Download The Parallel Development of Matrix and Wave Mechanics

University of Amsterdam
Bachelor Thesis in Physics and Astronomy
The Parallel Development of
Matrix and Wave Mechanics
1
Author:
J.G. de Swart1,2
Institute for Theoretical Physics, Faculty
of Science, University of Amsterdam
2
Institute for Interdisciplinary Studies,
Faculty of Science, University of
Amsterdam
Supervisor:
Prof. Dr. A.J. Kox
Secondary Supervisor:
Prof. Dr. E.P. Verlinde
Abstract
This paper elucidates the history of the development of a coherent formalism for quantum theory. It aims to describe this passage in the history of physics and reflect upon the
mathematical and conceptual steps which were taken. Starting with an incoherent and inconsistent patchwork of theorems and hypotheses concerning the quantum nature of physics,
in 1925 a consistent formalism was constructed independently by both Werner Heisenberg
and Erwin Schrödinger. Heisenberg’s Matrix mechanics was formed with the extensive use of
Bohr’s correspondence principle and led to a new multiplication rule for quantum-theoretical
observables. This multiplication rule is later recognised as the multiplication of matrices.
Schrödinger’s wave mechanics originated from the Hamilton-Jacobi equation, which gave a
new variational principle which is later to be justified through an analogy between optics and
classical mechanics. The well-known Schrödinger wave equation is the result of his work on
the variational principle and this analogy.
When both formalisms are compared it appears they are formally equivalent, yet stylistic
differences give rise to a philosophical interpretation of this interesting historical development.
Reflecting upon these considerations asks for a brief philosophical discussion on the structure
of scientific knowledge.
12 ECTS
Submitted August 31, 2010
[email protected]
Realised between 04-03-2010 and 27-08-2010
CONTENTS
CONTENTS
Contents
1 Introduction
1.1 What History? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
2
2
2 Background
2.1 Brief Conceptual Overview . . . . . . . .
2.1.1 Max Planck . . . . . . . . . . . . .
2.1.2 Albert Einstein . . . . . . . . . . .
2.1.3 Niels Bohr . . . . . . . . . . . . . .
2.1.4 1919-1925 . . . . . . . . . . . . . .
2.2 Old Quantum Theory: Dispersion Theory
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4
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3 Heisenberg and Matrix Mechanics
3.1 Background . . . . . . . . . . . . . . .
3.2 Calculation . . . . . . . . . . . . . . .
3.2.1 Deduction . . . . . . . . . . . .
3.2.2 Construction and Implications
3.3 Contextual . . . . . . . . . . . . . . .
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8
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4 Schrödinger and Wave Mechanics
4.1 Background . . . . . . . . . . . .
4.2 Calculation . . . . . . . . . . . .
4.2.1 Part I . . . . . . . . . . .
4.2.2 Part II . . . . . . . . . . .
4.3 Contextual . . . . . . . . . . . .
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15
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21
5 Discussion
5.1 Style and Stylistic Differences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 Scientific Knowledge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
22
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6 Conclusion
25
7 Acknowledgements
26
8 Bibliography
27
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1
1.0 Introduction
1 Introduction
1
1.1
Introduction
What History?
Quantum mechanics, it can be said, was one of the most fruitful physical theories of the twentieth
century. Modern physics would have a totally different character should this theory not exist, since
today the theory is used extensively all over physics and even in the field of chemistry. This means
also physics education has changed, now quantum mechanics is taught in first year graduate courses
in physics. Students have to learn about the characteristics of the micro world, about waves and
particles, probabilities, operators and hermitian matrices. Not to mention questions of interpreting
quantum theory, students are trained mathematically to cope with physical problems in this new
quantum mechanical way. Today the focus is on applying and using quantum mechanics, this
great discovery in the history of physics. This way however, the question how such a great theory
comes into play, is forgotten. Because where lies the origin of this complicated theory everybody
so eagerly applies?
Exactly these origins of quantum mechanics will be the subject of the present paper, and it
will appear that this subject is extremely interesting, and maybe even exciting to study. Since its
starting point in 1900 the theory has gone through various stages and constantly changed with
numerous physicists working on the completion of this new theory. Only more than twenty-five
years after the starting point physicists could say the theory got a rigid form, but still quantum
mechanics as we know it was far from finished at that time. In a historical study about the
development of quantum theory much can be learned about the theory itself, about the origins of
the concepts and mathematical tools we use today.
Also such a study can have a more philosophical side. When looking from up close how such
a scientific endeavour takes place, you can see aspects and characteristics of scientific enterprise
itself you would normally not see. For example the process which today is regarded as a discovery
does not at all look like something scientists quite passively take out of nature, like one-way traffic.
In context one sees the active interaction between many elements and the not yet schematized
stream of events which constructed the “true theory. And it maybe even shows that today the
process of discovery is viewed in a very rationalized way.
Trying to find a glimpse of these aspects of scientific enterprise, this paper focuses on a special
event in the history and development of quantum mechanics; the creation of a genuine mathematical formalism. That is: the moment where quantum theory turned into a quantum mechanics. This
truly consistent quantum mechanics was formed on a fairly unique way around 1925. Physicists Erwin Schrödinger and Werner Heisenberg independently came up with two radically different types
of mechanics to formalize all existing quantum-theoretical concepts into one system. Schrödinger
came up with a formalism called wave mechanics while Heisenberg introduced matrix mechanics.
At that time there was a parallel development of two formalisms which were both designed to
cover the same range of experience, this peculiar event shall be discussed in the present paper.
1.2
Outline
The main objective of this paper will be to create a general overview of the first period of the
formations of both the theory of wave and matrix mechanics. The mathematical construction of
the formalisms plays a central role in this objective. Several historical and technical details also
play part in this overview. The paper re-examines the formalisation of quantum theory and will try
to elucidate the conceptualisations that independently created a consistent quantum mechanics.
This way the paper intents for the reader to reach a better physical understanding of the subject
of quantum mechanics and the elements that played the major parts in the formation of both wave
and matrix mechanics, and thus quantum mechanics as a whole. A second aim of this paper is
to place the development in a more philosophical perspective concerning the structure of science.
