# When Heisenberg met Einstein

A mere 24 years old, Werner Heisenberg (1901-76) in 1925 developed a treatment of electron behavior based solely on directly observable quantities such as the frequencies of light that atoms absorb and emit. Recovering from hay fever on the island of Heligoland, *"still very uncertain about it"*, Heisenberg in July 1925 sent his manuscript to Max Born (1882-1970) for review. Upon reading Heisenberg’s paper, Born realized that in Heisenberg’s new formulation, the classical variables of position and momentum could be represented by matrices which can be multiplied together like numbers, albeit with the crucial difference that the order of multiplication matters. The first conceptually autonomous and logically consistent formulation of quantum mechanics had been born. Heisenberg’s formulation accounted for quantum jumps, thereby supplanting Niels Bohr (1885-1962)’s model of electron orbits by interpreting the physical properties of particles as matrices that evolve over time.

Albert Einstein (1879-1955), a proponent (and indeed pioneer) of the ‘old’ quantum mechanics disagreed strongly with the implications of Heisenberg’s paper. Although Einstein was only 46 years old at the time, the two men were a generation apart and indeed, in very different stages of their lives. Einstein had won the Nobel Prize in Physics in 1921, divorced and re-married, been promoted to full professor and entered into politics by joining the League of Nations as the delegate from Germany. Heisenberg had just completed his Ph.D. at the Ludwig Maximillian University of Munich under the supervision of Arnold Sommerfeld (1868-1951) who introduced him to Bohr in June of 1922. Heisenberg’s thesis had been on the topic of turbulence, discussing the stability of laminar flow and the nature of turbulent flow. Also a student of mathematics, he had studied with David Hilbert (1862-1943) at the Georg-August University of Göttingen. A *Privatdozent* there starting in 1924, he soon travelled to commence research with Bohr at the University of Copenhagen. This is where he wrote his seminal paper

**Heisenberg, W. 1925**. Über quantentheoretische Umdeutung kinematischer und mechanischer Beziehungen (“Quantum theoretical re-interpretation of kinematic and mechanical relations“).*Zeitschrift für Physik***33**(1), pp. 879-893.

## Their First Meeting (1926)

Heisenberg first secured an audience with Herr Einstein in early 1926. Heisenberg had been invited to speak at Max von Laue (1879-1960)’s famous physics colloquium on the ‘new’ quantum mechanics at the Humboldt University of Berlin (then Friedrich Wilhelm University), where Einstein was teaching. Referred to as the “citadel of physics” by Heisenberg, the University of Berlin was in 1926 also home Nobel laureates Max Planck (1858-1947), Otto Hahn (1879-1968) and Walther Nernst (1864-1941). As Heisenberg later wrote (Heisenberg, 1983),

“Einstein would thus be in the audience; I would make his personal acquaintance. It goes without saying, that I now prepared my lecture with the greatest care. For I wanted, in any event, to make myself intelligble, and more especially to get Einstein interested in the new possibilities.”

Heisenberg was invited to speak about his paper and its later further developments by Born and his assistant/former student Pascual Jordan (1902-80). Reading Heisenberg’s article, Born had been the one to recognize Heisenberg’s formulation as one which could be extended further in the systematic language of matrices. As Born later stated in his own Nobel lecture:

“Heisenberg's rule of multiplication left me no peace, and after a week of intensive thought and trial, I suddenly remembered an algebraic theory […] Such quadratic arrays are quite familiar to mathematicians and are called matrices, in association with a definite rule of multiplication. I applied this rule to Heisenberg's quantum condition and found that it agreed for the diagonal elements. It was easy to guess what the remaining elements must be, namely, null; and immediately there stood before me the strange formula.”

Heisenberg had developed this ‘entryway to matrix mechanics’ by rethinking the ‘old’ quantum physics (pioneered by Planck and Einstein) from the ground up, basing it *“exclusively on relationship between quantities that in principle are observable”*. Heisenberg’s breakthrough equation in the paper was the following (in general form):

The equation states that some term C may be computed by summing the products of some group of terms (A) by some related group of terms (B). There may potentially be an infinite series of A and B terms. Each multiplication has as its factors two measurements (n − a, n − b) that pertain to sequential downward transitions between the energy states of an electron:

“This type of rule differentiates matrix mechanics from the kind of physics familiar in everyday life because the important values are where (in what energy state or “orbital”) the electron begins and in what energy state it ends, not what the electron is doing while on one or another state.”

