We told in the previous post that on 27 January 1926 the article “Quantisierung als Eigenwertproblem” (Quantization as an eigenvalue problem), written by Erwin Schrödinger, was received in the editorial office of the Annalen der Physik, where the famous equation that today bears his name and which laid the foundations of what we know today as quantum wave mechanics appeared. Schrödinger’s work appealed to many physicists, because as a result, in principle, many of the unknowns of the newborn quantum theory could be explained. But, to be honest, what was most pleasing was that the mathematics behind Schrödinger’s theory was comprehensible to the vast majority of theoretical physicists at the time. Six months earlier, on 29 July 1925, the prestigious journal Zeitschrift für Physik had received an article signed by a young physicist trained in Munich and Göttingen named Werner Heisenberg with the strange title “Über die quantentheoretische Umdeutung kinematischer und mechanischer Beziehungen” (On a quantum theoretical reinterpretation of kinematic and mechanical relations). It was the first paper of what we now call Quantum Mechanics, but, unlike Schrödinger’s wave quantum mechanics, Heisenberg’s mechanics used a much stranger tool for physicists: the matrix calculus. We will deal with this matrix quantum mechanics in this entry, but first we must introduce our main protagonist: the German physicist Werner Heisenberg.
Heisenberg was born on 5 December 1901 in Würzburg, Bavaria and was the younger of two brothers. His father August was a teacher of classical languages at a high school (Gymnasium) and soon after at the very University of Würzburg. It seems that his father was an extremely authoritarian figure, very stern and rigid with his two sons, from whom, among other things, he demanded high academic standards and strict discipline. August was also extremely competitive, a competitiveness that he forced his two sons to practice (in lessons, sports, etc.) to such an extent that the brothers hardly spoke to each other (in fact, they practically stopped seeing each other during their entire adult life). As a curiosity, Erwin, Werner’s older brother, used to win at everything except mathematics, and this made Heisenberg concentrate especially on mathematics, something that, as will be seen, would serve him well in later years.
World War I caught Werner in Munich (they had been living there since 1910 when his father was appointed professor at the University of Munich) studying at the Gymnasium, but as it was occupied by German army troops, the classes were dispersed, which helped Werner to focus on physics and mathematics, while still participating in voluntary activities such as working on farms, among others. In 1919, his last year at the Gymnasium, he joined the Youth Movement (a kind of Boy Scouts), becoming the leader of one of its groups. Until 1933, when these groups were suppressed, Heisenberg spent much of his free time with his group on camping trips, mountain excursions, etc., a hobby that remained with him throughout his life.
After the war he entered the University of Munich. His original intention was to study mathematics with Lindemann, a celebrity in number theory, but the latter did not accept him as a student (there are several stories about this, the most likely being that Lindemann was not interested in having a first-year student and had only agreed to interview Werner because he was friend of his father). In this respect it is amusing to read what Heisenberg himself wrote in his memoirs [1].
I visited Lindemann, who also worked in the university administration, on the first floor of the university building, in a dark office furnished in a strangely old-fashioned way, which, because of the rigidity of its installation, provoked in me a feeling of a certain withdrawal. Before I spoke to the professor, who rose only slowly, I noticed a little black-haired dog crouching on the desk next to him, which in these surroundings immediately reminded me of the dog in Faust’s laboratory. The dark critter looked at me with hostility, considering me, no doubt, as an intruder ready to disturb his master’s peace. Somewhat confused by this, I stated my business stammeringly, and, just as I spoke, I realised the lack of modesty in my plea. Lindemann, a white-bearded old man, already looking rather tired, no doubt also noticed this immodesty, and the slight excitement which overcame him may have been the cause of the little dog’s sudden and terrible barking on the desk. In vain did his master try to calm him. The little animal translated his anger towards me into furious barking, which he repeated in successive waves, so that it became increasingly difficult to understand each other. Lindemann asked me what books I had studied lately. I quoted Weyl’s Space-time-matter. Amidst the uncontained fury of the little black guardian, Lindemann ended the conversation, saying: “Then you are, in any case, already lost for mathematics”. With that I was dismissed.
