Umdeutung paper
In the history of physics, "On the quantum-theoretical reinterpretation of kinematical and mechanical relationships"
, also known as the Umdeutung paper, was a breakthrough article in quantum mechanics written by Werner Heisenberg, which was published in Zeitschrift für Physik in July 1925.
In his article, Heisenberg described a new framework for quantum theory that was based on observable parameters, such as transition probabilities or frequencies associated with quantum jumps in spectral lines, rather than unobservable parameters, like the position or velocity of electrons in electron orbits. Thus, Heisenberg used two indices for his reinterpretation of position, corresponding to initial and final states of quantum jumps. Heisenberg used his framework to successfully explain the energy levels of a one-dimensional anharmonic oscillator.
Mathematically, Heisenberg used non-commutative operators in his new multiplication rule, i.e. generally for quantum quantities and. This insight would later become the basis for Heisenberg's uncertainty principle.
This article was followed by the paper by Max Born and Pascual Jordan of the same year, building on the conceptual ideas of the Umdeutung paper, and by the 'three-man paper' by Born, Heisenberg and Jordan in 1926. These following articles along with the Umdeutung paper laid the groundwork for matrix mechanics that would come to substitute old quantum theory, becoming the first mature mathematical formulation of quantum mechanics. Heisenberg received the Nobel Prize in Physics in 1932 for his work on developing quantum mechanics.
Historical context
The general narrative surrounding the Umdeutung paper states that Heisenberg, then 23 years old, worked on the article whilst recovering from hay fever on the largely vegetation-free island of Heligoland for about 10 days after arriving on June 6 1925, where he had a 'eureka moment' discovery that lead to mature quantum mechanics. However, this narrative has been criticised for stemming from Heisenberg's account in his 1969 to 1971 memoirs, where Heisenberg stated his book was written in broad historical terms without precise detail. It has been argued that Heisenberg's account is hard to reconcile with contemporary evidence and that the majority of the paper was likely written after Heisenberg's stay on Heligoland in the first weeks of July.It has also been argued that the idea of a 'eureka' discovery conflicts with Heisenberg's frequent correspondence with Wolfgang Pauli on his paper that expressed uncertainty. When asked for his opinion of the manuscript, Pauli responded favorably, but Heisenberg said that he was still "very uncertain about it". After returning from Heligoland, Heisenberg would discuss exclusively with Pauli using letters, making Pauli the only person aware of the successes and failures of Heisenberg's quantum mechanics. In July 1925, he sent the manuscript to Max Born to review and decide whether to submit it for publication. The idealisation of Heisenberg's work as a 'eureka' breakthrough has also been seen as disproportional when compared to the work other physicists made at the same time.
In classical physics, the intensity of each frequency of light produced in a radiating system is equal to the square of the amplitude of the radiation at that frequency, so attention next fell on amplitudes. The classical equations that Heisenberg hoped to use to form quantum theoretical equations would first yield the amplitudes, and in classical physics one could compute the intensities simply by squaring the amplitudes. But Heisenberg saw that "the simplest and most natural assumption would be" to follow the lead provided by recent work in computing light dispersion done by Hans Kramers. The work he had done assisting Kramers in the previous year now gave him an important clue about how to model what happened to excited hydrogen gas when it radiated light and what happened when incoming radiation of one frequency excited atoms in a dispersive medium and then the energy delivered by the incoming light was re-radiated sometimes at the original frequency but often at two lower frequencies the sum of which equalled the original frequency. According to their model, an electron that had been driven to a higher energy state by accepting the energy of an incoming photon might return in one step to its equilibrium position, re-radiating a photon of the same frequency, or it might return in more than one step, radiating one photon for each step in its return to its equilibrium state. Because of the way factors cancel out in deriving the new equation based on these considerations, the result turns out to be relatively simple.