Since two different theories were formulated parallel to each other it is stressed even more that a
study of the development of this event can bring deep insight in scientific activity. When reflecting
upon the development it will be tried to look at what we can learn from it about the position of
physicists in their field.
2
1.2
1 Introduction
Outline
To approach this study it is chosen to regard a specific set of texts by the authors W. Heisenberg
and E. Schrödinger. Looking at the small period of time where the quantum theory turned into
a real mechanics, only a specific set of articles are discussed in this paper. This of course also
has to do with considerations of length and the limited scope of this article. It means also
that some subjects will not be extensively treated, they will however be discussed sufficiently
enough to understand the main subject. The focus will be on founding papers of wave and matrix
mechanics, avoiding later discussions and additions in the further development of a formalization
of quantum theory. In the development of matrix mechanics Heisenbergs first article, Über der
quantentheoretische Umdeuting kinematischer und mechanischer Beziehungen (1925), is the only
one which is studied. In this article Heisenberg presents for the first time his new mathematical
style of thinking, in the line of earlier work on quantum theory. Many elaborations and reactions
on this article exist but they will not be considered here. Concerning wave mechanics two of
Schrödingers articles are studied; “Quantisierung als Eigenwertproblem” (1926) part I and II.
Because a conceptual clarification by means of an analogy which Schrödinger introduces in the
second part, both articles are used. Also these two articles have many follow ups, but they too will
not be discussed in this paper. The foremost secondary source used in this paper will be a book
by Max Jammer: “The Conceptual Development of Quantum Mechanics” (1966). This book is
used in many aspects throughout the paper to clarify the discussed development. Sometimes the
need for a better understanding of the context is prominently there and then often other papers
by physicists of that time will be used to clarify this context. In the reflection on the development
of wave and matrix mechanics I will use the philosophical vocabulary of the philosopher of science
and medical biologist Ludwik Fleck. In his book “Genesis and Development of a Scientific Fact”
(1935), Fleck describes the formation of a fact in a new, more of a sociological way. In his
discussion he uses his concept of Thought-Style, which also really fits into the discussion of this
paper. Needlessly to say, this paper will, due to it subject, be historical of nature. Therefore it will
have a more philosophical style of writing and reference than most articles in physics. This because
interest is paid in the thoughts of the physicists and the reason why equations are formulated, not
only in the equations.
The paper will be constructed as follows. The first section following to this introduction
will cover the background of the old quantum theory. This includes a short history of quantum theory where the most important contributions to quantum theory before its formalisation
by Heisenberg and Schrödinger will be treated. More extensively discussed will be Bohrs important correspondence principle. The chapter also contains a short theoretical treatment of an
important subject that laid the basis for Heisenbergs mathematical approach; dispersion theory.
Schrödingers theoretical influence, the Hamilton-Jacobi theory, is treated in section four with the
rest of Schrödingers material. Section three and four contain an extensive reproduction of the main
arguments of Heisenberg and Schrödinger respectively and how they constructed their formalism
mathematically. The background of section two is used to clarify the steps taken by the physicists.
The fifth section contains a philosophical discussion on the formation of quantum mechanics and
what such a parallel development shows. Here the philosophical vocabulary of Fleck is used to
describe the process quite elegantly. In section six a general conclusion is drawn regarding the
discussed historical development in the physical theory of quantum mechanics and the results are
summarized.
3
2.0 Background
2 Background
2
2.1
2.1.1
Background
Brief Conceptual Overview
Max Planck
The story of Quantum Theory can be set to begin in the year 1900. At that time classical physics
was unable to explain the experimentally observed energy distribution of black-body radiation.
The theory could in no possible way account for the relation between the frequency and the
energy of the black-body radiation. This inability created a need for a new way of looking at the
physical world. Max Planck initiated this turn of events; he was the first who was able to solve
the problem of black-body radiation. Planck was looking for a way to combine the results of the
“Rayleigh-Jeans radiation law”, which agreed with the mentioned experimental data in the region
of extremely low frequencies, and “Wien’s radiation law”, which agreed with the experimental
data in the region of high frequencies. Making an assumption based upon only the mathematical
properties of the equation he was working with, he found a perfect agreement with observation.
It was only when explaining this assumption Planck had implicitly to hypothesise the existence
of “energy elements”. This meant that in solving the problem, Planck considered energy of an
oscillator as a quantity which is “composed of a finite number of discrete equal parts” (Planck in
Jammer, 1966, p.20), and employed for that his famous natural constant h. When this constant is
multiplied by the frequency ν of a resonator it will give its energy. Hence, in this context energy
was no longer considered to be an infinite divisible quantity. It was here that the foundations of
a new physical theory were laid.
2.1.2
Albert Einstein
The generalization of the concept of elements followed in the work of Albert Einstein. In his 1905
paper on “the production and transformation of light”1 Einstein concluded that light, or more
generally, radiation behaved as if it consisted of a finite number of independent localized energy
“quanta” (Jammer, 1966, p. 30). This was a whole new kind of conceptual step which Einstein
made. A step which seemed to defy the wave-like nature of light; the overall accepted conception of
light. Particles and waves were considered as much as antonyms of each other, concepts that were
“fundamentally incompatible and mutually exclusive” (idem., p. 31). Einstein inserted a workable
new particle-like notion of radiation and thus came up with a new conception, the wave-particle
duality of light; light (i.e. radiation) can be seen both as a wave and a particle (or quantum).
Where Planck did not give his hypothesis any real physical significance, Einstein did and with this
development a real new physical concept came to be.
