It feels like cheating.

I continue to be amazed at the flimsiness of the heuristics that physicists use, often successfully, to make important theoretical progress. A particularly shocking example I’ve just read is Heisenberg’s “discovery” that systems with symmetric wavefunctions correspond to those that obey Bose-Einstein statistics, and that those with anti-symmetric wavefunctions correspond to those that obey Pauli’s exclusion principle. He does not refer to Fermi-Dirac statistics since this was before Dirac “discovered” them, and Fermi’s discovery was also published in German only later (although he had published it in Italian earlier).

Why it feels like cheating:

  1. The entire paper is based on the analysis of systems of coupled harmonic oscillators. He gives a quantum mechanical treatment of them that results in two groups of possible solutions, and shows that transitions between solutions can take place only between the members of each group. He then notes that we see only one of the two possible systems of ortho- and para-helium in nature, and suggests that this one-sidedness is due to the dichotomy he’d derived from his model. He then proceeds to generalise the dichotomy to all systems in nature:

    For the helium spectrum it is an empirical fact that only one system exists… that the other systems are not realised in nature. In fact it seems to me to indicate — if we assume, that the results we have derived for two systems can be generalised to arbitrarily many systems — on the one hand, an actual connection between the highlighted quantum mechanical indeterminacy [between which type of system exists], and on the other hand, the Pauli rule and the Einstein-Bose counting.

    (Pardon my amateur translation.) No justification for the generalisation exists in the paper.

  2. Shortly after he admits:

    Grounds that this is the only system, of all the possible quantum mechanical solutions, that occurs, will scarcely be derived from the simple quantum mechanical calculation.

  3. Despite all that, he still feels justified to extend the conclusions of his model into the realm of metaphysics:

    [The symmetric/anti-symmetric restrictions on wavefunctions] mean that it makes no physical sense to speak of the movement or the matrix representing the movement of an individual electron or of the matrix of any non-symmetric function of electrons in a system of atoms… Therefore e.g. the exchange relations in their familiar form also generally contain no physical sense

Of course, there is no question about the subsequent empirical success of relating anti-symmetric wavefunctions to Fermi-Dirac statistics and symmetric wavefunctions to Bose-Einstein statistics, but it remains amazing to me that such heuristics and casual generalizations as the ones Heisenberg uses are so successful. I do not think this is a one-off occurrence, either. In fact it seems to me that most theoretical development in physics proceeds this way, particularly the “revolutionary” developments.
Dirac, P. (1926). On the Theory of Quantum Mechanics Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character (1905-1934), 112 (762), 661-677 DOI: 10.1098/rspa.1926.0133
Fermi, E. (1926). Zur Quantelung des idealen einatomigen Gases Zeitschrift für Physik, 36 (11-12), 902-912 DOI: 10.1007/BF01400221
Heisenberg, W. (1926). Mehrkörperproblem und Resonanz in der Quantenmechanik Zeitschrift für Physik, 38 (6-7), 411-426 DOI: 10.1007/BF01397160

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3 Responses to It feels like cheating.

  1. wolfgang says:

    i think there are several factors.
    i) the toy model of the harmonic oscillator turned out to be a great tool in q.m.
    ii) there is some selection bias (we dont remember so well when they went wrong with their conclusions)
    iii) a lot of discussions in letters etc. which we do not see in their papers; the whole physics discussion (between Bohr, Heisenberg, Pauli, etc.) was not as fully referenced as it is nowadays.
    iv) but i also think the great physicists had a special feeling for what works (when they got older they often lost that special touch, e.g. Heisenberg with his ‘weltformel’)

  2. i) It turned out to be a great tool, but was it such a great tool by 1926 that Heisenberg was entitled to assume it was representative?
    ii) Interesting claim which demands more examination of papers around that time that made different generalisations that went wrong.
    iii) Fair enough.
    iv) merely pushes the explanatory burden back to how that ‘special feeling’ was acquired.

  3. […] It turns out that theoretical physicists can also get it wrong (though since it's theoretical, how can you know?). The Truth Makes me Fret discusses how Heisenberg wrote a paper based on…well…not much. […]

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