As we have seen in the last couple of posts, two very different mathematically based theories appeared almost at the same time to explain phenomena of the subatomic world. On one hand, quantum mechanics (as its creators Heisenberg, Born and Jordan referred to it) and, on the other hand, Schrödinger’s wave mechanics. In this entry we will tell a little more about both.
Although the first formulation of quantum mechanics was the matrix formulation, nowadays it is barely discussed. Why? There are two fundamental reasons, the first being that the underlying mathematics was tremendously complicated and unfamiliar to most physicists at the time. It is remarkable that Heisenberg, who was completely unfamiliar with this mathematical theory, was able to discover it. Max Born, then a professor at Göttingen, to whom Heisenberg was an assistant, had to be involved in its development. In the previous post we told how Heisenberg’s birth of quantum mechanics came about, so here we will tell you in a little more detail what happened over the following months.
As we have already described, on Heisenberg’s return from Heligoland and on his way to Cambridge, where he had received an invitation to give a lecture, he presented Born with his famous manuscript in early July 1925, explaining that he had not been able to go any further and, according to Born himself in his autobiography, suggesting that he should try to do so. When, after a few days, Born had read Heisenberg’s manuscript, he was fascinated by the idea that the theory should be based on explaining measurable physical quantities such as the amplitudes and frequencies of the spectral lines, rather than the position or velocity of the electron, which it was more than obvious to Heisenberg could not be observed in any experiment. He recounts this in his memoirs [1, p. 217]:
After I had submitted Heisenberg’s paper to the Zeitschrift für Physik for publication, I began to reflect on his symbolic multiplication, and soon became so involved in it that I thought about it all day long and could hardly sleep at night, because I felt that there was something essential behind it, the crowning achievement of our efforts of many years. And one morning, around 10 July 1925, I suddenly saw the light: Heisenberg’s symbolic multiplication was nothing but the matrix calculus, well known to me from my student days thanks to Rosanes’ lectures in Wroclaw.
In fact, during his university studies at the University of Wroclaw, Born was taught in 1900 by Jakob Rosanes, who introduced him to the theory of finite-dimensional matrices as an essential tool of analytical geometry. In 1904 he moved to Göttingen, where he attended Hilbert’s and Minkowski’s lessons. As a curious anecdote, Hilbert was so impressed by Born’s mathematical skills that he took him on as his personal assistant. This position, although unpaid, was fortunate for the young Born, as it enabled him to attend all of Hilbert’s lectures and then transcribe them. In other words, Born imbibed all the mathematical analysis that was being developed in Göttingen and which, in the end, turned out to be the mathematical tool of quantum mechanics.
With these ideas (matrix algebra) in his head on 19 July 1925, Born left for Hannover for a meeting of the Deutsche Physikalische Gesellschaft (German Physical Society), where he met his former assistant and collaborator Wolfgang Pauli, to whom he proposed to develop Heisenberg’s theory with him. But what he received was a cold and sarcastic refusal: “Yes, I know you like tedious and complicated formalisms. You will only spoil Heisenberg’s physical ideas with your useless mathematics”.
Luckily for everyone, Born decided to ignore Pauli and together with his other assistant Pascual Jordan, who did join the project, developed Heisenberg’s idea using matrix calculus and together they wrote a paper on quantum mechanics, Zur Quantenmechanik, which was received in Zeitschrift für Physik at the end of September 1925. This second paper was essential, as it gave a mathematical formalism to Heisenberg’s preliminary ideas, providing them with the formal framework on which to work. It was in this paper that one of the most famous expressions of quantum mechanics appeared for the first time: the commutation relations between the “variables” position and momentum.
$$p\, q – q\, p= \frac{h}{2\pi i}I,$$
where \(p\) and \(q\) were infinite matrices representing the position and momentum amplitudes of the subatomic particles and \(I\) was the matrix with ones on its diagonal and zero elsewhere. It contained Planck’s constant \(h\) and the imaginary unit \(i\). This relation would be the key to everything. The whole of quantum theory was condensed in it, and it even contained Heisenberg’s famous uncertainty principle, to which we will devote a future article. Born himself, many years later, acknowledged that the discovery of this relation was the climax of his scientific career (it appears in his epitaph) stating:
I was moved by this result in the same way that a sailor, after a long and wandering journey, is moved by the sight of the land he was looking for; my only regret was that Heisenberg was not there.