2.1.3
Niels Bohr
After the introduction of both Planck’s energy elements, which was related to the interaction
between radiation and matter, and Einstein light quanta, which was connected to radiation in
transit, elaborations of the concept appeared. Quantum conceptions were applied to other problems in the field of physics. This also happened in the explanation of line spectra. It was Niels
Bohr who in 19132 wrote an article called “On the constitution of atoms and molecules” in which
he made a very important contribution to quantum theory. In his search for a consistent model
of the atom Bohr combined Rutherford’s atomic model with the quantum conception introduced
by Planck. Bohr made a bold postulation about the atom to cope with its size and stability and
said there exists a discrete set of permissible stationary orbits for the electron in the hydrogen
atom; and as long as the electron remains in any stationary orbit, no energy is radiated (idem.,
p. 77). A consideration which he later also seems to justify. Note that this is a crucial step in the
development of quantum theory. Bohr’s seemingly random hypothesis of stationary states was not
1 “Einstein, A.(1905) On a heuristic viewpoint concerning the production and transformation of light.Annalen
der Physik ”
2 He wrote two more on the some subject in 1914 and 1915.
4
2.1
2 Background
Brief Conceptual Overview
seen before and gave rise to a whole new interpretation of the atom. Furthermore Bohr postulated
another great hypothesis that appeared very useful. He states that in transitions between these
stationary states the observable radiation is emitted with a frequency ν of h1 ∆E, with ∆E the
difference between the energy of the stationary states. With these two assumptions in his theory
Bohr was able to describe the hydrogen atom, account for the origin of line-spectra and deduce
the Balmer formula for the hydrogen spectrum perfectly. He made use of Planck’s quantum conceptions. You should notice that due to the high density of postulates, Bohr’s treatment can be
called quite ad hoc.
In his 1913 paper Bohr fully explains his theory and the above mentioned concepts, and
introduces a new idea, which he elaborates in more detail in 1918. This idea had an enormous
heuristic value in the further development of quantum mechanics. To make this idea clear let us
start with an enunciation of the two fundamental assumptions of quantum theory we just saw;
1. An atomic system can only exist permanently in a discontinuous series of ‘stationary’ states.
2. The radiation absorbed or emitted during a transition between two stationary states possesses
a frequency ν given by
E 0 − E 00 = hν
(2.1)
(van der Waerden, 1967, p5)
Bohr now remarks that it has been possible to account for the phenomenon of radiation coming
from the atom by classical electrodynamics, but only in a special limiting case of slow vibrations.
This is the old theory. A new theory which considers the transition between two stationary
states as quantum theory tells us, should however account for the radiation without being a
limiting case. Therefore Bohr writes that “we may expect that any theory capable of describing
this phenomenon in accordance with observation will form some sort of natural generalisation of
the ordinary theory of radiation” (Bohr, 1918 in van der Waerden, p. 6). Bohr comes up with some
kind of correspondence between the ‘ordinary’ theory of radiation and a new ‘quantum’ theory
of radiation. The ordinary theory is only capable to describe the radiation of the hydrogen atom
correctly in a limiting case, a new theory which could describe the radiation in the full range of
observation should than be a “natural generalisation” of the ordinary theory. In his 1918 paper
Bohr specifies this:
“[T]he conditions which will be used to determine the values of the energy in the stationary states are of such type that the frequencies calculated by [equation [2.1]], in
the limit where the motions in successive stationary states differ very little from each
other, will tend to coincide with the frequencies to be expected in the ordinary theory
of radiation from the motion of the system in the stationary states.” (Bohr, 1918 in
van der Waerden, 1967, p.6)
So what does this mean? Bohr states here that the frequency of the radiation from an electron in
orbit calculated with electrodynamics (classically) corresponds with the frequency of the radiation
coming from a transition between stationary states of an atom (quantum-theoretically)! This is
only valid in the limit where the difference between the classical orbits of successive stationary
states is very small.
Bohr quantifies his principle with the use of Fourier series. He uses the series to describe the
classical orbits of the electron. From this and further considerations Bohr gives a condition for
this transition from stationary state n to a neighbouring state n0 . He writes that the stationary
states are related in the following way: n0 = n − α. Here α is the element of the Fourier series
over which is summed. We now have a “far-reaching correspondence between the various types
of possible transitions between the stationary states on the one hand and the various harmonic
components of the motion on the other hand.” (Bohr, 1920 in Jammer, 1966, p.111). Eventually,
from the above mentioned considerations, we get a relation between the quantum-theoretical and
5
2.2
2 Background
Old Quantum Theory: Dispersion Theory
classical frequency:
νquantum (n, n − α) = νclassic (n, α)3
(2.2)
On the one side we have the frequency from the radiation which is due to a transition between
stationary states, from n to n − α, thus ν depends on both variables. And on the other side we
have the dependence on the αth Fourier component of the frequency of the radiation due to an
electron in a orbit.
This is the “Principle of Correspondence”4 and it eventually comes to drive most of the research
on quantum theory in the years to follow.
2.1.4
1919-1925
As shown above, the old quantum theory was characterized by rather ad hoc conditions which
should be satisfied by quantum theory. Planck used an innovative hypothesis, but Bohr also
came up with assumptions that should govern the hydrogen atom, of which his idea of stationary
orbits is the most noticeable. The research following the work of these physicists is characterized
by a same ‘attitude’ towards this new born quantum theory. With classical mechanics as their
foundation, they were slowly creating the building blocks of a complete new theory. Eventually
all the work and research on quantum theory combined resulted in a large patchwork of ideas,
hypotheses, principles, theorems and computational rules one could use to make the world turn
‘quantum’. In short; Quantum theory anterior to 1925 was a matter of translating. Classical
solutions of quantum systems could be put into this inscrutable machine which implemented the
right conditions to create the good quantum-theoretical solutions. The main tool used for this
translation was Bohr’s correspondence principle. One could even say that at that time quantum
theory “may be described as systematic guessing, guided by the Principle of Correspondence”(van
der Waerden, 1967,p8).
For realising the significance of this paper’s subject, it is important to understand the situation
of quantum theory prior to 1925. It allows you to see the steps which are taken to create a fully
fledged scientific theory more clearly. The point Heisenberg and Schrödinger find themselves is
precisely the point a genuine theory is created and scientists try to pin down the meaning of a
new concept.
I would like to end this short overview with last remark which should be made. The history of
the origin of quantum theory, of course, is much more elaborate than pictured above. However, a
full review is far beyond the scope of this article. It is only to indicate the process which involves
the formation of such a theory and a general introduction into the physics of that time. Also it
gives a good indication of the atmosphere there was in the field of physics which was present in
the formulation of a quantum mechanics.5 To describe this formulation we first need to go into one
subject more elaborately for it is used by Heisenberg in the development of his formalism.
