Dragons and Unicorns

When I was an undergraduate I was often told by lecturers that I should find quantum mechanics very difficult, because it is unlike the classical physics I had learned about up to that point. The difference – or so I was informed – was that classical systems were predictable, but quantum systems were not. For that reason the microscopic world could only be described in terms of probabilities. I was a bit confused by this, because I already knew that many classical systems were predictable in principle, but not really in practice. I blogged about this some time ago, in fact. It was only when I had studied theory for a long time – almost three years – that I realised what was the correct way to be confused about it. In short, quantum probability is a very strange kind of probability that displays many peculiarities and subtleties  that one doesn’t see in the kind of systems we normally think of as “random”, such as coin-tossing or roulette wheels.

To illustrate how curious the quantum universe is we have to look no further than the very basic level of quantum theory, as formulated by the founder of wave mechanics, Erwin Schrödinger. Schrödinger was born in 1887 into an affluent Austrian family made rich by a successful oilcloth business run by his father. He was educated at home by a private tutor before going to the University of Vienna where he obtained his doctorate in 1910. During the First World War he served in the artillery, but was posted to an isolated fort where he found lots of time to read about physics. After the end of hostilities he travelled around Europe and started a series of inspired papers on the subject now known as wave mechanics; his first work on this topic appeared in 1926. He succeeded Planck as Professor of Theoretical Physics in Berlin, but left for Oxford when Hitler took control of Germany in 1933. He left Oxford in 1936 to return to Austria but fled when the Nazis seized the country and he ended up in Dublin, at the Institute for Advanced Studies which was created especially for him by the Irish Taoiseach, Eamon de Valera. He remained there happily for 17 years before returning to his native land at the University of Vienna. Sadly, he became ill shortly after arriving there and died in 1961.

Schrödinger was a friendly and informal man who got on extremely well with colleagues and students alike. He was also a bit scruffy even to the extent that he sometimes had trouble getting into major scientific conferences, such as the Solvay conferences which are exclusively arranged for winners of the Nobel Prize. Physicists have never been noted for their sartorial elegance, but Schrödinger must have been an extreme case.

The theory of wave mechanics arose from work published in 1924 by de Broglie who had suggested that every particle has a wave somehow associated with it, and the overall behaviour of a system resulted from some combination of its particle-like and wave-like properties. What Schrödinger did was to write down an equation, involving a Hamiltonian describing particle motion of the form I have discussed before, but written in such a way as to resemble the equation used to describe wave phenomena throughout physics. The resulting mathematical form for a single particle is

i\hbar\frac{\partial \Psi}{\partial t} = \hat{H}\Psi = -\frac{\hbar^2}{2m}\nabla^2 \Psi + V\Psi,

in which the term \Psi  is called the wave-function of the particle. As usual, the Hamiltonian H consists of two parts: one describes the kinetic energy (the first term on the right hand side) and the second its potential energy represented by V. This equation – the Schrödinger equation – is one of the most important in all physics.

At the time Schrödinger was developing his theory of wave mechanics it had a rival, called matrix mechanics, developed by Werner Heisenberg and others. Paul Dirac later proved that wave mechanics and matrix mechanics were mathematically equivalent; these days physicists generally use whichever of these two approaches is most convenient for particular problems.

Schrödinger’s equation is important historically because it brought together lots of bits and pieces of ideas connected with quantum theory into a single coherent descriptive framework. For example, in 1911 Niels Bohr had begun looking at a simple theory for the hydrogen atom which involved a nucleus consisting of a positively charged proton with a negatively charged electron moving around it in a circular orbit. According to standard electromagnetic theory this picture has a flaw in it: the electron is accelerating and consequently should radiate energy. The orbit of the electron should therefore decay rather quickly.

Bohr hypothesized that special states of this system were actually stable; these states were ones in which the orbital angular momentum of the electron was an integer multiple of Planck’s constant. This simple idea endows the hydrogen atom with a discrete set of energy levels which, as Bohr showed in 1913, were consistent with the appearance of sharp lines in the spectrum of light emitted by hydrogen gas when it is excited by, for example, an electrical discharge. The calculated positions of these lines were in good agreement with measurements made by Rydberg so the Bohr theory was in good shape. But where did the quantised angular momentum come from?

