Wednesday, April 15, 2015

Max Born's Nobel lecture

I decided to read Max Born's 1954 Physics Nobel Prize lecture,
The statistical interpretation of quantum mechanics (PDF),
in some detail. Even though it was written and spoken more than 60 years ago, it makes such a perfect sense. Unsurprisingly, the lecture is a combination of physics and history. Let's look at those 12 pages.

We like to think about the Copenhagen school. Born makes it clear that Niels Bohr was a guru in his eyes. But he also prefers to talk about three schools, the Copenhagen school; the Göttingen school; and the Cambridge school. He was the boss of the Göttingen school in 1926-27; the Cambridge school had Dirac as its epicenter.

On the first page, he tells us he wants to describe how physics solved the "crisis". He also enumerates some important people who have always remained uncomfortable with the standard interpretation of quantum mechanics – including Planck, Einstein, de Broglie, and Schrödinger.

On the second page, we're reminded about Planck's 1900 calculation of the black body spectrum. In the early 1920s, it was already generally accepted that the electromagnetic energy at frequency \(f\) simply has to come in packets \(E=hf\): this assumption was able to explain so many things. The corpuscular theory of light was partially revived and people got used to some form of wave-particle duality although they didn't quite understand why these two aspects of light didn't contradict one another.

Also, in 1913, the 20-year-old Niels Bohr presented his model of the atom. The hydrogen atom had discrete states and light is emitted when a transition occurs. Born uses a rather modern quantum mechanical language to discuss Bohr's idea – but it's plausible that so did Bohr. Born already sketches matrices relatively to the energy eigenstate basis and suggests that Bohr was already thinking in similar ways in 1913. Also, Bohr was able to see that the principle of correspondence had to hold. In modern words, when the quantum numbers (integers) are much larger than one, physics must reduce to its classical limit.

On the third page, he sketches how people managed to reconcile the discreteness of the electromagnetic energy with the principle of correspondence. The number of photons at a given frequency had to be integer; however, when this integer was high, one should have gotten classical physics where light may be emitted at any intensity or frequency.

How can one describe emission – the process that becomes continuous and allows an arbitrary intensity in the classical limit – if the number of photons is discrete, \(n\in\ZZ\)? Born credits Bohr, Kramers, Sommerfeld, Epstein, and especially Einstein with the idea that the intensity has to be replaced by transition probabilities. The number \(n\in\ZZ\) cannot change continuously. But what can be continuous is the probability that it changes to \(n\pm 1\), among other things.

When non-orthodox "interpreters" of quantum mechanics (Bohmian mechanics, MWI, objective collapses etc.) try to claim that the proper formulation of quantum mechanics is unnecessary, they love to talk about one electron, a fermion that obeys the Pauli exclusion principle. And they design various tricks and extra stories that are meant to replace quantum mechanics.

It never works properly. But if they considered emission or absorption of radiation, it could have been much clearer to them that the intrinsically probabilistic description is absolutely unavoidable. It's unavoidable for a simple reason. The energy – of electromagnetic radiation or the atom – often has the discrete (or mixed) spectrum. So the change of the energy (like the emission or absorption of a photon by an atom) has to be a discrete operation, too. It either happens (one) or doesn't happen (zero) and because the underlying equations of physics are continuous, the only continuous relevant quantity that physics may calculate for this (or any) discrete process is the probability \(p\) that the process will take place!

In Göttingen, Born, Ladenburg, Kamer, Heisenberg, and Jordan did the hard work for the case of atoms. For the first time, it became clear that they needed to talk about the probability amplitude (which is a square root of the probability), and not the probability itself. There were lots of confusion about the right formulae for quite some time. They localized the correct one mainly because they were demanding the principle of correspondence – the correct classical limit – at every point.

Here I am praising lots of people, and not just Born, with the "probability amplitude". So why do we only celebrate Born for \(\rho(x)=\abs{\psi}^2\)? Well, the amplitudes in the previous paragraphs were only appreciated for photons' being there or not (intensity), and they were not functions of space. Even though the probabilistic description of "something" was already being adopted in physics, it wasn't universal yet.

A place where nobody dared to go, except for Heisenberg, as Max Born's granddaughter put it. I had to embed it here because they're just playing Xanadu on the radio now.

As discussed on the fourth page, in 1925, Heisenberg – despite his hay fever – did the most important step when he described observables such as \(x\) by "arrays" and replaced the multiplication of those observables by the matrix multiplication. Heisenberg had never heard of matrices before (although they had obviously existed in mathematics at that time) and it seems that no one else among these founders of quantum mechanics did, either. Well, Born was able to remember that he had heard about matrices somewhere at school. He was insanely excited about Heisenberg's ideas and sure that they were on the right track. And suddenly, Born could even see\[

pq-qp = \frac{h}{2\pi i},

\] if I use the original notation. Observables were non-commuting which is cool and deep. Heisenberg's first paper didn't have the complete quantum mechanics and lots of work must have been done – to prove that the off-diagonal entries drop from certain expressions but not others, and so on. Born together with his "pupil" Pascual Jordan, Born invited himself to become a collaborator of Heisenberg. ;-) Well, it was a remote collaboration because Heisenberg was out of town.

As the fifth page reminds us, they soon published the paper with three authors which had the full "matrix mechanics" in it. They already recognized that Heisenberg's "arrays" were mathematicians' "matrices". Before it got published, a paper by Dirac appeared and it got almost equivalent results, I think already in the bra-ket notation. It wasn't a complete coincidence that the papers appeared simultaneously: Dirac's work was inspired by a Cambridge talk by Heisenberg.