2.2
Old Quantum Theory: Dispersion Theory
Now some results of research on dispersion theory will be considered to get familiarized with
Heisenberg’s mathematical background, without going deeper into the actual calculations.
We start with looking at the polarization of an atom caused by a vibrating electrical field.
Classically there is a simple relation between the polarization and the field:
~
P~ = αE
(2.3)
Where α is empirically well determined. Note that the polarization becomes high when the
frequency of the vibrating field is near to the absorption frequency νi of the atom.
The factor α can also be derived by classical theory on the assumption that the atom contains
a certain number of oscillators, whose frequencies are equal to the absorption frequencies νi (van
3 Valid
for large n and small α.
It would however not get this fancy name until Bohr’s paper in 1920.
5 For a very elaborate overview of the chain of events leading to the formation of Quantum Mechanics I advise
you to read the nicely constructed “The Conceptual Development of Quantum Mechanics”(1966) by Max Jammer.
4 ‘Korrespondenzprinzip’.
6
2.2
2 Background
Old Quantum Theory: Dispersion Theory
der Waerden, 1967, p.9). This idea was extended to the quantum-theoretical regime by Ladenburg
in 1921:“. . . we may say that Ladenburg replaced the atom, as far its interaction with the radiation field is concerned, by a set of harmonic oscillators with frequencies equal to the absorption
frequencies νi of the atom” (idem, p.11). Oscillators which Bohr later called ‘virtual harmonic
oscillators’. Ladenburg used this in his attempt to connect the experimental results on dispersion
with quantum-theoretical considerations on the probability of transitions.
Using Bohr’s correspondence principle, it was Hendrik Anthony Kramers who went beyond
Ladenburg’s result to find a quantum-theoretical expression for α. In the context of the correspondence principle this meant that in the region of large quantum numbers the interaction
between the atom and the radiation field tends to coincide with that expected on classical theory (idem, p.14). This consideration led to an expression which will not be discussed here. His
expression does result in a possible formal substitution, equivalent with that of Ladenburg:
“The reaction of the atom against the incident radiation can (. . . ) formally be compared
with the action of a set of virtual harmonic oscillators inside the atom, conjugated with
the different possible transitions to other stationary states.” (Kramers, 1967[1924],
p.179)
Born extrapolates the method used by Kramers to interactions between two mechanical systems, instead of between a radiation field and an atom. Born uses the above idea and assumes
again that an atom in a stationary state n may be replaced, as far as the calculation of emission,
absorption and dispersion is concerned, by a set of ‘virtual oscillators’ of frequencies
ν(n, n0 ) =
1
[E(n) − E(n0 )] .
h
(2.4)
(Van der Waerden, 1967, p.15)
And now we have come across a formula which likely heavily influenced Heisenberg’s way of
thinking. This has to do with the fact that in the above set of ‘virtual oscillators’ every possible
transition between stationary states is present. That is to say; in this set n and n0 take every
possible value, so all of the transition frequencies of an atom are present in equation [2.4]. So
one can formally compare the atom with a set of virtual oscillators which take on every possible
transition frequency. Keep this in mind, for it will return in the next section.
Now for a transition between frequencies one can speak of a ‘virtual resonator’. Then, according
to the correspondence principle, every virtual resonator must correspond with the element of the
classically computed Fourier series of the motion in state n. If Born then again looks at this
correspondence he remarks that this formally means replacing a differential with a difference.
The classical frequency can be expressed as a differential of the total energy6 . And thus a new
correspondence rule is born to translate classical formulas into quantum-theoretical analogues:
α
∂Φ(n)
↔ Φ(n) − Φ(n − α)
∂n
(2.5)
Where Φ(n) can be any arbitrary function. This is an important correspondence which is extensively used by Heisenberg.
Two important results thus appeared in this very short review of quantum-theoretical dispersion theory. First we saw that when calculating on mechanical processes of the atom, for
calculating purposes one could replace the system by a set of ‘virtual harmonic oscillators’. Secondly, to go from a classical process to a quantum-theoretical one must replace all the differential
coefficients with difference quotients. Because Heisenberg collaborated with Kramers on a paper regarding these subjects, this is knowledge important for the perpetuation of Heisenberg’s
theoretical research.
6 This
is a result of earlier work combined with Planck’s E = hν. See also section 3.2.
7
3.0 Heisenberg and Matrix Mechanics
3 Heisenberg and Matrix Mechanics
3
3.1
Heisenberg and Matrix Mechanics
Background
Werner Heisenberg was the first to publish a paper in which a new type of mechanics was introduced
to coherently describe quantum phenomena. His paper was received the 29th of July in the German
‘Zeitschrift fur Physik’. Also publishing work with Kramers on his dispersion theory, Heisenberg
places himself in a tradition with the likes of Bohr, Born, Kramers and Slater, whose papers laid
the foundation for Heisenberg’s Matrix Mechanics.
Heisenberg started his search for the basis for a theoretical quantum mechanics with a philosophical presupposition which guided his way through the process. The foremost important thing
according to Heisenberg was, to create a theory which is founded only upon observable entities:
“. . . it seems more reasonable to try to establish a theoretical quantum mechanics, analogous to
classical mechanics, but in which only relations between observable quantities occur” (Heisenberg,
1967[1925], p262). He emphasizes the “the failure of the quantum-theoretical rules as a deviation
from classical mechanics”, which is the step that lead to his urge to create a new quantum mechanics from scratch, only using quantities which in principle are observable. Heisenberg thus avoids
quantities like the the orbit of an electron, which can not be observed directly. The observables
Heisenberg worked with are the frequency and amplitude of the radiation emitted by an electron
in orbit. This radiation was both described classically and quantum-theoretically, and both descriptions corresponded to each other through Niels Bohr’s correspondence principle, a tool we
saw was of inexhaustible heuristic value for the development of quantum mechanics. Using this
correspondence principle and the concept of a Fourier expansion, Heisenberg wants to inquire in
his paper “. . . about the form higher order terms would assume in quantum theory”(idem). He
refers to the higher order terms of the electromagnetic field strengths induced by an electron moving in an orbit. Classically these terms are easy to calculate, but quantum-theoretically there was
at that time no way to compute this.