Upon reading Heisenberg’s paper, Born’s application of the method lead him to the formula

where **Q** is the matrix for the displacement, **P** is the matrix for momentum, *i* is the imaginary unit (the square root of negative one) and *h* is Planck’s constant.

### Heisenberg’s Lecture

Heisenberg’s lecture to the Berlin colloquium went well, as Heisenberg later recalled, *“More or less as desired; there were good and helpful questions asked in the discussion that followed. I saw that I had secured Einstein’s interest, when immediately afterwards he invited me to come home with him” *(Heisenberg, 1983 p. 113)*.*

In 1926 Einstein was living at 5 Haberlandstrasse in the Bavarian Quarter of Berlin with his wife Elsa. The walk from the University of Berlin is about an hour long. On the walk, Einstein questioned Heisenberg about his background and studies with Sommerfeld, Heisenberg’s Ph.D. doctoral advisor. However, *“on arrival, he at once began with the central question about the philosophical foundation of the new quantum mechanics”.*

One of Einstein’s issues with Heisenberg’s formulation of quantum mechanics was *“the notion [that] an ‘electron path’ did not occur at all, but that in a cloud-chamber the track of the electron can of course be observed directly.” *It seemed to Einstein absurd to claim that there was an electron path in the cloud-chamber, but none in the interior of the atom (as Heisenberg’s model asserted).

“I defended myself to begin with by justifying in detail the necessity for abandoning the path concept within the interior of the atom. I pointed out that we cannot, in fact, observe such a path; what we actually record are frequencies of the light radiated by the atom, intensities and transition-probabilities, but no actual path. And since it is but rational to introduce into a theory only such quantities as can be directly observed, the concept of electron paths ought not, in fact, figure in the theory”.

In his later writings on the topic, Heisenberg makes clear that this assertion, that *“it is rational to introduce into a theory only such quantities as can be directly observed”*, indeed was inspired from Einstein’s own writing. In discussing the so-called Compton and Stern-Gerlach effects, Heisenberg writes:

*“I thought of an idea I had read in Einstein’s work, namely the requirement that a physical theory should contain only quantities that can be directly observed”.*Applying this principle, so the idea goes, provides a guarantee that there are connections between mathematical formulae and the phenomena described.

“To my astonishment, Einstein was not at all satisfied with this argument. He (now) thought that every theory in fact contains unobservable quantities. The principle of employing only observable quantities simply cannot be consistently carried out”.

Sitting there in the apartment of his hero Professor Albert Einstein in Berlin, the 24-year-old Heisenberg argued back that he had* “merely been applying the type of philosophy that Einstein, too, had made the basis of his special theory of relativity”.* To his great disappointment, Einstein’s reply was simply to brush this off by saying *“Perhaps I did use such philosophy earlier, and also wrote it, but it is nonsense all the same”* (Heisenberg, 1983 p. 114).

### Electron Behavior

Heisenberg and Einstein’s conversation next turned to the topic of electron states. In particular, Einstein brought up the “special question” of what happens in the passage of an electron from one stationary state to another:

“The electron might suddenly and discontinuously leap from one quantum orbit to the other, emitting a light-quantum as it does so, or it might, like a radio transmitter, beam out a wave-motion in continuous fashion. In the first case there is no accounting for the interference phenomena that have often enough been observed; in the second, we cannot explain the fact of sharp line-frequencies.”

Heisenberg’s response was to re-iterate Niels Bohr’s argument that such phenomena lie *“far beyond the realm of everyday experience, and so cannot be expected to be describable in terms of traditional concepts”*. Einstein, famously did not find this “excuse” satisfactory. He challenged Heisenberg to explain in what quantum state, then, the continuous emission of a wave was supposed to take place.

“I then produced the comparison with a film, in which the transition from one picture to another often does not occur suddenly, the first picture becoming slowly weaker, the second slowly stronger, so that in an intermediate state we do not know which picture is intended”

Heisenberg is alluding to the paradox of quantum superposition, the ‘Schrödinger’s cat’ thought experiment later devised by Erwin Schrödinger (1887-1961) in a 1935 discussion with Einstein:

“A cat is penned up in a steel chamber, along with the following device (which must be secured against direct interference by the cat): in a Geiger counter, there is a tiny bit of radioactive substance, so small, that perhaps in the course of the hour one of the atoms decays, but also, with equal probability, perhaps none; if it happens, the counter tube discharges and through a relay releases a hammer that shatters a small flask of hydrocyanic acid. If one has left this entire system to itself for an hour, one would say that the cat still lives if meanwhile no atom has decayed. The first atomic decay would have poisoned it. The psi-function of the entire system would express this by having in it the living and dead cat (pardon the expression) mixed or smeared out in equal parts.”