The Schrödinger equation describes some form of wave; its solutions \Psi(\vec{x},t) are generally oscillating functions of position and time. If we want it to describe a stable state then we need to have something which does not vary with time, so we proceed by setting the left-hand-side of the equation to zero. The hydrogen atom is a bit like a solar system with only one planet going around a star so we have circular symmetry which simplifies things a lot. The solutions we get are waves, and the mathematical task is to find waves that fit along a circular orbit just like standing waves on a circular string. Immediately we see why the solution must be quantized. To exist on a circle the wave can’t just have any wavelength; it has to fit into the circumference of the circle in such a way that it winds up at the same value after a round trip. In Schrödinger’s theory the quantisation of orbits is not just an ad hoc assumption, it emerges naturally from the wave-like nature of the solutions to his equation.

The Schrödinger equation can be applied successfully to systems which are much more complicated than the hydrogen atom, such as complex atoms with many electrons orbiting the nucleus and interacting with each other. In this context, this description is the basis of most work in theoretical chemistry. But it also poses very deep conceptual challenges, chiefly about how the notion of a “particle” relates to the “wave” that somehow accompanies it.

To illustrate the riddle, consider a very simple experiment where particles of some type (say electrons, but it doesn’t really matter; similar experiments can be done with photons or other particles) emerge from the source on the left, pass through the slits in the middle and are detected in the screen at the right.

In a purely “particle” description we would think of the electrons as little billiard balls being fired from the source. Each one then travels along a well-defined path, somehow interacts with the screen and ends up in some position on the detector. On the other hand, in a “wave” description we would imagine a wave front emerging from the source, being diffracted by the screen and ending up as some kind of interference pattern at the detector. This is what we see with light, for example, in the phenomenon known as Young’s fringes.

In quantum theory we have to think of the system as being in some sense both a wave and a particle. This is forced on us by the fact that we actually observe a pattern of “fringes” at the detector, indicating wave-like interference, but we also can detect the arrival of individual electrons as little dots. Somehow the propensity of electrons to arrive in positions on the screen is controlled by an element of waviness, but they manage to retain some aspect of their particleness. Moreover, one can turn the source intensity down to a level where there is only every one electron in the experiment at any time. One sees the dots arrive one by one on the detector, but adding them up over a long time still yields a pattern of fringes.

Curiouser and curiouser, said Alice.

Eventually the community of physicists settled on a party line that most still stick to: that the wave-function controls the probability of finding an electron at some position when a measurement is made. In fact the mathematical description of wave phenomena favoured by physicists involves complex numbers, so at each point in space at time \Psi is a complex number of the form \Psi= a+ib, where i =\sqrt{-1}; the corresponding probability is given by |\Psi^2|=a^2+b^2. This protocol, however, forbids one to say anything about the state of the particle before it measured. It is delocalized, not being definitely located anywhere, but only possessing a probability to be any particular place within the apparatus. One can’t even say which of the two slits it passes through. Somehow, it manages to pass through both slits. Or at least some of its wave-function does.

I’m not going to into the various philosophical arguments about the interpretation of quantum probabilities here, but I will pass on an analogy that helped me come to grips with the idea that an electron can behave in some respects like a wave and in others like a particle. At first thought, this seems a troubling paradox but it only appears so if you insist that our theoretical ideas are literal representations of what happens in reality. I think it’s much more sensible to treat the mathematics as a kind of map or sketch that is useful for us to do find our way around nature rather than confusing it with nature itself. Neither particles nor waves really exist in the quantum world – they’re just abstractions we use to try to describe as much as we can of what is going on. The fact that it doesn’t work perfectly shouldn’t surprise us, as there are are undoubtedly more things in Heaven and Earth than are dreamt of in our philosophy.

Imagine a mediaeval traveller, the first from your town to go to Africa. On his journeys he sees a rhinoceros, a bizarre creature that is unlike anything he’s ever seen before. Later on, when he gets back, he tries to describe the animal to those at home who haven’t seen it.  He thinks very hard. Well, he says, it’s got a long horn on its head, like a unicorn, and it’s got thick leathery skin, like a dragon. Neither dragons nor unicorns exist in nature, but they’re abstractions that are quite useful in conveying something about what a rhinoceros is like.

It’s the same with electrons. Except they don’t have horns and leathery skin. Obviously.