Without looking at the cause of the motion, Heisenberg thus engages himself in the kinematics
of quantum theory. To simplify his theoretical investigation in this region of kinematics, Heisenberg
poses a new question which covers his previous interest in ‘higher order terms’; “If instead of a
classical quantity x(t) we have a quantum-theoretical quantity, what quantum-theoretical quantity
will appear in place of x(t)2 ?” (idem, p263).
3.2
3.2.1
Calculation
Deduction
To look for a quantum-theoretical x(t)2 we start with the observable quantum-theoretical quantities which are known from earlier work on quantum theory. Heisenberg first looks how these known
quantities act in quantum phenomena. These quantities are the quantum-theoretical frequency
of the radiation emitted by a transition between stationary states of an atom and the classical
frequency of an electron in an orbit, as discussed above. Like Bohr and Born, Heisenberg looks
at both the classically and quantum-theoretically computed frequency of the radiation. How does
this quantity look from these different points of view?
First let us see what the frequency looks like an a classical context. In equation [2.2] we saw
the classical frequency has the following form.
νclassical = ν(n, α)
(3.1)
This is the αth harmonic frequency (or the frequency of the αth Fourier-element) of the fundamental frequency ν(n, 0) = ν(n). Classically it is possible to write this harmonic as a linear
combination of the fundamental frequency.
ν(n, α) = αν(n)
8
(3.2)
3.2
3 Heisenberg and Matrix Mechanics
Calculation
This is quite clear. However there is another feature of the classical frequency which is not
immediately clear. It looks like this:
ν(n, α) = αν(n) =
α dH
h dn
(3.3)
Where H is the Hamiltonian or the total energy. This is a result I will not discuss, but it is
important to notice the differential in the above equation.
Now take a look at how the frequency is constructed in a quantum context. We have already
seen how Bohr constructed the frequency, and Heisenberg makes great use of this result. In the
terminology used above we get:
ν(n, n − α) =
1
[H(n) − H(n − α)]
h
(3.4)
The frequency depends, after the transformation to quantum mechanics, also on (n − α). As
shown, this is according to Bohr the frequency of the radiation coming from a transition of an
electron from stationary state n to stationary state n−α. In retrospect, however, this could also be
the consequence of Born’s correspondence rule treated in subsection 2.2, where for any arbitrary
function the differential in a classical concept makes place for a difference in quantum theory.
So if we compare quantum theory with classical theory, we are thus dealing with the following
analogue;
αν(n) ⇐⇒ ν(n, n − α)
(3.5)
To go from a classical quantity to a quantum-theoretically one only replaces a dependence on (n)
with a dependence on (n − α). In quantum theory it is no longer possible to write a frequency
as the αth overtone times the base frequency. The quantum-theoretical frequency always depends
on two variables which both are equally important. It is from this point on that we can work our
way to how one must manipulate quantum mechanic observables.
The relation we have for the quantum-theoretic frequency in equation [3.4], results in new
combination rules for this quantity. This is due to the terms of H which are able to cancel each
other out when adding the different frequencies.
ν(n, n − α) + ν(n − α, n − α − β) = ν(n, n − α − β)
(3.6)
These combination rules will appear to be very important in Heisenberg’s reasoning towards a
mechanics of quantum theory. Also the amplitude, like the frequency, has a correspondence
between classical and quantum phenomena, and thus quantum-theoretical amplitudes are also
functions of the two values n and n−α. Now these features of both quantum-theoretical quantities
are defined, they can be applied in other cases. How could the amplitude and frequency help us
to construct a general quantum-theoretical quantity? Heisenberg looked at the Fourier-series as
representations of a quantity, which consist of exactly these two variables.
We introduce a new quantity x(t), which we assume can be written both classically and
quantum-theoretically. This we will do as Heisenberg suggests in his article, using a Fourier
expansion. Classically this will be just a normal Fourier expansion as still used all over modern
physics7 ;
∞
X
x(n, t) =
Aα (n)eiαω(n)t
(3.7)
α=−∞
Here we have Aα (n) and αω(n) which are, respectively, the amplitude (or Fourier-coefficient) and
the angular frequency of the αth Fourier component. So the expansion can be divided in into an
amplitude and a phase part. This is a normal quantity as it exists in classical theory.
7 This paper restricts itself in looking at the the Fourier sum, and will not cover the Fourier integral. This is due
to considerations of length. However, the procedure using Fourier integral can be done exactly analogous to the
procedure done in this paper with a Fourier sum. Which one to use obviously depends on whether the quantity
x(t) is periodic or not.
9
3.2
3 Heisenberg and Matrix Mechanics
Calculation
Is there a possible quantum-theoretic analogue? And if so, how is such an expansion constructed quantum-theoretically? In answering this question Heisenberg employs the analogue we
explicated in equation [3.5], but now we have to be careful. Replacing (n) with (n, n − α), an
expansion as done above becomes nonsensical. Because both the frequency and the amplitude
depend on n and n − α, an analogous expansion would give us an expression without any physical
meaning. This is because classically you could say the representation for a quantity x(t) is a summation over all possible harmonics of the base frequency in the phase, with an amplitude which
also depends on this harmonic. With the introduction of ν(n, n − α) 8 this interpretation is no
longer possible, it can no longer be said that the phases of all possible frequencies are added up
since both n and n − α are evenly important in computing the frequency. Hence physically such
an quantum expansion does not mean anything like the classical Fourier-series. Disregarding this,
Heisenberg makes the following hypothesis as a quantum analogue to equation 3.79 ;
∞
X
x(n, t) =
A(n, n − α)eiω(n,n−α)t
(3.8)
α=−∞
It must be noted again that this quantum representation of x(t) is not a meaningful expression since
the two variables are equally weighted. But when computing x(t)2 , Heisenberg really neglected
this fact, and thought this representation could just do the trick. This is an important conceptual
step.
It is almost cogent that Heisenberg here got influenced by early dispersion theory. As we saw
in section 2.2 Born came up with a new way to describe the atom, one could formally replace
the atom with a set of virtual oscillators. This set consisted of oscillators that together possessed
all of the possible transition frequencies in an atom. This is exactly what Heisenberg does here!
