In the history of physics, "On the quantum-theoretical reinterpretation of kinematical and mechanical relationships" (), also known as the Umdeutung (reinterpretation) 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 focused exclusively on observable parameters, parameters that could be measured in scientific experiments, such as transition probabilities or frequencies associated with quantum jumps in spectral lines. This contrasted with earlier quantum theories that focused on unobservable parameters, like the position or velocity of electrons in electron orbits. In contrast to other quantum theories, Heisenberg assumed that issues stemmed from the kinematics (how position and velocity were defined) in the old quantum theory, not classical equations of motion.
Mathematically, Heisenberg used two indices for his reinterpretation of position, corresponding to initial and final states of quantum jumps. In addition, Heisenberg introduced non-commutative operators in a new multiplication rule, i.e. generally for quantum quantities and . This insight would later become the basis for Heisenberg's uncertainty principle. Heisenberg used his framework to successfully explain the energy levels of a one-dimensional anharmonic oscillator.
The Umdeutung paper was followed by another paper by Max Born and Pascual Jordan in 1925 that built on the Umdeutung paper's conceptual framework, and by the 'three-man paper' () by Born, Heisenberg and Jordan in 1926. These articles described matrix mechanics that would come to substitute old quantum theory. It was the first mature mathematical formulation of quantum mechanics.
For his work on developing quantum mechanics, Heisenberg received the Nobel Prize in Physics in 1932.
Historical context
In early 1924, a year before Heisenberg's paper was published, a few physicists were already calling for abandonment of unobservable quantities in electron orbits; for instance, Arnold Sommerfeld abandoned them in discussing complex spectral lines.
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 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 Classically, an electron's orbit in stationary state can be expressed as a Fourier series for some characteristic frequency where to ensure that is real-valued ( is the complex conjugate of ).
Heisenberg reinterprets this equation in terms of quantum jumps from state to all other states
where represents the frequency of light emitted/absorbed and represents an intensity of the light from the jump, relating to the probability of it transitioning.
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 , corresponding to the frequency of the electron's periodic motion, 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 where classically , 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 (in what energy state or "orbital") 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 amplitudes of quantum transitions, 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 to state , and to that add the product found by multiplying the frequency for a change of energy from state to state by the frequency for a change of energy from state to state , and so forth. Symbolically, that is:
(According to the convention used, represents a higher energy state than , so a transition from to would indicate that an electron has accepted energy from an incoming photon and has risen to a higher orbital, while a transition from to would represent an electron falling to a lower orbital and emitting a photon.)
Heisenberg originally devised this equation to enable himself to multiply two measurements of the same kind (amplitudes), 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, , and displacement, , then "a significant difficulty arises". It turns out that multiplying a matrix of by a matrix of gives a different result from multiplying a matrix of by a matrix of , with their difference being where is the reduced Planck constant.
Motion Heisenberg also notes that, in the old quantum theory, the observable quantities are solved from Newton's second law
and the Wilson–Sommerfeld quantisation rule where is the force per unit mass function, where Heisenberg makes the assumption that, with his new Fourier reinterpretation of , the same Newtonian equations may be used to solve for observable quantities.
The idea that these same classical equations may continue to be used in quantum mechanics has been viewed as one of the most important ideas of Heisenberg's paper.
Quantisation
As a part of Heisenberg's justification for a theory of mechanics that is only reliant on the intensities and frequencies that are measurable, Heisenberg derives the Thomas-Reiche-Kuhn sum rule found from studying dispersionfor .
Quantum anharmonic oscillator