Heisenberg and Schrödinger’s thought experiments both illustrate the question* “ when does a quantum system stop existing as a superposition of states and become one or the other? *

“Thus, in the atom also, a situation could arise in which for a time we do not know which quantum state the electron is in.”

Einstein was not satisfied with Heisenberg’s film analogy, replying that the state of the electron could not possibly be a matter of physicists’ knowledge of the atom in question. Two different physicists might have two different sets of information, what then would the conclusion be?

Einstein’s objection to the thought experiment (and indeed both Heisenberg and Schrödinger’s approaches) was that they fundamentally—in principle—acknowledged the statistical character of natural laws because in statistics, findings are a function of one’s incomplete knowledge of a system. Einstein wanted nothing to do with this, although, as Heisenberg later wrote *“He himself, in his paper of 1918, had introduced such statistical concepts”. *At this point in the conversation, Heisenberg acquiesced.

“I had no idea what to do just then, and we separated in the common conviction that a great deal of work still needed to be done before reaching a full understanding of the quantum theory”

“We agreed to disagree, as the British say.”

## Their Second Meeting (1927)

By the time of their second meeting a year later, in the fall of 1927, significant progress had been made in the development of quantum theory. Schrödinger had succeeded Max Planck’s chair at the University of Berlin and proven that his new wave mechanics (based on the earlier attempts by Louis de Broglie) were mathematically equivalent to Heisenberg’s quantum mechanics (addressing the wave-particle duality concept).

Amazingly, practically all of the names mentioned in this article so far—from both the ‘old’ quantum physics and the new—in October of 1927 gathered at the Fifth Solvay International Conference to discuss the topic of “electrons and photons”. Of the 29 attendees at the meeting, 17 had or would go on to win the Nobel Prize in physics or chemistry. Among them was also Marie Curie (1867-1934) the first person to win both:

Nobel Laureates in Physics in attendance:Hendrik Lorentz (1902), Marie Curie (1903), Lawrence Bragg (1915), Max Planck (1918), Albert Einstein (1921), Niels Bohr (1922), Arthur Compton (1927) C.T.R. Wilson (1927), Owen Richardson (1928), Louis de Broglie (1929), Werner Heisenberg (1932), Paul Dirac (1933), Erwin Schrödinger (1933), Wolfgang Pauli (1945) and Max Born (1954).

Nobel Laureates in Chemistry in attendance:Marie Curie (1911), Irving Langmuir (1932), and Peter Debye (1936).

The old guard was represented by Planck, Einstein and Lorentz, who chaired the conference. There too, was the new guard, including Bohr, Born, Pauli, Schrödinger, de Broglie, Dirac and of course, Heisenberg.

### Uncertainty Relations

By the spring of 1927, six months prior to the conference, Heisenberg had formalized what would later be known as Heisenberg’s uncertainty principle. Later in life, Heisenberg repeatedly highlighted his talk with Einstein in Berlin as the ‘crucial junction’ on his journey to its formulation (Kumar, 2008). On February 23rd, 1927 he wrote a fourteen-page letter to Pauli summarizing his findings.

Heisenberg began work on the formulation in the beginning of February 1927. Bohr had left Copenhagen for a skiing holiday in Norway. Heisenberg, left alone, set out to clear up the difficulty that arose when concepts such as the ‘position of the electron’ and the ‘path of the electron’ were used in the classical sense, despite the fact that Born’s commutation relation

meant something quite different in quantum mechanics (Rigden, 1987). At the beginning of the month, he wrote Pauli that he was *“again occupied all the time with the logical foundations of the whole pq — qp swindle”*. As the days went on, he probed the meaning of the words ‘position of an electron’ and, again inspired by Einstein, the notion that *“it is the theory which decides what can be observed”*. Replacing the assertion ‘position of an electron’ with instead the question *“How does one determine the position of the electron?”*, his work eventually culminated in the letter to Pauli written on February 27th, which Heisenberg used as a draft *“to clarify his own thinking”* before writing the now famous paper:

**Heisenberg, W. 1927.**Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik (“The Physical Content of Quantum Kinematics and Dynamics”).*Zeitschrift für Physik***43**(3-4), pp. 172-198.

