© European Physical Society, EDP Sciences, 2025

On 29 July 1925, Max Born, Professor of Theoretical Physics at the University of Göttingen, submitted a paper to the Zeitschrift für Physik. The paper was by his 23-year old assistant (we would nowadays say: postdoc) Werner Heisenberg and it contained the foundations of what would become the theory of quantum mechanics [Heisenberg 1925].

In this paper, Heisenberg aimed to construct a theory for calculating spectroscopic quantities, that is the frequencies and intensities of the spectral lines of atomic and molecular systems¹. He called these spectroscopic quantities observable and contrasted them with the unobservable quantities describing the motion of the electrons within these microscopic systems. It should be noted that this labeling does not fully correspond to the use of the term“observable” in modern quantum mechanics. Nowadays, the position of an electron is considered an “observable,” at least in the formal sense of being represented by a hermitian operator. But its observability is limited, due to the Heisenberg Uncertainty Principle. In 1925, this finer-grained analysis of observability was still two years in the future.

Instead of looking to the future, let us briefly look a bit further into the past. In classical electrodynamics, there is of course an intimate physical connection between the motion of the system emitting radiation (Heisenberg’s unobservables) and the spectral properties of the emitted radiation (Heisenberg’s observables). It had been clear for a while, ever since Niels Bohr’s atomic model of 1913, that this intimate physical connection could not be upheld in quantum theory – the frequencies of the motion were not equal to the frequencies of the radiation. Bohr himself had proposed to replace the physical connection between motion and radiation with a more formal one, which went over into the established physical connection in the classical limit. This was Bohr’s correspondence principle. It left open what the formal connection between motion and radiation at the quantum level was supposed to look like in detail. Heisenberg’s 1925 paper was an attempt to fill Bohr’s Correspondence Principle with life, building on a number of previous attempts to do so.

The formal connection between motion and radiation that Heisenberg proposed was the following: the spectroscopic observables of the radiation are represented by a novel mathematical object, the so-called quantum Fourier series x, where, following Heisenberg, we restrict ourselves to systems with one degree of freedom. This object x now obeys the equations of motion of the system emitting the radiation. So whileHeisenberg used the equations of motion, the quantity x in those equations now no longer represented the position of an electron, but rather the properties of the emitted radiation. The unobservable motion was thus eliminated from the theory, and the equations of motion were now relations between observables, which could be solved to determine the spectroscopic frequencies and intensities.

The great challenge for Heisenberg – and this had been the starting point for his endeavor some two months earlier – was representing the spectroscopic observables as a quantum Fourier series x. The general idea was clear enough. Writing the Fourier series as

$x\left(t\right)=\underset{\tau =-\infty }{\overset{\infty }{\mathrm{\Sigma }}}{a}_r{e}^{i\tau {\omega }_{0}t}$(1)

the frequencies τω are to be identified with the spectral frequencies and the coefficients a_τ are to be identified with the corresponding amplitudes, the square of the amplitude determining the spectral intensity. However, as opposed to a classical Fourier series, the spectral frequencies are not integer multiples (overtones) of a fundamental frequency ω₀ and cannot be labelled by a single integer τ (the order of the overtone)². Instead, in the BohrModel, the spectral frequencies are labelled by two integers, the quantum number of the initial state (n) and of the final state (m). Heisenberg now identified the order of the overtone τ with the magnitude of the quantum jump, τ = n − m. τω was to be replaced by ω(n, n − τ), the frequency for the transition n → n −τ ; a_τ was to be replaced by a(n, n − τ), the corresponding transition amplitude. At this point, the modern reader can see the concept of a matrix slowly emerging.

This substitution is good and well when one is just talking about x(t), but one runs into trouble when trying to write down x(t)², as it appears, e.g., in the force term of the equation of motion for the harmonic oscillator. The problem is that, when multiplying two terms in the series, the frequencies in the complex exponentials simply add up. However, the sum of two spectral frequencies is not automatically a spectral frequency again – only under specific conditions. These conditions had first been formulated by Walther Ritz in 1908 (Ritz Combination Principle), in Bohr’s atomic model they had received a physical interpretation: the sum of two spectral frequencies is again a spectral frequency if the final state of the one transition is the initial state of the other. Heisenberg then constructed a new multiplication rule for his quantum Fourier series that would ensure that the product would only contain sums of spectral frequencies that obey the Ritz Combination Principle.

It is intuitively clear that this will give something like matrix multiplication: in a matrix product, two elements are multiplied only if the second index of the first element is equal to the first index of the second element. Famously, Heisenberg did not make this connection, which is not as surprising as it seems in hindsight: before the advent of quantum mechanics, linear algebra played a far smaller role in physics education than it does today.

