with the laws of nature and that technological inability is an ever shrinking circle as this recent Stanford experiment represents.

So, what happens when we open the box? We either see the cat dead or alive. But how is our seeing the cat dead or alive different from the cat inhaling or not-inhaling poisonous gas (in a closed chamber). The answer according to QM is that it isn’t – human experimenter’s sight is no more special that the cat’s inhaling of gas. So, actually when we open the box, it is not that the experiment collapses with the dead or alive cat, but now involves the experimenter-who-saw-dead-cat and experimenter-who-saw-alive-cat simultaneously. In other words, the experimenter becomes a part of the goings-on of the experiment, and the information about the goings-on spreads luminally (at the speed of light).

So, in world 1 the experimenter saw the dead cat, wrote an obituary about the cat in the newspaper, which was read by other people across the world who mourned the cat; and in world 2 the experimenter saw the alive cat, wrote a triumphal piece about the cat’s indestructibility in the newspaper, which was read by cat lovers who loudly cheered – both happening simultaneously. We all become part of the experiment (i.e. entanglement) and the effect spreads out at light speed, in effect creating causally separate worlds of parallel goings-on. The Many Worlds!

The Many Worlds thus created quickly lose superposition, almost like heat loss, and are said to decohere from each other. The analogy with heat loss is quite apt and we can mathematically calculate how quickly this coherence loss (i.e. world-splitting) happens as the number of quantum particles in a system goes up. Once decoherent, the worlds are in physical sense causally separate, independent versions of reality. The probabilistic nature of QM that we seem to observe comes not from actual probabilities existing in nature, but an emergent effect of these many worlds – all of which are completely  deterministic as Einstein would have preferred. But what about the uncertainty we face in quantum experiments? After all we don’t know which slit we’ll measure the single photon at, in any given instance. Well, that uncertainty comes about because the experimenter does not know which instance of the many-experimenters (one in each of the many worlds) he/she is.

So, how do we resolve the double-slit result? Well, the photon we measure at the point of firing, at each slit and the one that hit the wall is not the same photon! Every time we measure the photon, we are splitting along with the experiment. So the trajectory we plot out of the photon from the point of firing to the point it lands on the wall (through either slit), is really an ensemble of instances of photons in the different many-worlds. When we do not measure, all these worlds (given by the wavefunction) literally exist and interact – just like how a large number of photons in one world exist – and we see the interference pattern as a result of a photon being guided by its equally real many-world counterparts.

The same is true of my Boron problem. The electron captured in the first snapshot is not the same as the one in the second, though they are functionally inter-changeable with each other (i.e. fungible). We always count 1 photon/electron in every world we inhabit, but each of those worlds are causally-separate parallel goings-on that effectively split at the point of measuring (or entanglement). The many-worlds in their entirety are purely deterministic and Einstein’s dice worry would no longer hold. The wave-function simply represents the relative magnitudes of some worlds vs others, i.e. some worlds are more numerous than others but they all happen if there’s a truly quantum experiment.

While MWI is certainly one of top 2 or 3 contenders of the “correct” picture of Quantum Mechanics, we should bear in mind that it is not the only one. There are also other – more fringe – interpretations I have not discussed here (like Quantum Bayesianism etc). That said, MWI is certainly the most parsimonious of all, in that it does not assume anything other than the ontology of the mathematical wavefunction – no pilot waves or stochastic collapse postulates. The Many Worlds are not a priori assumption, but an a posteriori result of following through the logic (in much the same way, existence of planets on other stars is a result of planetary formation theory even when we had not observed/detected any directly). At the risk of sounding like a cheap proselytizer (maybe I am), I should add that MWI is also the fastest growing of all interpretations of QM and more details on it can be found here and here 🙂

[*] Nothing to do with the movie of course, but is a reference to linear algebra (matrix transformations in Hilbert space) typically used to express the mathematics of QM.

[**] It is not surprising how much the language of Physics has German embedded in it. Germany (Prussia) was a scientific superpower of the world and the Prussian academy of sciences had some of the brightest minds with near total creative freedom. The Wars wrought on/by the German people were a veritable catastrophe for its Science, which modern Germany has still not recovered from.

PS: I take popularization of scientific ideas very seriously, so would much appreciate your feedback (especially negative).

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15 Replies to “Quantum Mechanics Simplified”

  1. I dont have phd in physics but have masters, It would be good to start off with linear states to begin with so that people know that measurement is xi*xi. That mathematical tip at the beginning will clear this up that inspite of the physical strangeness of it, the fact that basis states of various possibilities already exist as xi means that physicists got it covered with mathematics. Now the question is whether the math is totally true or not and requires modification. Next, hilbert space allows for infinite dimensions, but some new studies show that quantum mechanics seems to not exhibit itself in that way ( dont remember). Finally quantum decoherence . That measurement doesnt mean that things have collapsed, it just means quantum aspect of various states have been lost to the environment. Hence new ideas of Zurek is very important because the questions he asks might lead to new ground.