Luckily for everyone, Heisenberg ended up under the tutelage of one of the great physicist-mathematicians of the time, Arnold Sommerfeld. Thus, in October 1920, he began his studies in theoretical physics, but without neglecting mathematics (in case things went wrong). There he met and befriended another of Sommerfeld’s disciples, Wolfgang Pauli, who led him irremediably into the world of quantum physics. In 1922 he attended Bohr’s famous course in Göttingen, where he met another of the great physicists of the time, Max Born. As a result, during Sommerfeld’s stay in the United States, Heisenberg accepted Born’s invitation to go and work with his team in Göttingen. That trip undoubtedly changed his life in every way, for there he devoted himself to research in the still “mysterious” quantum physics. He returned to Munich to defend his doctoral thesis on fluid mechanics (turbulence problems). The defence of his thesis was undoubtedly one of his first frustrations. It was obligatory for the doctoral candidate, after explaining his original contributions, to face an oral examination by the leading professors, one of whom was Wilhelm Wien, one of the most important experimental physicists, who to Heisenberg’s disgrace was opposed to quantum theory and therefore disdainful of Sommerfeld’s work in this area. That Werner was so openly dismissive of experimental physics did not help, so after a disastrous exam (Heisenberg was unable to answer almost any of the questions posed by Wein) he finally passed by the skin of his teeth, as Wein passed with the minimum grade (pass, I), which after Sommerfeld’s pressure remained at an average of cum laude (II), far from the maximum summa cum laude (IV). This was a humiliation for someone like Heisenberg, but, as we will tell on another occasion, he learned his lesson, as he studied Wein’s questions, which later helped him to prepare a mental experiment that led him to the famous uncertainty principle that today bears his name.
After obtaining his doctorate, Heisenberg left Munich and moved to Göttingen as an assistant to Max Born, who had no hesitation in accepting him (arguing that Wein had been a bit too high in his grade) and devoted himself entirely to research in the newborn atomic physics. At that time, the Dane Niels Bohr, together with his assistants Kramers and Slater, was trying to resolve the latest contradictions that had arisen when studying the interaction of light (electromagnetic waves) with atoms. The BKS theory (after the initials of its authors) was a resounding failure and showed Bohr’s unkind side (he almost lost the friendship of more than one of his former colleagues because of the totally irrational way in which he defended this theory, which called into question causality and the conservation of energy, the sacred pillars of physics), but it gave Heisenberg an idea that would eventually lead to the discovery (or invention) of matrix mechanics: it was not important how the electrons moved in the atom (atomic structure), but what could be observed about them, which in this case, was the frequencies and intensity of the spectral lines. He then recalled (or, if you prefer, rescued) Fourier’s theory. He himself recalled it years later: “The idea itself suggested that one should write the mechanical laws, not as equations for the positions and velocities of the electrons, but as equations for the frequencies and amplitudes of their Fourier expansion”.
And he set to work. It is better to hear it from David Lindley [2].
It was not hard for Heisenberg to write down, in a formal mathematical way, equations expressing the position and velocity of an electron as combinations of the fundamental oscillations of an atom. But when he inserted these composite expressions into standard equations of mechanics, what he created was tremendous confusion. Individual numbers became lists of numbers; simple algebra exploded into pages of repetitive and confusing formulae. For weeks, Heisenberg tried different calculations, played algebraic games with the Fourier series, fought futilely against the confusion, and then stopped when a monstrous outbreak of hay fever clogged his brain.
To recover, on 7 June 1925 he went to the island of Helgoland, a barely populated island where not a single plant grew. There he stayed in the middle of nowhere with his accounts, his physics books and some books by Goethe. Every day he went climbing, took long walks, and read Goethe, until his brain cleared. Heisenberg’s problem was that the frequencies and intensities were not simple numbers, but tables of numbers, tables which in mathematics are called matrices and which, believe it or not, Heisenberg had no idea existed. The problem Heisenberg faced was very simple: if we have two numbers \(a\) and \(b\), the multiplication of both is the number \(a\cdot b\), but what if what we have is a table of numbers?
Let’s take an example:
Let’s suppose we have the square tables (matrices).