With the new multiplication rule in hand,Heisenberg had all of the tools he needed to recast the equations of motion as equations for the quantum Fourier series x(t). But there was still one thing missing to solve them. For the classical equations of motion one needs initial conditions, in addition to the equations themselves, in order to single out a particular solution. In Bohr’s atomic model, the initial conditions had been replaced by the quantum condition, which singled out a discrete set of possible orbits. Particular orbits were then simply labeled by their quantum numbers. In its most sophisticated form, due to Sommerfeld, it read

$\oint pdq=nh$(2)

where the integration is performed along an orbit described by the phase space variables q and p. This integral expression could not be translated unambiguously using Heisenberg’s new prescription. In order to bring the quantum condition into his new scheme, he had to first differentiate it with respect to the quantum number n and then invoke an additional translation rule (first suggested by Born a year earlier) namely that derivatives with respect to n should really be taken as finite differences, since n could only take integer values. The exact prescription that Heisenberg used was

$\tau \frac{df}{dn}\to \left(f(n+\tau ,n)-f(n,n-\tau )\right)$(3)

where f is an arbitrary function and τ an integer (the order of the overtone or the magnitude of the quantum jump, as above). With this prescription,Heisenberg was able to translate the entire set of equations of the Bohr model into equations for the spectral transition amplitudes, which could be solved for a number of simple systems.

He was also able to calculate the energies of the stationary states, by taking the quantum Fourier series that solved the equations of motion and plugging them into the expression for the energy as a function of x and its derivatives. In principle this could have delivered an arbitrary quantum Fourier series, and it would then have been hard to give a physical interpretation of the many terms involved. However, it turned out that the quantum Fourier series for the energy only contained constant terms, which could easily be interpreted as the energies of the states. In more modern parlance, Heisenberg obtained a diagonal Hamiltonian matrix.

Upon studying Heisenberg’s manuscript, Born realized that Heisenberg’s new multiplication rule was in fact matrix multiplication and that Heisenberg’s quantum Fourier series x was a matrix. With Heisenberg on vacation, Born and his student Pascual Jordan (who was working towards the postdoctoral habilitation degree) reworked Heisenberg’s ideas in matrix language. Just two months later, on 27 September 1925, they submitted their paper Zur Quantenmechanik (On Quantum Mechanics) to the Zeitschrift für Physik [Born and Jordan 1925]. It was the first time this term was used to designate the emerging theory; Heisenberg had not yet spoken of “quantum mechanics” in his paper.

The most striking feature of the new matrix notation was that Heisenberg’s quantum condition now appeared as the soon-to-be iconic canonical commutation relation between the coordinate matrix q and the momentum matrix p:

$\text{pq}-\text{qp}=\frac{h}{2\pi i}.1$(4)

This turned the non-commutativity of the new multiplication rule from a somewhat confusing novelty into a defining feature of the theory. This was highlighted even further by the British physicist Paul Dirac, another early adopter of Heisenberg’s novel ideas, who, in a paper submitted on 7 November 1925, formulated quantum mechanics in terms of an abstract non-commutative algebra (rather than an explicit matrix representation of that algebra) and pointed out the connection between commutators and classical Poisson brackets [Dirac 1925].

Born and Jordan’s work was more than a mere reformulation of Heisenberg’s equations in terms of matrices, however. Using their new formalism, they were able to give a general proof that the energy matrix would be diagonal if p and q obeyed the quantum equations of motion. And these equations of motion were no longer simply taken over from the classical theory, but were obtained as a quantum version of the canonical equations of motion of the classical Hamiltonian formalism:

$\frac{ih}{2\pi }\dot{\text{q}}=\left[\text{q,H}\right]$(5)

where H is the Hamiltonian matrix.

In addition, Born and Jordan asked a question that the readers of this short article may also have asked themselves: why should the elements of the x (or, in Born and Jordan’s phase space formulation, q) matrix be interpreted as transition amplitudes for the emission of radiation? In the last section of their paper, Born and Jordan, tried to derive this interpretation by coupling the atomic system, described by p and q, to the quantized electromagnetic field. This was not entirely successful – it was not clear howbest to represent field quantities (which were themselves functions of, potentially non-commuting coordinates) as matrices, and it would take a long time for a workable theory of quantum electrodynamics to be developed. But the general idea of how to connect the elements of the coordinate matrix to radiative transitions is already present in Born and Jordan’s last section: q is taken to represent an oscillating electric dipole, which then serves as the source term in Maxwell’s equations, leading to the emission of electromagnetic radiation.

With the work of Born and Jordan, and even more with the subsequent Dreimännerarbeit (Three-Man Paper), which Heisenberg, Born and Jordan wrote together in late 1925, quantum mechanics starts looking more and more recognizable to the modern reader [Born et al. 1925]. Still, the following two years saw manifold further developments in what was arguably the most rapid foundational transformation in the history of science. The next big step was the development of the Schrödinger equation, the centenary of which we will be celebrating next year.

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