Equation [3.8] is equivalent to a sum over a set of oscillators with amplitude A and frequency ω,
just like what happened in equation [2.4]. Since ω(n, n − α) is a transition frequency, summing
over α means all these transitions come to pass. One could say that here instead of summing
over all types of waves with amplitudes and frequencies, we now sum over all possible transitions
with the accompanying frequencies. Heisenberg worked with Kramers on dispersion so it is most
likely that he got his inspiration for his quantum-theoretical x(t) from this work. The result: The
hypothetical expansion of x(t) is one of the founding elements of Heisenberg’s Matrix Mechanics.
With this new expansion the tools are available to try to calculate x(t)2 . Again, first the
squared value of x(t) is computed classically. And this we will do again with a Fourier-expansion.
The Fourier-series of x(t)2 gives us:
2
x(t) =
∞ X
∞
X
β=−∞
Aα (n)e
iαω(n)t
α=−∞
×
∞
X
i(β−α)ω(n)t
Aβ−α (n)e
(3.9)
α=−∞
This requires some further explanation. The second term and the summation over the new variable
β is due to the fact the Fourier transform of a product has a special feature. If a function can
be decomposed as the product of two square integrable functions (here x(t) and x(t)), then the
Fourier transform of the function is given by the convolution product of the respective Fourier
transforms. This is why the β − α is introduced in the second term, with β the new variable over
which is summed.10 We can write the product [3.9] tidier when we bring terms together.
x(t)2 =
∞
X
∞
X
Aα (n)Aβ−α (n)eiαω(n)t ei(β−α)ω(n)t
(3.10)
β=−∞ α=−∞
This way we notice that the power of e after multiplication drops out α, and only depends on
β. The phase having only one variable over which is summed gives rise to a beautiful coherent
− α) = π1 ω(n, n − α)
in 3.5 the amplitude can also be replaced
in an analogous way:Aα (n) ⇐⇒ A(n, n − α)
P∞
10 F [p(t)q(t)](α) = F [p(t)](α) ∗ F [q(t)](α) =
β=−∞ F [p(t)](α)F [q(t)](β − α)
8 ν(n, n
9 As
10
3.2
3 Heisenberg and Matrix Mechanics
Calculation
representation of x(t)2 , since only the amplitude changed.11 We can write this like Heisenberg
did, in the following way:
∞
X
x(t)2 =
Bβ (n)eiβω(n)t
(3.11a)
β=−∞
with
∞
X
Bβ (n)eiω(n)βt =
Aα (n)Aβ−α (n)ei(α+(β−α))ω(n)t
(3.11b)
α=−∞
It is now possible to notice even better that x(t)2 only differs from its square root in the amplitude
B, the phase again is a sum over all possible frequencies in a power of e. That the classical x(t)2
Fourier expansion has a phase which eventually only depends on the the variable β will turn out
to be an important fact when trying to create a quantum analogue of x(t)2 .
The past transformation of x(t) was, within the classical frame of reference, a mathematical
manipulation which did not introduce anything new for physicists of that time. Heisenberg’s big
step forward was to copy this same manipulation, and instead of the classical Fourier expansion of
x(t) he wanted to do it for his hypothetical quantum equivalent of this expansion, equation [3.8].
For the quantum-theoretical x(t)2 Heisenberg suggested the following:
x(t)
2
∞
∞ X
X
=
=
β=−∞
∞
X
iω(n,n−α)t
A(n, n − α)e
α=−∞
∞
X
×
∞
X
A(n − α, n − β)e
iω(n−α,n−β)t
α=−∞
A(n, n − α)A(n − α, n − β)ei(ω(n,n−α)+ω(n−α,n−β))t
(3.12)
β=−∞ α=−∞
Now take a good look at equation [3.9] again and notice the insertion of (n, n − α) for the classical
dependence on (n) and Fourier-element α, but also the insertion (n − α, n − β) for the classical
dependence on n and Fourier-element β − α which came from the convolution product. Using the
combination relation of [3.6] we are able to again get a nice expression for x(t)2 in which the phase
is the same as that of x(t). With the combination relation it is possible to rewrite ω in equation
[3.12], to get a phase factor which again only depends on β.
ω(n, n − α) + ω(n − α, n − β) = ω(n, n − β)
(3.13)
And thus we get
x(t)2 =
∞
X
∞
X
A(n, n − α)A(n − α, n − β)eiω(n,n−β)t
(3.14)
β=−∞ α=−∞
Which is a really nice expression. Its relation with the quantum-theoretical x(t) is strikingly
analogous to how the classical x(t)2 and x(t) relate to each other.
3.2.2
Construction and Implications
Equation [3.14] is definitely not the ending point, this is only where Heisenberg’s sets his story to
begin. But before we look at what follows on his treatment of quantum-theoretical observables the
expression above needs further explanation. Therefore I will now stop to reconstruct Heisenberg’s
article and try to find a justification for the steps Heisenberg took and look at what exactly he
did.
Without further explication on how he came up with exact form of the second sum over β
in [3.12], Heisenberg regarded this relation as the new quantum-theoretical x(t)2 . He only notes
that this equation “...is an almost necessary consequence of the frequency combination rules”
(1967[1925], p265). This is rather strange since it is the crucial step that eventually will lead
11 Multiplying
quantities with the same phase should result in a new quantity with again the same phase
11
3.2
3 Heisenberg and Matrix Mechanics
Calculation
to Heisenberg’s famous multiplication rule. The first part of equation [3.12] is clear, here he did
exactly the same as when computing the quantum theoretical x(t). However, the construction of
the second part is not directly visible. So then what could be the reason of this exact substitution
Heisenberg performed?
?
Aβ−α (n)ei(β−α)ω(n)t =⇒ A(n − α, n − β)eiω(n−α,n−β)t
(3.15)
A short explanation is necessary. Looking at the substitution performed in exactly the same
line as Heisenberg determined x(t) quantum-theoretically from the classical representation, using
equation 3.5, we would also try to compute x(t)2 quantum-theoretically. But instead of only α we
now also have the element β − α. We then get the following transition from classical to quantum
theory:
Aβ−α (n)ei(β−α)ω(n)t =⇒ A(n, n − (β − α))eiω(n,n−(β−α))t
(3.16)
Applying the correspondence principle we did the substitution (β − α)ω(n) → ω(n, n − (β − α)).
