Translation as heuristics: Heisenberg׳s turn to matrix mechanics
Alexander Blum*, Martin J€ahnert, Christoph Lehner, Jürgen Renn
Max Planck Institute for the History of Science, Boltzmannstraße 22, 14195 Berlin, German
2017
As we will show, this transformation emerged from the attempt
to solve a specific problem (the spectroscopic intensity problem)
using the tools of the old quantum theory (the correspondence
principle). In the course of this transformation a toy model (the
anharmonic oscillator) came to play a central role by connecting the
specific problem with a generalizable framework (classical me-
chanics). The elaboration of the implications of this constellation
guided the transformation of this framework into the new theory
(matrix mechanics).
the anharmonic oscillator,
which, unlike the purely harmonic motions occurring in the spe-
cific physical problems they had investigated earlier, showed the
decisive feature of allowing transitions between all possible states.
This facilitated the integration of the theoretical resources of clas-
sical mechanics (Fourier analysis of motion, perturbative con-
struction of equations of motion) with the already existing partial
structures of quantum theory.
The old quantum theory was based on Bohr׳s two
fundamental postulates: the stability condition, which demanded
that quantum states were stable non-radiating states defined by
additional quantum constraints on classical mechanics, and the
frequency condition, which postulated that transitions between
quantum states were accompanied by the emission or absorption of
electromagnetic radiation with a frequency given by the Planck
relation between the frequency and the energy difference between
the stable orbits; neither the transition from one orbit to another
nor the emission/absorption process could be described by classical
radiation theory. The frequency condition alone, however, did not
give a complete description of the process either, most importantly
because it did not specify the intensity and polarization of the
emitted radiation.
looked very much like the later noncommutative product of
matrix mechanics, was not interpreted by Heisenberg in quantum
mechanical terms as a “matrix product” or “even a commutator.”
Nonetheless, it offered important insights to Heisenberg:
he would keep in mind two essential characteristics of the
extended theory of dispersion: quantum theoretical amplitudes
appeared to play a fundamental role, and they combined only
through products of the type [aik bkj], where the stationary states
corresponding to the middle indices are identical (Darrigol,
1992a, 230).
This implies that dispersion theory led to the insight that transition probabilities had to be multiplied in a specific manner that would later be recognized as matrix multiplication.
that results from dispersion theory played in Heisenberg׳s construction of the quantum condition is his use of Born׳s principle of turning differential into difference equations, as we will discuss in Section 6.
He specified that the system be an anharmonic oscillator and thus returned to the paradigmatic example for which he (together with Kronig and Pauli) had derived a quantum expression
This expression again only contains frequencies which are multiples of the fundamental frequency (overtones)....In the quantum case, however, the transition frequencies of the
system are not multiples of some fundamental frequency (except for the simple case of the harmonic oscillator), and consequently the sums of transition frequencies are not in general transition frequencies of the system. The quantum calculational rule for constructing x2 [squared] thus had to be different from the classical rule, in order to ensure that x2 only contain transition frequencies...all the frequencies appearing in the Fourier expansion of the square were again transition frequencies. To fulfill this requirement, Heisenberg invoked,
as he had in the case of dispersion, the combination principle of Ritz, i.e., the idea that each radiation frequency could be written as a difference of two terms, interpreted as the energies of quantum levels in the old quantum theory. ...this implied that the frequency of a radiative transition was equal to the sum of the frequencies occurring in a two-step transition via an intermediate state.
This was entirely in keeping with the original correspondence principle according to which the Fourier coefficient belonging to the function cosaut corresponded to the transition amplitude aðn; n aÞ. It is, however, in contrast to matrix mechanics in the Born-Jordan formulation, where one cannot say that the system is in a particular state: The matrix always represents the totality of transitions between all possible states.
the next step for Heisenberg was to insert the classical Fourier series into the equation
of motion. Heisenberg thus obtained recursive relations between
the Fourier coefficients: To understand the solution it is helpful to
go through this calculation in detail.41 From the ansatz 14, Hei-
senberg could write down the second derivative €x:..The x2 -term is obviously a lot more complicated, containing an infinite series for each individual overtone. Heisenberg limited
himself to leading order in perturbation theory for each overtone.
This result would have been trivial using complex Fourier series; using Heisenberg׳s real Fourier series it relied on the physical insight that the transition from n to n 3 can
be split up in two distinct ways.
Directly written in terms of the (complex) Fourier coefficients at of the coordinate x of a simply periodic system, the usual quantum condition reads:
This equation defied Heisenberg׳s translation scheme from the Kronig letter, because only the frequency tu could be translated into the transition frequency ..., while the “order of the overtone” by itself had no obvious quantum counterpart.
(the transition from differentials to differences) in empirical evidence from dispersion
theory, i.e. independent of the spectroscopic context from which his scheme had emerged.
In any case, with the inclusion of the new quantum conditions, Heisenberg׳s calculational scheme for transition amplitudes was essentially complete.
the coefficients multiplying the negative frequency terms. Consequently, Heisenberg started to move away from the real Fourier representations (using only cosine terms and transitions from n to a lower state)
The transition from a still classical conception of the system being in a state, where, as discussed in the previous section, x is given by the set of all possible transitions starting from that state, to the full abandonment of the state concept in matrix mechanics, where x involved all transitions possible for the system, was only completed by Born and Jordan.
It is only at this point that the orbits of the Bohr model were replaced by an alternative method of determining energies that relied only on the transition amplitudes and made no more reference to the actual coordinates.
Nevertheless, Heisenberg was still skeptical about the viability of the scheme, as he wrote to Pauli. Both the physical interpretation and the ultimate consistency of his new scheme remained open questions. Concerning the latter point, an essential question was
whether the energies would be constant in time.
