In the history of physics, "On the quantum-theoretical reinterpretation of kinematical and mechanical relationships"
(German: Über quantentheoretische Umdeutung kinematischer und mechanischer Beziehungen), 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
A
B
≠
B
A
{\displaystyle AB\neq BA}
for quantum quantities
A
{\displaystyle A}
and
B
{\displaystyle B}
. 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' (German: Dreimännerarbeit) 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.
Contents
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
n
{\displaystyle n}
can be expressed as a Fourier series for some characteristic frequency
ω
(
n
)
\omega (n)
x
=
∑
α
=
−
∞
∞
X
→
α
(
n
)
e
i
α
ω
(
n
)
t
,
{\displaystyle x=\sum _{\alpha =-\infty }^{\infty }{\vec {X}}_{\alpha }(n)e^{i\alpha \omega (n)t},}
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
ω
(
n
,
α
)
+
ω
(
n
,
β
)
=
ω
(
n
,
α
+
β
)
,
{\displaystyle \omega (n,\alpha )+\omega (n,\beta )=\omega (n,\alpha +\beta ),}
where classically
ω
(
n
,
α
)
=
Motion
Heisenberg also notes that, in the old quantum theory, the observable quantities are solved from Newton's second law
x
¨
+
f
(
x
)
=
0
{\displaystyle {\ddot {x}}+f(x)=0}
and the
Wilson–Sommerfeld quantisation rule
∮
p
d
q
=
∮
m
x
˙
d
x
=
∮
m
x
˙
2
d
t
=
J
=
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
x
=
∑
α
=
−
∞
∞
a
(
n
,
n
−
α
)
e
i
ω
(
n
,
n
−
α
)
t
{\displaystyle x=\sum _{\alpha =-\infty }^{\infty }a(n,n-\alpha )e^{i\omega (n,n-\alpha )t}}
.
Quantum anharmonic oscillator
Heisenberg additionally studies the energy levels of the anharmonic oscillator described by
x
¨
+
ω
0
2
x
+
λ
x
2
=
0
{\displaystyle {\ddot {x}}+\omega _{0}^{2}x+\lambda x^{2}=0}
, as the conceptually more simple harmonic oscillator would not have required his new multiplication rule. Whereas under a classical consideration he finds that
E
n
=
n
ℏ
ω
0
{\displaystyle E_{n}=n\hbar \omega _{0}}
, but under a quantum consideration using the theory developed in his paper, up to order
λ
2
{\displaystyle \lambda ^{2}}
Legacy
Development of matrix mechanics
The one thing that people at that time most wanted to understand about hydrogen radiation was how to predict or account for the intensities of the lines in its spectrum. Although Heisenberg did not know it at the time, the general format he worked out to express his new way of working with quantum theoretical calculations can serve as a recipe for two matrices and how to multiply them.
The Umdeutung paper does not mention matrices. Heisenberg's great advance was the "scheme which was capable in principle of determining uniquely the relevant physical qualities (transition frequencies and amplitudes)" of hydrogen radiation.
After Heisenberg wrote the Umdeutung paper, he turned it over to one of his senior colleagues for any needed corrections and went on vacation. Max Born puzzled over the equations and the non-commuting equations that Heisenberg had found troublesome and disturbing. After several days he realized that these equations amounted to directions for writing out matrices.
By consideration of ... examples. .. [Heisenberg] found this rule ... This was in the summer of 1925. Heisenberg ... took leave of absence ... and handed over his paper to me for publication ... 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
Q
P
−
P
Q
=
i
h
2
"Magical paper"
Steven Weinberg would in 1993 refer to Heisenberg's paper as "pure magic" due to Heisenberg's lack of written motivation that he takes in justifying each mathematical step, leaving the reader "mystified" despite the successful predictions of Heisenberg's framework. A number of attempts have been made to explain his paper more accessibly.
Hundredth anniversary
UNESCO celebrated the 100th anniversary of the publication of Heisenberg's paper by pronouncing 2025 as an observance year on quantum science and technology.