19 Responses to “Dragons and Unicorns”

  1. i have to admit when i got a book to study quantum mechanics that the mathematics were basically impossible for me to follow. the wave function, gamma, looks only vaguely familiar. perhaps it is because of my present state of ignorance that i still perceive wave particle duality to be a paradox. it is unreasonable for me to comprehend of such a thing as a physical phenomenon that transforms between particle and wave, especially without actually seeing a working example. sure i can accept the double slit experiment, but what is actually going on? i don’t particularly know much about waves either, i got a degree in business technology, and dabbled a bit in computer science, but from what i understand about waves, is that they need a medium. every wave i can think of is energy moving through a medium.

    i don’t know, the more i think about it, the more it seems that space itself is a medium. from what i know about general relativity, space bends, but how can nothing bend?

    i of course have zero experimental evidence for these random thoughts, and no matter how hard i rack my brain, i can’t conceive of an experiment that would test any of these wacky ideas. and even if i could, i wouldn’t have the resources to implement. i mean, is it possible to build a particle decelerator?

    i guess the question does lie in whether or not language can utterly describe and explain reality. if it can, then i would very humbly suggest that these theories are more than just maps. they are more like a blueprint of how the whole universe works, or at least, i’m hoping that’s what they can be, because i don’t have much faith for humanity’s (or my) future prospects, otherwise.

  2. Anton Garrett Says:

    I’m just back from Lords. Can you bribe the particle to go through a particular slit?

  3. It helps if you think of the wave function as the “real” thing and the particle is just the way it interacts with the world. Then asking where the particle is when it isn’t observed is like asking where the wind is when it isn’t blowing. The very question is clearly nonsensical. Well you still have strange correlations to deal with and you should be careful how seriously you take your ontological commitments.

    In the end our monkey brains evolved in a classical world. It isn’t really a big surprise that we are poorly equipped to deal with things on the quantum level. Its kinda cool that things are so different.

    • telescoper Says:

      Actually, most of the confusions arising from quantum mechanics arise from assertions about what is “real”, whatever that means. Physicists are very likely to say foolish things when they start talking ontology.

  4. > Physicists are very likely to say foolish things when they start talking ontology.

    But, in my hubble opinion, not as foolish as philosophers when they go on about how physics works and what we do.

    • Anton Garrett Says:

      Well said Cusp! Popper’s doctrine of falsifiability is a perversion of the truth. Whoever believes that scientists find fulfilment by seeing their theories proved *wrong*? Rather, theories have to be testable – meaning that the probabilities we assign to them are capable of being changed by experimental data. In a test of Newtonian vs relativistic mechanics, the data haved driven the probability of the latter to 0.9999999999… and of the former to 0.00000000001, and we use the words True and False as a shorthand. But, someday, fresh anomalies might emerge and a new theory be found which supersedes Einstein in the same way. Popper could not put it like that, because he rejected inductive logic, and inductive logic IS probability theory provided that it is done correctly. Popper never understood that; he accepted probability but denied induction, and hid his misunderstanding beneath a mass of disingenuous rhetoric (expertly dissected by the philosopher David Stove). Hence his unhappy notion of falsifiability. Thomas Kuhn also rejected induction and concluded that ‘paradigms’ (such as Newtonian and Einsteinian) come and go as arbitrarily as fashions in clothing. The better fit of relativistic mechanics to the data ultimately means nothing to him (although again, he never admits it so starkly). He was a good historian of science but a lousy philosopher of it. Today this strand of thought has merged, via PK Feyerabend, with the wider postmodernist movement which denies that there is any such thing as truth. My response to that claim (which it is itself touted as true!!) is to point out that people clearly live their lives as if certain things are true.

  5. > in my hubble opinion

    Am currently sitting here fitting globular cluster profiles in ACS data – so a Freudian slip there on my part.

  6. Garret Cotter Says:

    Peter: I learned only recently that Young’s Slits has been demonstrated with Fullerene, which begged the question in a discussion with a friend, should one do a double-slit experiment with double-decker buses, how far away would the screen have to be to show fringes? That would make a good first-year undergraduate estimation question.

    Anton: I’m confused by your last argument. You seem to be responding to those of us who say “The map is not the land” by saying “Well some people say it is”? Do I misunderstand you? (And I remind you I’m a committed Bayesian!)