However, attempts to prove the conservation of energy in general were unsuccessful. Here, the problem of the noncommutativity of Heisenberg׳s scheme first showed itself: A term of the form ẋ x, which he probably encountered in the energy expression of the Zeeman effect, did not have a unique quantum mechanical counterpart, calling into question not just the physical but the mathematical consistency of the scheme.
This observation also invalidates the widely held notion that introducing noncommutativity was the seminal insight that brought Heisenberg to the new theory.
Heisenberg׳s own positivistic reading of his new theory as merely establishing
relations between observable quantities, and Born and Jordan׳s interpretation of the theory as a full mechanical theory (to be discussed below)
The question of how to calculate the energy in cases where non-commutativity came into play still remained unresolved.
Using the concomitant phase space formulation,
the quantum condition now appeared as a commutation relation
between canonically conjugate dynamical variables, a universal
addition to the Hamiltonian equations of motion. Born and Jordan
referred to this new quantum condition as the “sharpened quan-
tum condition,” taking up Born׳s idea from 1924 that the corre-
spondence principle was to be sharpened by a discretization of
continuous quantities.
In this way, the matrix mechanics of Born and Jordan represents
in a much stronger sense a new, independent dynamical theory
that actually replaces classical mechanics. Consequently, Born and
Jordan identified their new framework as a “quantum mechanics,”
and thereby as a realization of Born׳s own pre-Umdeutung program,
formulated in 1924. Heisenberg on the other hand, had not used the term quantum mechanics up to this point.
By bringing the equations of motion into his frame-work, he went beyond pure kinematics, introducing what would eventually be seen as dynamical quantum equations of motion. It is, however, quite clear that this interpretation was not fully performed by Heisenberg himself and is really due to Born and Jordan.
In his letter to Pauli, Heisenberg had posed precisely this question: What do the equations of motion mean, once they are reinterpreted as relations between transition amplitudes?
The reinterpreted equations of motion serve the function of giving relations between directly observable quantities. This is a far cry from seeing them as the (quantum) dynamical equations of motion of the system in question;
the initial translation from classical quantities to quantum matrices (quantization, in modern parlance) was to be all the correspondence one needed. From this perspec-
tive, the fact that the quantum mechanical equations of motion
were equations relating transition amplitudes became something
that was in need of a demonstration within the new formalism. In
the final section of their paper, Born and Jordan thus developed a
tentative argument from a yet to be formulated quantum theory of
electrodynamics to show that the classical relation between the
Fourier coefficients of motion and the radiation intensities would
survive in the quantum theory.
The Born-Jordan formulation departed even further from
Heisenberg׳s positivist interpretation of his equations. The
attempt to construct a fully autonomous quantum mechanics
involved taking over as much of the complex structure of Hamil-
tonian classical mechanics as possible. In order to replicate the
canonical phase-space structure of Hamiltonian mechanics, Born
and Jordan introduced the momentum matrix. This is the canonical conjugate of the position matrix in an abstract sense; as
Born and Jordan explicitly stated, it need not always coincide with the time derivative of the position matrix. Consequently, the momentum matrix did not have an immediate interpretation in
terms of observables.
Most importantly, partial differentiation proved to be incredibly cumbersome in the matrix
formalism, a problem only alleviated by Dirac׳s insight that the
easiest way to translate partial differentiation into matrix me-
chanics is via Poisson brackets. Nevertheless, Born and Jordan׳s
work presented a major breakthrough on the way to an autono-
mous quantum mechanics and definitely switched the agenda for
good, back to Born׳s program of constructing a quantum mechanics.
Heisenberg׳s notion of quantum kinematics faded.
First, it originated from a different physical problem than is usually assumed,
not from the context of dispersion, but rather from the context of
the intensity of (multiplet) spectral lines.
the reconstruction offered here
shows how the search for a multiplication rule grew out of Heisen-
berg׳s work on the intensity problem, how the formulation of the
rule was a natural consequence of physical considerations (Ritz׳s
combination principle), and how its application to the equations of
motion was a direct continuation of his earlier work on the transition
intensities of the anharmonic oscillator.
By first stepping back and reinterpreting merely the kinematics, Heisenberg
had provided the essential element lacking in Born׳s program, i.e., a
reinterpretation of the quantities appearing in the mechanical
equations, which, in hindsight, was a necessary first step towards the
reinterpretation of those equations themselves.
The present reassessment of Heisenberg׳s work and his
pathway puts us in a position to better understand the intrinsic
weirdness of matrix mechanics. It can be traced to the unsettled tension between foundational pillars of his Umdeutung: the
operational basis in spectroscopic experiments (Heisenberg׳s new
semantics), on the one hand, and the strong structural connection
to classical mechanics, on the other.
They could not yet, however, provide a new semantics for the theory that might replace Heisenberg׳s operational basis in spectroscopic intensities. Indeed, the
semantics of the new quantum mechanics was the essential
element in the ensuing interpretational debates of 1926/1927, and
we can now clearly see how and why this was an essential issue
already within matrix mechanics itself, independently of the rise of
wave mechanics. As is well known, the emerging Copenhagen
interpretation of quantum theory, and most importantly Bohr׳s
notion of complementarity implied a return to a semantics based
on classical mechanics, albeit involving epistemological restrictions. This completed the shift from Heisenberg׳s spectroscopic theory to a quantum mechanics that deserved that name on all accounts.
In this regard, Heisenberg׳s statement in the Kronig letter that “everything can be calculated from the amplitudes” (a rather trivial truth in classical mechanics) plays a similar role of an Archimedean point for theory transformation as Einstein׳s principle of equivalence does in the development of general relativity.
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