Description
Heisenberg's reinterpretation of quantum theory
One of the key aspects of Heisenberg's paper in forming a new quantum theory was its deliberate choice to not utilise unobserved quantities such as the electron's position and period but only to utilise observable parameters like transition probabilities or spectral frequencies. Whilst this has been seen as one of the succeeding innovations of Heisenberg's paper in moving away from classical mechanics, others have viewed the correspondence principle, the idea that quantum systems will behave like classical systems in the classical limit, as the primary driver of Heisenberg's argument. His letter to Ralph Kronig one month before the publishing of the Umdeutung paper closely follows the same structure as the paper but emphasises the correspondence of the Fourier coefficients rather than the philosophical focus on observable quantities.Fourier reinterpretation
, a periodic orbit for an electron in stationary state may be described as using a Fourier series, where for to be real-valued.Heisenberg decides to reinterpret this for his quantum theory to be
where represents the frequency of light emitted/absorbed and represents an intensity relating to transition probabilities when there is a transition from stationary state to. Heisenberg mentions that may be found using the Bohr model calculation of where is the energy of a given stationary state.
A justification for Heisenberg's Fourier reinterpretation can be seen when comparing classical and quantum expressions for the orbital frequencies. Classically, using action-angle variables, it can be found that as ; in the Bohr model, by contrast, it is found that
Using the correspondence principle it can be found using from the old quantum theory. This correspondence between and also suggests a correspondence between the intensities and in the connection
This Fourier reinterpretation was also justified in Heisenberg's view so that the electron in its periodic motion, which the Bohr model predicted, would not have a single characteristic radiation frequency in stationary state that had never been experimentally measured. Under Heisenberg's reinterpretation, the motion of the electron would only have the experimentally observed spectral frequencies as components of its periodic motion. Heisenberg notes, however, that by the publishing of the paper that the electronic orbit was abandoned as physically meaningful.
Non-commutative multiplication rule
Heisenberg uses the Rydberg–Ritz combination principle as a fundamental part of how frequencies should behave in his quantum theory. Rather than the classical rule Heisenberg writes out the Ritz combination principle.Classically, multiplying two position Fourier series together forms a new Fourier series: where
Heisenberg analogised from this classical case that multiplying two quantum series should give another quantum series of the same form. In other words, Heisenberg finds that if a quantum series may be written then
Whilst a normal multiplication would not work to form a new quantum series, Heisenberg deliberately reinterprets multiplication to be now following the new quantum multiplication rule
This is so that using the earlier Rydberg-Ritz Combination Rule so that all the frequencies are correct for the new series.
Generalizing from the earlier multiplication case of, Heisenberg wrote out his new, non-commutative multiplication rule that is the quantum mechanical analog for the classical computation of intensities:
There will potentially be an infinite series of terms and their matching terms. Each of these multiplications has as its factors two measurements that pertain to sequential downward transitions between 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 the electron begins and in what energy state it ends, not what the electron is doing while in one or another state.
If and both refer to lists of frequencies, for instance, the calculation proceeds as follows:
Multiply the frequency for a change of energy from state to state by the frequency for a change of energy from state n − a to state n − b, and to that add the product found by multiplying the frequency for a change of energy from state n − a to state n − b by the frequency for a change of energy from state n − b to state n − c, and so forth. Symbolically, that is:
It would be easy to perform each individual step of this process for some measured quantity. For instance, the boxed formula at the head of this article gives each needed wavelength in sequence. The values calculated could very easily be filled into a grid as described below. However, since the series is infinite, nobody could do the entire set of calculations.
Heisenberg originally devised this equation to enable himself to multiply two measurements of the same kind, so it happened not to matter in which order they were multiplied. Heisenberg noticed, however that if he tried to use the same schema to multiply two variables, such as momentum, p, and displacement, q, then "a significant difficulty arises". It turns out that multiplying a matrix of p by a matrix of q gives a different result from multiplying a matrix of q by a matrix of p. It only made a tiny bit of difference, but that difference could never be reduced below a certain limit, and that limit involved the Planck constant, h.