    • Anton Garrett Says:

      Hi Garret, you are using an analogy so I’m not sure exactly what part of my argument you are questioning – do say more – but if you mean my passing shot at postmodernism, then I mean that if you get to know somebody well you will find that they are passionately committed to certain deep ideals by which they live their life. This is so even if they are not intellectual enough to speak out those ideals in words. They live their lives on the basis that those ideals are true. I assert that postmodernists are no exception and that their claims that truth does not exist are at odds with the way they live their lives. Their life axioms might well be rather different from mine, but they *do* have them.

      Re your response to Peter, I think that the classical limit is still a live issue. More than one parameter is set to zero in the classical limit of quantum mechanics, and how the (conceptual) limiting process is done can be important, eg h –> 0 and a –> 0 (‘a’ is the size of the system) but does a/h remain constant or itself –> 0 or infinity? Then there is the issue of decoherence and how isolated from its environment the system is. Complex issues!


    • telescoper Says:

      I did some work a few years ago on trying to use Schrodinger’s equation to describe classical (compressible) fluids. It’s not as mad as it seems, because Madelung showed that by transforming variables you can write the Schrodinger equation as an Euler equation coupled to a Poisson equation for the potential. The fluid density turns out to be basically Psi^2, so if you do perturbation theory on Psi you will always get a positive density, which isn’t the case if you do perturbative calculations with rho directly. It’s fun, but has yet to catch on and change the world. I mention it here because how you take the classical limit in that case is indeed rather subtle.

    • Garret Cotter Says:

      Hi Anton,

      All I wanted to say, really, (and it’ll probably get lost in the heat and noise of the Hawking business) is that it’s possible to have a worldview without axioms, as we have touched on here before. Unless, perhaps, “We can never know the truth for sure” is counted as an axiom, but I think in formal logic it isn’t!

      I completely agree with your point that many people have implicit axioms that they don’t admit. And as a convinced subjective Bayesian I could never say to a theist such as yourself that they were wrong, simply that I find their worldview, personally, utterly improbable. I just “don’t get it”; which I think, to cross-reference to some of the followups to Peter’s blog entry on the Hawking debacle, is what Feynman’s attitude was.

      And I certainly won’t attempt to make ethical decisions based on models of basic physics; but I have to admit that I have to work them out myself, and _there_ one has to start thinking of axioms. But ethical axioms are not, I feel, welded into some “truth” of nature.

      Does that make me a some sort of post-modernist relativist, though? Well, if you choose to live by doubt, you have to spend a lot of time worrying about these things. I certainly do. And I definitely don’t think there should be a free-for-all on ethics. But do you think there are any ethical axioms constant in time and space?

      Peter: are you talking about taking probability current and making it compressible? Is there an elementary writeup anywhere? Looks quite neat.

    • Anton Garrett Says:

      Hi Garret; you say that it is possible to have a worldview without axioms, but you agree with me that many people have implicit axioms that they don’t admit.

      I don’t think it is possible to have a worldview without axioms. Secular people have axioms as much as theists, and they take those axioms from the prevailing culture. For that very reason their axioms are as hard to discern as glass immersed in water (having similar refractive index).

      To make the point: Can you give examples of public figures whom you believe lived their lives with explicit axioms; with implicit axioms; and with no axioms? The first category is easy, but how do you dsitinguish between the 2nd and 3rd categories?


    • telescoper Says:


      You can find an overview of this approach in a conference talk I gave some years ago:



    • Garret Cotter Says:

      Peter – thanks!

      Anton – from the point of view of ethics, I think we’re in agreement. But I originally intended to focus strictly on the physical world. To take your question, I would say that the “Feynman” view is in your third category; do you see hidden axioms in it, to make it actually in the second?

  7. Anton Garrett Says:

    Peter: Where the Navier-Stokes equations predict negative density they obviously fail, but surely that is a warning that you should correct them by imposing rho=0 by hand within the predicted negative-density region? To overcome the problem by changing equations to one which always has a smoothly varying rho seems to me to lose the capability to model important phenomena such as cavitation.

    • Yes indeed, but this was a specific fix for weakly non-linear perturbative calculations where it’s difficult to enforce positivity. It’s definitely the case that you can’t model strongly non-linear phenomena such as shocks using this approach. In fact it only works in the case I was interested in because the expanding background makes the growing mode of instability quite slow, so you can get a lot with 1st order perturbations.

  8. […] few days ago I posted what was intended to be a fun little item about the wave-particle duality in quantum mechanics. Basically, what I was trying to say is that […]

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