Heisenberg learned of his colleagues’ results during his holidays and asked Jordan for the details so that he could catch up and not repeat the same calculations as them, and set to work with the new matrix formalism in September while in Copenhagen, a formalism that was unknown to him at the time, as he told Jagdish Mehra years later [2]:
I didn’t know that those things I had used in my article were matrices. I had never taken a course on matrix theory. Of course, I knew how to solve linear equations in a trivial way as you learn in school, but I had never learned the general scheme of matrices and I didn’t know that matrices can represent groups. So when I learned from Born that what I had done was really an example of matrix multiplication, I was very interested, but it was new to me. […] In my paper, the fact that the product XY was not equal to YX was very unpleasant to me. I felt that this was the only point of difficulty in the whole scheme, otherwise I would be perfectly happy. But this difficulty had troubled me and I was not able to solve it.
This difficulty was solved together by Born, Heisenberg and Jordan in their only joint paper on quantum mechanics (or matrix mechanics, as the rest of the physicists called it). In that paper, known in the physics world as the Dreimännerarbeit, (three men’s paper) is the complete development of matrix mechanics. If the reader masters modern analysis, he will be astonished to discover how Born and his two assistants use in 1925 Hilbert’s results on what we now call Hilbert spaces, results discovered and published by Hilbert and his disciples in the previous 15 years. In particular, they realise that they are dealing with infinite-dimensional spaces and the associated problems, that the quadratic forms they use are not bounded, that the matrices (or their corresponding operators) have a continuous spectrum and not just a discrete one (as is the case for finite-dimensional matrices), and so on. In other words, they were aware of the mathematical problems behind their formalism.
To give the reader an idea of what we are talking about, we will show the example of the spectrum. Let us take a matrix \(A\) of size \(3 \times 3\) (in general \(N \times N\)). Its spectrum is the set of \(\lambda\) such that
$$ A \cdot x = \lambda x,\quad x\neq 0, $$
where \(x\) is a vector \(x=(x_1,x_2,\dots x_n)\) (remember that we are using matrix multiplication). It is a well-known result of linear algebra that there exist exactly \(N\) (complex, in general) numbers \(\lambda_ 1,\lambda_ 2,\dots,\lambda_N\) (not necessarily distinct) such that they satisfy the above equation. In the present case of quantum mechanics, the \(A\) matrices are symmetric (in general hermitic), so these \(\lambda\) are real numbers. For example, for the matrix
$$A=\begin{pmatrix}
3&1&0\cr
1&3&2\cr
0&2&3\cr
\end{pmatrix},
\quad
\lambda_1=3, \quad
\lambda_2=3-\sqrt5,\quad
\lambda_3=3+\sqrt5.
$$
The \(\lambda\) values are called the eigenvalues of \(A\) and the corresponding \(x\) vectors are the eigenvectors. However, if the dimension is infinite, the situation is much more complex and one must speak of the spectrum of \(A\) instead of the eigenvalues. For example, the spectrum of the infinite matrix
$$ \begin{pmatrix}
3 & 1 & 0 & 0 & 0 & \dots \cr
1 & 3 & 2 & 0 & 0 & \dots \cr
0 & 2 & 3 & 1 & 0 & \dots \cr
0 & 0 & 1 & 3 & 2 & \dots \cr
0 & 0 & 0 & 2 & 3 & \dots \cr
\vdots & \vdots & \vdots & \vdots & \vdots & \ddots\cr
\end{pmatrix}$$
consists of a discrete “eigenvalue” \(\lambda_1=3\) and an infinite set of \(\lambda\); those belonging to the interval \([0,2]\cup[4,6]\) (continuous spectrum).