$$A=\left(\begin{matrix}
a_1 & b_1 \\
c_1 & d_1
\end{matrix}\right)\quad\mbox{y}\quad
B=\left(\begin{matrix}
a_2 & b_2 \\
c_2 & d_2
\end{matrix}\right),$$
What is the value of the product of both? It could be, for example,
$$\left(\begin{matrix}
a_1 & b_1 \\
c_1 & d_1
\end{matrix}\right) \cdot
\left(\begin{matrix}
a_2 & b_2 \\
c_2 & d_2
\end{matrix}\right)=\left(\begin{matrix}
a_1 a_2& b_1 b_2 \\
c_1 c_2& d_1 d_2
\end{matrix}\right),$$
or else
$$\left(\begin{matrix}
a_1 & b_1 \\
c_1 & d_1
\end{matrix}\right) \cdot
\left(\begin{matrix}
a_2 & b_2 \\
c_2 & d_2
\end{matrix}\right)=\left(\begin{matrix} b_{1}\,
c_{2}+a_{1}\,a_{2}&b_{1}\,d_{2}+a_{1}\,b_{2}\cr c_{2}\,d_{1}+a_{2}\,
c_{1}&d_{1}\,d_{2}+b_{2}\,c_{1} \end{matrix}\right),$$
to give two examples from among the many options we could come up with. Heisenberg’s problem was that he could not find a way of defining multiplication that would lead him to something that made physical sense. Interestingly, it was physical reasoning (and not mathematics) that led him to discover that the latter of the two ways shown above of multiplying matrices was the one that made sense (interestingly, that is the “good” definition of multiplying matrices from a mathematical point of view). What happened next is recounted by Heisenberg himself in his memoirs [1]:
A few days were enough to get rid of the mathematical ballast that always appears at the start in such cases and to find a simple mathematical formulation for my problem. A few days later it became increasingly clear to me that in such a physics, instead of the quantum conditions of Bohr and Sommerfeld, only the observable quantities were to play their role. I could clearly see that with this additional condition the central point of the theory had already been formulated, and that from this point on there was no room for further freedom. But I realised that there was no guarantee that the mathematical framework thus created could in any way be applied without contradiction. In particular, it was completely uncertain whether the principle of the conservation of energy was still valid in this scheme, and it could not be hidden from me that, without the principle of energy, the whole system would become worthless. On the other hand, there were many signs in my calculations that the mathematics I had outlined could really be developed without contradiction and in a coherent way if the energy principle could be demonstrated in it. Accordingly, my work concentrated more and more on the question of the validity of the energy principle, and one night I got so far ahead that I was able to determine every single term in the energy table […] When I saw that in the first terms the energy principle was really confirmed, I fell into a kind of excitement, which made me make mistakes in all the following calculations. It was three o’clock in the morning when the final result of the calculation was complete before my eyes. The validity of the energy principle had been demonstrated in all its terms, and, as this result had offered itself – without any violence – I could no longer have any doubts about the lack of mathematical contradictions or about the complete unity of quantum mechanics hinted at here. At the first moment I was deeply shocked. I had the feeling that through the surface of atomic phenomena I was looking into an underlying background of fascinating inner beauty, and I almost lost my senses at the thought that I now had to go after this multitude of mathematical structures that nature had opened up before me. I was so impressed that I could not fall asleep. So at the first light of dawn I left the house and made my way to the southern tip of the plateau, where a solitary tower-like rock jutting out into the sea had awakened in me the desire to climb it. I managed to climb the tower without much difficulty and sat on top of it waiting for the sun to rise.
As David Lindley nicely puts it, “It was with delight and bewilderment that he discovered that his strange mathematics did in fact produce a consistent result for the energy of a system, but only so long as that energy belonged to a restricted set of values. His new form of mechanics was, in fact, a quantized form of mechanics.” Indeed, up to that point, every attempt to construct quantum physics had to be connected in one way or another with Plack’s quantization rule, but Heisenberg had not done that, he had made two physical systems quantize themselves, without any external help, and one of the two systems was none other than one of the paradigmatic systems of physics: the harmonic oscillator. In fact, Heisenberg obtained that the energy of the oscillator would have to be given by the expression
$$E_n= \frac{h\omega_0}{2\pi}\left( n+\frac12\right), \quad n=0,1,2,\ldots$$
The only concern Heisenberg had was that his multiplication rule did not respect the order, i.e., \(A B \neq B A\), the product was not commutative.
On his return to Göttingen, Heisenberg passed through Hamburg, told his friend Pauli what he had found, and Pauli urged him to write it down as soon as possible. When he had done so, he showed it to Born, who unhesitatingly sent it for publication to the Zeitschrift für Physik, where it was received on 29 July 1925. But Born guessed something hidden in Heisenberg’s reasoning. On 19 July Born left for Hannover for a meeting of the Deutsche Physikalische Gesellschaft (German Physical Society) and on the way he realised that Heisenberg’s strange multiplication law was, in fact, nothing more than the law of multiplication of matrices (the second of the multiplication formulae shown above) and that if he denoted by \(p\) the matrix of momentum and by \(q\) that of position then what he got was a somewhat strange formula
$$p\, q – q\, p= \frac{h}{2\pi i}I,$$
where \(h\) was Planck’s constant, \(i=\sqrt{-1}\) the imaginary unit and \(I\) the unit matrix (all zeros except the diagonal elements, which are ones). This relationship, which played a tremendously important role in the development of quantum mechanics, was the key that allowed Born, along with his other student Jordan (Pauli, who had been invited by Born to collaborate, turned away from the problem), to decipher Heisenberg’s enigmatic idea. The paper was not long in coming, and was in turn sent for publication to Zeitschrift für Physik, where it was received on 27 September 1925, under the title “Zur Quantenmechanik” (On Quantum Mechanics).