Heisenberg later described the main idea of the paper, what he called ‘uncertainty relations’, in the following way (Heisenberg, 1983 pp. 116):

“[Uncertainty] relations involves the statement that two determinants of a system, which must both be known at once in classical physics, in order to determine the system completely, cannot, in quantum theory, be exactly known at the same moment; and hence that between the uncertainties or inexactitudes of these quantities there are mathematical relations which prevent an exact knowledge of both quantities”

Before sending the article for publication, he gave a draft to Bohr who disagreed with Heisenberg’s assessment of the source of the uncertainty (particle collisions), suggesting instead that it stemmed from the dual nature of particle-waves. Despite Bohr’s objections, following a bit of drama ending with Heisenberg supposedly breaking out in tears, Heisenberg eventually sent the article for publication near the end of March. He also sent a copy to Einstein, but as far as we know, did not receive a response. Heisenberg’s paper appeared in print in May and lead to his accepting of an offer for a professorship at Leipzig University. He left Copenhagen in June.

By the time of the conference in October, Heisenberg’s uncertainty relations and Schrödinger’s work on wave mechanics had been circulated among all the top physicists in Europe. Indeed, as Heisenberg later wrote, *“they formed the main top of the discussion in Brussels”. *Once again, Einstein inserted here himself into the discussion on the side of the opponents, refusing to accept any statistical interpretation of nature:

“Einstein […] kept trying, during the congress, to refute the uncertainty relations by means of counter-examples, which he formulated in the shape of thought experiments. We were all living in the same hotel, and Einstein was in the habit of bringing along to breakfast a proposal of this kind, which then had to be analysed.”

Einstein, Bohr and Heisenberg were the main interlocutors:

“[We] would usually make our way to the congress hall together, so that even on this short walk a beginning could be made on analysing and clarifying the assumptions. In the course of the day, Bohr, Pauli and I would frequently discuss Einstein’s proposal, so that already by dinner-time we could prove that his thought-experiments were consistent with the uncertainty relation, and so could not be used to refute them.”

The details of this ‘debate of the century’ (including Einstein’s thought experiments and Heisenberg’s role) is discussed in a separate newsletter, available to paid subscribers here.

## Their Third Meeting (1954)

Heisenberg met Einstein a third and final time, in 1954. On a lecture tour in the United States, Einstein invited the now-53-year-old Heisenberg to his *“pleasantly unpretentious one-family house with a small garden“ *at 112 Mercer Street on the edge of the Princeton University campus in New Jersey.

“I had been warned beforehand that my visit should last only a short time, since Einstein was obliged to spare himself, on account of a heart condition. Einstein, however would have none of this, and with coffee and cakes I was made to spend almost the whole afternoon with him”.

Einstein’s sole interest in talking to Heisenberg seems to have been the latter’s by-then broadly accepted interpretation of quantum theory *“which continued to disturb him, just as it had done in Brussels twenty-five years before”. *Throughout their conversation, according to Heisenberg, Einstein’s remark *“but you cannot believe, surely, that God plays at dice”* was *“several times repeated, almost as a reproach” *(Heisenberg, 1983).

Those interested in reading more about the relationship between Heisenberg and Einstein should see the following article by Holton (2000):

**Holton, G. 2000.**Werner Heisenberg and Albert Einstein.*Physics Today*53 (7).

I recommend everyone interested in the history of of quantum mechanics buy the excellent and beautiful book:

**Purrington, R.D. 2018.***The Heroic Age. The Creation of Quantum Mechanics, 1925-1940.**Oxford University Press.

Thank you for reading this week’s newsletter.

Sincerely,

Jørgen

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### References

**Heisenberg, W. 1983.***Encounters with Einstein. And other Essays on People, Places, and Particles*.*Princeton University Press.**Kumar, M. 2008.***Quantum. Einstein, Bohr and the Great Debate about the Nature of Reality**. Icon Books.**Rigden, J. S. 1987**. Editorial: Heisenberg, February 1927, and physics.*American Journal of Physics***55**(107).

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