    Suggest everyone to read this article.

    https://aeon.co/essays/the-quantum-view-of-reality-might-not-be-so-weird-after-all

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    1. Thanks bharat. The idea of decoherence is a good one. Though I tangentially mention it, I realised I don’t do so explicitly. I have updated the text.

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      1. yes, there are criticisms of mwi. Most importantly that this would mean some timelines would be absurd. Also, that deriving probabilities I hear is an issue. back then i was wondering whether one could experiment with borns rule. See if there are contributions other than that from borns rule in 3 slit experiment. And it was done and they didnt see any other contribution.

        I was wondering about what was relational quantum mechanics,(independently). However, my math doesnt support me to follow through on things I wonder though.not so bright.

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        1. https://www.youtube.com/watch?v=3ZoeZ4Ozhb8

          sean carroll and rovelli.
          carlo rovelli whose ideas i stumbled on independently twice((ofcourse long after he made his contribution already) ,on time and relational qm, but ofcourse there is huuuuge difference between a kid who asks a question and a scientist who puts it all into mathematical form.

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  2. Good piece. Enjoyed reading! I myself adhere to the Copenhagen interpretation, and in my Weltanschauung, it is sufficient and more parsimonious than anything else. I can see how some might find it unsatisfying and welcome alternative interpretations. I recently read Adam Becker’s book from earlier this year that covers much of the same ground that you do here, although he makes some interesting attacks on Popperian falsifiability.

    Btw, in your view of EPR, is QM incomplete or is it non-local?

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    1. // in your view of EPR, is QM incomplete or is it non-local? //

      Neither. EPR is quite misunderstood. John Cramer and David Albert show that what EPR really implies is that every quantum interpretation/theory must either violate contra-factual definiteness (CFD) or locality – NOT completeness or locality. MWI is the only properly local interpretation of Quantum Mechanics. See more here.

      Also I’m sorry to say but you are mistaken about the parsimony of CI 🙂 At a minimum, for CI to hold it has to assume:
      a) Schroedinger’s wave equation, i.e. wave-function transport
      AND
      b) Stochastic collapse postulate

      CI needs (b) to explain how the wave-function converts to a classical single-world observation. MWI only ever needs (a).

      PS: Falsifiability is an epistemological *guidance* not a law of nature. A falsifiable theory is likelier to be true. Often times people who take pot-shots at falsifiability bring down a strawman version of it. Not read Becker so can’t tell if he’s doing it too, but I will read it (thanks!).

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  3. Excellent piece worth pondering slowly like a glass of prosecco. I did masters in the subject in 1974 and after that all the readings Feynman lectures, Schiff, Herman Weyl and scores of others have receded into the darkest places of my brain. Posts like this rekindle some of those passions. Slapstick is to be commented for making an abstruse subject into layman’s language without diluting it. Reminds me Physics for Fun by Ya.Perelman and other Soviet books.

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  4. MWI may solve the local problem of simultaneous existence of many photons. But I think it is just pushing the problem to the next level that there could be infinite worlds.
    It seems we accept the deterministic explanation by kind of agreeing that we can’t be measuring all the other worlds simultaneously and compile all of it from outside of this physical universe.

    I don’t see how this is any different from probabilistic interpretation because in both cases we can’t measure it with any confidence or have an accurate predictive model of the world. But that’s just practical interpretation of engineers anyway.😀

    I really liked the PhD comics presentation of quantum mechanics based on Caltech grad students explanations. It is great for visual learners. The question it raises is basically we might be seeing different projections of the same reality rather than reality itself being different.

    Thanks for the post. Finally something fun.

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  5. I have no background in this field and I found your write up to be quite understandable — even all the way to the end.

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  6. Thanks Slapstick, very readable and posted on my FB page (link below). Hopefully will get comments from non science types.

    You dragged info from the recesses of my mind. I last did some QM back in 1980 for a general science degree. I think we did p and (s?) orbitals at A/L chemistry too an obviously for basic bachelors.

    Nice to read an update on the theories to explain the experimental theories.

    Thanks again.

    https://www.facebook.com/permalink.php?story_fbid=489142201562925&id=100014014474339

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  7. Thanks for the kind words sbarrkum, Ikram, VijayVan, Zach, Violet 🙂
    I’m glad it reminded you of your physics education.

    @Violet
    Yes, you could think of QM in probabilistic / information-theoretic terms. In the end, these approaches are mathematically equivalent. However, in my humble opinion, MWI is the only true ontology – the shit that’s really happening out there…

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