Thus Born and his colleagues were aware that dealing with the infinite case required some care. The remarkable twist is that the mathematical theory was developed for \(A\) matrices corresponding to the case of bounded quadratic forms, but Born was aware that this was not his case. In this respect it is worth quoting the sentence included in that article in a footnote to chapter 3 (a chapter entirely written by Max Born): “But here we are only interested in non bounded forms. However, we can assume that, in general, the rules are executed in the same way”. This has been, in many cases, the usual way of working of many theoretical physicists (which in many cases is still maintained today), to assume that what works for certain cases will work in general, something that is not necessarily true (in fact it is often false). Fortunately for everyone, the results assumed by Born turned out to be true, although they required a significant mathematical effort. Among those who solved many of the mathematical problems of the nascent quantum theory was John von Neumann, another brilliant mathematician who spent time in Göttingen. But let Norman Macrae, who wrote a superb biography of John von Neumann, tell the story:
In Johnny’s first few weeks in Göttingen in 1926, Heisenberg gave a lecture on the difference between his theory and Schrödinger’s. The elder Hilbert, a professor of mathematics, asked his assistant in physics, Lothar Nordheim, what the hell this young Heisenberg was talking about. The elder Hilbert, a professor of mathematics, asked his physics assistant, Lothar Nordheim, what on earth the young Heisenberg was talking about. Nordheim sent the professor a paper that Hilbert did not understand. To quote Nordheim himself, as recorded in Heims’ book: “When von Neumann saw [Heisenberg’s] paper, he transformed it in a few days into an elegant axiomatic form to Hilbert’s liking”. To Hilbert’s delight, Johnny’s mathematical exposition made much use of Hilbert’s own concept of Hilbert space.
The ignorance of the physicists of the time about matrix techniques is evident in the letter Einstein wrote to his friend Paul Ehrenfest on 20 November 1925:
Heisenberg has laid a big quantum egg. In Göttingen they believe so (I don’t). “Heisenberg hat ein grosses Quantenei gelegt. In Göttingen glauben sie daran (ich nicht)”.
Pauli himself wrote to Ralph Kronig in October 1925:
Heisenberg’s mechanics has once again given me hope and joie de vivre. It has not yet provided the solution to the enigma, but I believe that, once again, it is possible to move forward. First we must succeed in freeing Heisenberg’s mechanics from the formal scholarship of Göttingen [i.e. Max Born] and better expose its physical essence.
Surprisingly, Heisenberg, who had learned of Pauli’s opinion from Kronig himself, responded angrily to him – a rare occurrence for the young Heisenberg – on 12 October 1925:
With regard to your last two letters, I must give you a speech and ask your pardon for continuing in Bavarian: it really sucks that you can’t help but get into a smear fight. Your eternal insult to Copenhagen and Göttingen is an outrageous scandal. You will have to concede to us in any case that we do not intend to ruin physics by malice. When you blame us for being such big jackasses that we have never produced anything new in physics, that may well be true. But of course, you are also an equally big jackass because you haven’t succeeded either… (The dots denote a curse of about two minutes’ duration!) Don’t think badly [of me] and many regards.
Incidentally, on 23 October 1925, Heisenberg apologised to Pauli for his angry reply, excusing himself for having spent three days trying to understand the problem of the continuous spectrum that Born had explained without understanding anything (in other words, he was very stressed). The consequence of the letter of 12 October was soon to follow. Pauli, who had exceptional mathematical skills (and no doubt hurt in his self-esteem), took three weeks to send Hesienberg the solution to the problem of the spectrum of the hydrogen atom, that is, the explanation using the new quantum mechanics of Bohr’s atomic theory. The new mechanics had just passed its first big test.
But the mathematics of quantum mechanics remained obscure to most physicists, so when Schrödinger published his wave mechanics, they breathed a sigh of relief. Einstein was full of praise for Schrödinger, and most physicists were delighted. The scientific community was divided between the advocates of one or the other. We will not repeat here some of the juicy anecdotes (not to say fights) between the advocates of one formulation or the other. What we will do is mention that, curiously, Schrödinger published an article “proving” the equivalence of both formalisms entitled Über das Verhältnis der Heisenberg-Born-Jordanschen Quantenmechanik zu der meinem (On the relation between Heisenberg-Born-Jordan quantum mechanics and mine) published on 4 May 1926 in Annalen der Physik (also Jordan and Dirac would “prove” the equivalence). In it he wrote:
I am not aware of any relation [of wave mechanics] to Heisenberg’s matrix mechanics. Naturally, I knew about his theory, but I was discouraged, if not repulsed, by what seemed to me very difficult methods of transcendental algebra, defying any visualisation.