The initial paper of Heisenberg didn't have the final theory with all the right interpretations but it had the right "advice" what kind of mathematics should be used to describe observables. And I think – the history seems to confirm this opinion – that because of the existence of smart people with a good intuition, Heisenberg's initial paper already made the imminent discovery of the full quantum mechanics inevitable.

Shortly after the three-men paper, Wolfgang Pauli used quantum mechanics to calculate the spectrum of the hydrogen atom. From that moment, Born says, there could have been no doubts about the validity of matrix mechanics. I agree with that. It was a completely new formalism that was "deduced" by looking at totally different phenomena than the hydrogen atom. Nevertheless, it was able to calculate all the energy levels of the hydrogen atoms accurately – while the character of the energy eigenstates was highly inequivalent to Bohr's orbits. This agreement with the experiments really couldn't have been a coincidence.

Interpretational confusions remained after the three-men paper. To complicate things further, in a year, Schrödinger published his wave mechanics – a more complex (potential energy allowing etc.) variation of the ideas by de Broglie. Schrödinger himself proved the mathematical equivalence of wave mechanics to the old good matrix mechanics, which quickly ended the strange "schism" in early 1926.

Wave mechanics quickly became more popular than the Göttingen and Cambridge versions of quantum mechanics. However, to make things dirty, Schrödinger's insights came with lots of physical gibberish. He was convinced that one could keep determinism and that \(e\abs{\psi}^2\) was the electric charge density \(\rho\) that one should simply insert to the classical Maxwell's equations.

Born informs us on the sixth page that his Göttingen school already found those remarks "unacceptable". More precisely, they clearly contradicted some (already) well-established empirical knowledge. Particles could have already been counted and precisely located – which contradicted the continuous character of the charge that would follow from Schrödinger's philosophy.

On seventh page, Born says that Schrödinger's picture seemed to lead to a misleading interpretation for bound electrons, so Born himself tried to get the right interpretation by working on matrix mechanics. During a neat collaboration with Norbert Wiener at MIT, he was finally able to return to wave mechanics and combine some ideas about the wave function with Einstein's picture of transition probabilities for the photons – and to make the simple and bold claim that \(|\psi|^2\) is the probability density for an electron.

Science doesn't end with such statements. It's an intriguing hypothesis and most of the work is about proving it. One great argument, completed by Wentzel, reproduced Rutherford's scattering formula from Born's interpretation of the wave function. Our modern scattering theory and S-matrices came from those advances. However, Heisenberg's paper coining the uncertainty principle contributed more to this right interpretation of the wave function. On the eighth page, Born thanks a few early authors such as Faxén, Holtsmark, Bethe, and my granduncle Nevill Motl in England. ;-)

Together with Fock of Russia, Born also formulated the probabilistic interpretation of \(|c_n|^2\) in the case of the discrete spectrum, especially that of a harmonic oscillator. Dirac formulated the transition probabilities within the final quantum framework more elegantly.

At the end of the eighth page, Born asks once again why Einstein et al. couldn't embrace those advances. On the ninth page, he divides these psychological obstacles to the attachment to determinism; and to realism. The determinism is less serious, the ninth page explains, because a small inaccuracy has led to indeterminism in practice even in classical physics. It was always operationally nonsensical to ask whether the location \(x\) is equal to the transcendent number such as \(\pi\), for example. One can't recognize those. Even in classical mechanics, one needs to formulate things probabilistically if we use any real-world measurements etc. Positions are uncertain and have distributions etc.

So the determinism isn't terribly deep and "hard to abandon". It's the realism that is the source of more serious objections. What prevents us from measuring both position and momentum if we can measure them separately? Niels Bohr developed the theory of measurement especially as a sequence of replies to Einstein. The principle of complementarity became a part of it. Objects allow to be studied by many experiments, but two experiments often mutually exclude each other.

The last, eleventh page (if I don't count the references on the twelfth page) becomes rather philosophical – and interesting. It tells us that Bohr tried to extend his complementarity to very distant disciplines such as the "brain" and the "free will". Born also says things about physics itself. He wants to preserve the concept of a particle. Why do particles exhibit wave properties etc.? Born's favorite way of talking about these matters was the following one:
Every object that we perceive appears in innumerable aspects. The concept of the object is the invariant of all these aspects. From this point of view, the present universally used system of concepts in which particles and waves appear simultaneously, can be completely justified.
It's pretty wise. When we talk about particles, we are discussing some "aspect" of the objects. For those aspects, it's totally sensible to talk about "particles". But the objects also have other aspects that allow us to talk differently and there is no contradiction. To switch from particles to waves is a form of a transformation and the essence of the object is unchanged by this transformation.

In the last paragraph, he speculates that for further progress, we will have to eliminate some "concept of our doctrine" that is still being used although it's unjustified by experience. A modification of the mathematics – the Hilbert space and the Hamiltonian – won't be enough, Born says. He justified those speculations by particle physics and especially hadrons. In the mid 1950s, the zoo of the strongly interacting particles already began to emerge and Born's last paragraph is an example of the desire of the people at that time to make a "new revolution" similar to the quantum one in order to understand the hadrons.

As we know today, no conceptual revolution was needed to describe the interior of the protons etc. QCD – the first correct description of hadrons we had – is just a slightly more technical sibling of QED. In recent years, we also found the dual stringy/holographic description of these things which may perhaps be classified as more than an evolutionary advance. But I guess that he meant that some "complete new quantum-like revolution" would be absolutely necessary to describe the hadrons, and this wasn't the case.

The last paragraph seems to be the only one in which this 60-year-old lecture may be considered outdated, I think.

No comments:

Post a Comment