But now, unlike in Heisenberg’s procedure, we do not get a phase factor which corresponds with
the phase of the quantum-theoretically constructed x(t), which was the case in the classical x(t)2 .
We could try to use the combination relations to transform ω(n, n − (β − α)) into something more
convenient, but this appears to be futile. From here there is no possible way to end with the
frequency ω(n − α, n − β) which Heisenberg so mysteriously introduces.
When looking closer it is however possible to get exactly the relation Heisenberg called so
“necessary”. It is but a small step we have to do, to get an glimpse of Heisenberg’s computation
of x(t)2 . Starting with rewriting the left-hand side of equation [3.16]:
Aβ−α (n)ei(β−α)ω(n)t = Aβ−α (n)ei(βω(n)−αω(n))t
(3.17)
Now from the correspondence principle [3.5] we can make the transition to quantum theory and
get:
βω(n) ⇒ ω(n, n − β)
αω(n) ⇒ ω(n, n − α)
Our quantum theoretical expression for the phase of the latter part of x(t)2 , the second sum over
α in [3.12] hence becomes:
ei(ω(n,n−β)−ω(n,n−α))t
Following directly from the combination rules stated above, or the relation explicitly stated in
equation [3.13], we get this expression:
ω(n, n − β) − ω(n, n − α) = ω(n − α, n − β)
(3.18)
For the phase we then get:
ei(β−α)ω(n)t ⇒ ei(ω(n−α,n−β))t
(3.19)
So it appears that there is at least one justification possible for the expression of the frequency
in equation [3.14], which is a result of the multiplication of the two phases in [3.12] of which
the expression of one of the phases has been justified above. This gives an indication of how
Heisenberg could have come up with his quantum-theoretical x(t)2 . However, the expression for
the amplitude cannot be justified in the same manner.
If we just extrapolate the result to the amplitudes and using this in the classical expression
for x(t)2 of equation [3.9] we get a familiar expression. The combination relations can be applied
again to compute the quantum-theoretical x(t)2 and we get:
x(t)2 =
∞
X
∞
X
A(n, n − α)A(n − α, n − β)eiω(n,n−β)t
β=−∞ α=−∞
Which is exactly the expression Heisenberg states in his article, equation [3.14].
12
(3.20)
3.3
3 Heisenberg and Matrix Mechanics
Contextual
We now have a quantum-theoretical x(t)2 which is not lacking physical meaning; Heisenberg
notes that the phases have just as a physical significance as their classical analogues(Heisenberg,
1967[1925], p. 265). One can see this even better in the notation of Heisenberg:
∞
X
2
x(t) =
B(n, n − β)eiω(n,n−β)t
(3.21a)
β=−∞
with
B(n, n − β)eiω(n,n−β)t =
∞
X
A(n, n − α)A(n − α, n − β)eiω(n,n−β)
(3.21b)
α=−∞
So what can be said reflecting on the equation we now have? We can conclude Heisenberg constructed a new type of higher order term x(t)2 , which does not behave anything like classical
higher order terms. It is a new expression which is satisfied by every observable in the quantum
theory, it is a general form. With this expression, in this new theory, Heisenberg totally changed
what an observable represents in physics. It therefore also a beautiful event in the development
of scientific knowledge, where foundations are laid for a ‘true’ formalism which can be applied at
all times. Quantum theory at this point was accompanied by a genuine new quantum mechanics.
Taking off from this theoretical hypothesis, it is possible to construct a new type of multiplication rule in which the amplitude is constructed in for quantum-theoretical observables. This can
be done for any x(t)n , but also in a more general sense with two different quantum-theoretical
quantities, p(t) and q(t). If we characterize p(t) with amplitude P and q(t) with amplitude Q we
can compute the representations of a simple product; p(t)q(t), which we will call R.
Classical:
∞
X
Rβ (n) =
Pα (n)Qβ−α (n)
(3.22)
α=−∞
Quantum-Theoretical:
R(n, n − β) =
∞
X
P(n, n − α)Q(n − α, n − β)
(3.23)
α=−∞
Where the last relation is Heisenberg’s most (in)famous result. From this we can conclude that
quantum-theoretically, observable quantities do not behave as we are classically used of them. The
product between two observable quantities p(t)q(t) now is non-commutative!
p(t)q(t) 6= q(t)p(t)
or,
∞
X
P(n, n − α)Q(n − α, n − β) 6=
α=−∞
3.3
∞
X
Q(n, n − α)P(n − α, n − β)
α=−∞
Contextual
In the theory of quantum mechanics formalized by Heisenberg a giant conceptual step has been
made. After many years passed where physicists constructed new relations, based up on a hypothesis in which the property of continuous natural quantities was replaced by a system of
discrete values, Heisenberg introduced a radical new approach. An approach in which he defied
the classically determined laws of nature and opened an age with a new style of thinking. The
whole conceptual scheme of classical mechanics was thrown aside by his ‘Quantum-Theoretical
Re-Interpretation’, and older quantum conceptions were also disposed of. This departure from
old methods, we could say, was due to the specific style of Werner Heisenberg. His institutional
background, the people he worked with, his collaborations on early dispersion theory and even his
13
3.3
3 Heisenberg and Matrix Mechanics
Contextual
philosophical preferences12 determine his perception of physics and its problems, and thus also
the way one should approach these problems. In section 5 these consideration will be elaborated
some more.
In his article Heisenberg takes his new kinematics of quantum theory and then turns to solving dynamical problems with his new mechanics. He applies his expression to the equations of
motion which can now be solved quantum-theoretically and finally gives an example regarding
an anharmonic oscillator. This won’t be covered here since this only concerns application. After
the deduction of his multiplication rule Heisenberg also made other contributions to quantum
mechanics and worked on with the general rules he found and collaborated with other physicist
to find solutions to other, more complicated problems. Before this, Max Born and mathematical
physicist Pascual Jordan worked out the findings of Heisenberg even more elegantly. Most noticeably the multiplication rule following from Heisenberg’s formalism they recognised as the same
rule on which matrix multiplication takes place. It is exactly what happens when you multiply
‘rows times columns’. The next equation tries to exemplify this feature.