Heisenberg himself was very unhappy about this and confided to Pauli in a letter of 8 June 1926:
One more unofficial comment on physics, by the way: the more I think about the physical part of Schrödinger’s theory, the more I find it appalling. What Schrödinger writes about the visualisation of his theory is “probably not quite right” [an echo of Bohr], in other words, it’s bullshit. […] But excuse this heresy and don’t tell anyone about it.
It must be said that the proofs of Schrödinger, Jordan and Dirac were not mathematically rigorous enough and it was again von Neumann who solved the problem in a series of papers between 1927 and 1929, culminating in the aforementioned “Mathematische Grundlagen der Quantenmechanik” (Mathematical Foundations of Quantum Mechanics) first published in 1932 and translated from German into English in 1955. The explanation is too technical to be included here, as it requires refined theorems of functional analysis in Hilbert spaces. The interested reader is referred to the excellent paper [3].
Quarrels aside, both Heisenberg and Schrödinger received the Nobel Prize for their contributions to quantum mechanics. Heisenberg in 1932 and Schrödinger together with Dirac in 1933. Max Born had to wait until 1954, which annoyed him a lot (he always complained that only Heisenberg’s work was recognised and not his work with Jordan). Pauli received it in 1945. Jordan was never awarded for his work (it is speculated that this was due to his close association with the Nazi party, as he joined the party in 1933 and was even a member of Hitler’s SA, although he was never part of the Deutsche Physik movement). Max Born was on the fisrt victims of the Nazi persecution of the Jews and had to escape Germany in 1933 when he read in the newspaper that he was going to be fired from the University. From his exile, Born helped many Jewish colleagues to escape Germany.
As we have seen in this post, mathematics proved essential in unravelling the mysteries of the quantum world. Knowledge of the appropriate mathematical techniques (matrix theory for the Göttingen physicists or the theory of differential equations and boundary problems in the case of Schrödinger – the case of Dirac deserves a separate chapter) was essential to be able to construct a complete theory of the microworld. However, as many physicists realised (among them the renowned Einstein or Sommerfeld to name some), there was still much to be understood from the physical point of view in both cases. There was the problem of the “visualisation” of which Schrödinger spoke and which so annoyed Heisenberg, and, of course, the aforementioned problem of what this Schrödinger wave function was, to which was also added the interpretation of the \(x\) eigenvectors of the Heisenberg matrices. But that was about to be solved by… Max Born, who by using both approaches together (the matrix and the wave) was able to give an explanation which, together with Heisenberg’s famous uncertainty principle, changed not only the way of looking at the World (nature) but philosophy itself. But that, my reader friend, will have to wait for another post.
I want to end this one with a statement of principle by Niels Bohr on what he expected about the relationship between physics and mathematics, the timeliness of which is more than evident, and which appeared at the end of his article published in Nature, vol. 116, pp. 845-852 (1925) under the title Atomic Theory and Mechanics:
It will be of interest for mathematical communities to know that the mathematical tools created by higher algebra play an essential role in the rational formulation of the new quantum mechanics. Thus, the general proofs of the conservation theorems of Heisenberg’s theory by Born and Jordan are based on the use of matrix theory, which goes back to Cayley and was developed by Hermite. It is to be hoped that a new era of mutual stimulation between mechanics and mathematics has begun. To the physicist it will seem, in the first place, regrettable that in atomic problems we have apparently encountered such a limitation of our usual means of visualisation. This regret, however, will have to give way to gratitude that mathematics, also in this field, gives us the tools to pave the way for further progress.
In other words, it is much more interesting when Physics and Mathematics go together rather than separately, but we already talked about that here and here.
References
[1] Max Born, My life. Recollections of a Nobel laureate. Taylor & Francis.
[2] Jagdish Mehra, Helmut Rechenberg-The Historical Development of Quantum Theory. Vol. 3 The Formulation of Matrix Mechanics and Its Modifications, 1925-1926, Springer, 1982.
[3] Carlos M. Madrid Casado, De la equivalencia matemática entre la Mecánica Matricial y la Mecánica Ondulatoria, Gaceta de la Real Sociedad Matematica Española, Vol. 10, Nº 1, 2007, págs. 103-128.
You can read more about Heisenberg and the early days of quantum mechanics in
D.C. Cassidy, Beyond Uncertainty. Heisenberg, Quantum Physics, and The Bomb. Bellevue Literary Press, 2009.
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