On 28 July 1925, Heisenberg presented his results at the Kapitza club in Cambridge, where a young engineer (yes, my dear reader, engineer) was studying. After graduating, he decided to become a mathematician (no, not a physicist), finishing his degree in mathematics in 1923 and coming first in his class, which opened the doors to Cambridge. After graduating, Dirac tried to work in the area of general relativity with E. Cunningham, but when Cunningham turned him down, he ended up working with the physicist and astronomer R. Fowler, who was very interested in the nascent quantum theory. Although Dirac did not hear Heisenberg’s lecture and was not aware of the latter’s work until Fowler passed him the proofs of the German’s paper in September, after only a couple of weeks, Dirac not only understood Heisenberg’s idea but developed it further by realising that Heisenberg’s relations were very similar to one of the algebraic structures of mechanics: the Poisson bracket. Thus, on 7 November, a paper by Dirac entitled “On the Theory of Quantum Mechanics” (the third paper in chronological order on quantum mechanics) was received in the Proceedings of the Royal Society A just nine days before Heisenberg, this time together with Born and Jordan, sent the complete theory of matrix mechanics for publication. Thus it was that the paper entitled “Zur Quantenmechanik II”, which is often referred to as the Dreimännerarbeit (Three Men’s Work), reached the editorial office of Zeitschrift für Physik on 16 November 1925. What followed was an explosion of work (most notably that of Pauli who managed to solve the problem of the hydrogen spectrum using the Dreimännerarbeit theory) which in little more than a year culminated in a theory that was both physically and mathematically sound.
But how was matrix quantum mechanics accepted? Roughly speaking, as Sánchez-Ron describes in his excellent Historia de la física cuántica, there were two groups: the moderns, led by men like Born, Bohr and Pauli, who accepted it, and the ancients, who rejected it. Among the latter was, oddly enough, Einstein, who wrote to his friend Ehrenfest on 28 August 1926 that “with admiration and distrust at the same time they watched the development of quantum mechanics”. The problem, of course, was mathematics. In 1925-1926 very few physicists were familiar with matrix calculus, and even less so if it involved infinite matrices. We have already seen that Heisenberg himself did not know them, and the fact that Born and Jordan did was due to the fact that Born had been an assistant in Göttingen to Hilbert himself, the most important mathematician in the world at that time.
It was perhaps Peter Debye who, in an interview many years later, gave the best answer to the question of whether physicists paid attention to the quantum mechanics developed in the Dreimännerarbeit:
Yes, yes. Physicists paid attention to it. There was the feeling “why this complicated formalism?”, but as soon as it was put into a differential equation, related to mechanics [this is the Schrödinger equation that we talked about in the previous post]… one felt very relieved. I felt that I understood things then, whereas with Heisenberg I didn’t understand them very well.
But there was still much to be understood… such as, for example, what the meaning of the famous Schrödinger wave function was, which, as we have already mentioned, would bring probability to physics in a way never seen before, and not only that, Heisenberg still had to discover the principle that would make him famous (much more than matrix mechanics): the uncertainty principle. But we will talk about that in our next post.
Learn more
Heisenberg’s autobiography is published in Spanish as
[1] Werner Heisenberg, Diálogos sobre la física atómica. La Editorial Católica. Madrid. 1975. Also available in the Colección Universal de Círculo de Lectores en el volumen titulado “Física cuántica ( Werner Heisenberg, Niels Bohr, Erwin Schrödinger)” Círculo de Lectores. Barcelona. 1996.
On the life and work of Heisenberg we have:
[2] David Lindley, Incertidumbre. Einstein, Heisenberg, Bohr y la lucha por la esencia de la ciencia,
Ariel, 2008.
And as a general reference book,
[3] José Manuel Sánchez Ron, Historia de la física cuántica: I. El período fundacional (1860-1926), Drakontos, 2001.
Excelente trabajo, felicitaciones