It is possible to define two matrices to get the following relation:

 
P1,n−α
Q1,1
 Q2,1
P2,n−α 
 
..  ×  ..
.   .
Pn,1 Pn,2 . . . Pn,n−α
Qn−α,1
P
P
Pn−α P1,n−α Qn−α,1 Pn−α P1,n−α Qn−α,2
 n−α P2,n−α Qn−α,2
n−α P1,n−α Qn−α,2


..
..

.
.
P
P
P
Q
P
n−α n,n−α n−α,1
n−α n,n−α Qn−α,2
P1,1
 P2,1

 ..
 .
P1,2
P2,2
..
.
...
...
..
.
Q1,2
Q2,2
..
.
...
...
..
.
Q1,n−β
Q2,n−β
..
.



=

Qn−α,2 . . . Qn−α,n−β
P

. . . Pn−α P1,n−α Qn−α,n−β

...
n−α P2,n−α Qn−α,n−β 

..
..

.
.
P
...
P
Q
n−α n,n−α n−α,n−β
(3.25)
In this equation β is constant, and n is a variable which is chosen constant. The term on the nth
row and (n − β)th column of the final matrix, we see, is represented by:
X
X
Pn,n−α Qn−α,n−β =
Pn,n−α Qn−α,n−β
(3.26)
n−α
α
Which is a general expression for all of the elements in the matrix, it is the definition of matrix
multiplication! The multiplication of a n by n − α matrix times a n − α by n − β matrix. Notice
that the summation in the earlier expression deduced by Heisenberg sums from −∞ to ∞ , which
means the observables are represented by an infinite dimensional matrix. The matrix which is
showed above then does not correctly represent the quantum observable, but it they gives a good
indication.
The multiplication of matrices is a rule which had already been formulated in 1855 by a the
French Mathematician Arthur Cayley, but never in the history of physics was this mathematical
tool used before. In a paper by Max Born and Pascual Jordan, which was the first reaction
on the work of Heisenberg, they explicated this fact. Their paper, ‘On Quantum Mechanics’
developed Heisenberg’s approach into “a systematic theory of quantum mechanics with the aid of
the mathematical matrix methods” (Born and Jordan, 1967[1925], p 277). It was from here big
steps were made towards a greater understanding of this new quantum mechanics. The further
development of this specific ‘Matrix Mechanics’ carries with it the names of Paul Dirac and
Wolfgang Pauli.
But as mentioned before, Heisenberg was not the only one who started a project for a new
mechanics of quantum theory.
12 Heisenberg is said to be impressed by Kant’s Critique but also by Wittgenstein’s writings (Jammer, 1966, p.
198).
14
4.0 Schrödinger and Wave Mechanics
4 Schrödinger and Wave Mechanics
4
4.1
Schrödinger and Wave Mechanics
Background
Only a short time after the publication of Heisenberg’s classical paper, Erwin Schrödinger also
submitted a paper in which he introduced a new way of looking at quantum theory13 . In the
79th volume of the Annalen der Physik Schrödinger wrote a paper on the deduction of the energy
spectrum of the hydrogen atom, and in this paper Schrödinger used an idea which is totally
unrelated to Heisenberg’s conception of quantum mechanics. Unlike Heisenberg, Schrödinger does
not stand in a tradition connected with the old quantum theory. His earlier works were on the
topics of kinetic theory of gases, statistical mechanics, elasticity and was more generally working
on the physics of continuous media (Jammer,1966, p. 255, 256).
Schrödinger was not part of the development of quantum theory as Heisenberg was, but it is
known that Schrödinger was influenced by two well-known persons in the field of quantum physics:
Albert Einstein and Louis de Broglie (idem., p. 257). The works of both physicists had great
implications for the constitution of matter and light. Where, as discussed in subsection 2.1.2,
Einstein introduced us with a conception of light which could both we wave-like and particle-like,
de Broglie did the same for the conception of matter. De Broglie suggested “that any moving body
may be accompanied by a wave and it is impossible to disjoin motion of body and propagation of
wave.”(De Broglie, 1924 in Jammer, 1966, p. 244). These physicists gave Schrödinger important
clues for his approach.14
In his paper series “Quantisation as a Problem of Proper Values” Schrödinger introduces a
mechanics for quantum theory based upon these considerations about the wave-like character of
matter and particle-like character of light. He states that he wants to show that “the customary
quantum conditions can be replaced by another postulate”(Schrödinger, 1926, p. 1). The notion
of quantum numbers should follow from this physical consideration, and Schrödinger writes that
these whole numbers appear like the “node numbers of a vibrating string.” 15 (idem.) He uses an
equation called the Hamilton-Jacobi equation to construct a variational principle which gives rise
to a differential equation which can be solved for a new introduced function Ψ. This means that
the earlier quantum conditions and hypotheses “ are replaced by this variation principle.”(idem.,
p.2). After he reached the differential equation Schrödinger tries to deduce the energy spectrum
of the hydrogen atom, this will however not be treated in this paper.
4.2
4.2.1
Calculation
Part I
In his paper Schrödinger almost immediately starts with his main deduction of a new mechanics,
a formalism which could explain the ad hoc conditions of the old quantum theory. He begins with
the Hamilton-Jacobi differential equation and this needs some explanation.
The Hamilton-Jacobi equation is a is a reformulation of classical mechanics, just like Lagrangian
or Hamiltonian mechanics. It comes forth out of a specific canonical transformation in Hamiltonian
mechanics. A canonical transformation in Hamiltonian mechanics is a transformation from one set
of co-ordinates (qi , pi , t) to a new set (Qi , Pi , t), under which the form of the Hamilton equations is
preserved without necessarily preserving the Hamiltonian itself (Goldstein, 1950, p.237,238). This
is done with the use of a, so-called, generating function S;
S (q1 , . . . , qn , t)
(4.1)
13 Heisenberg submitted his paper on the 29th of July 1925 whereas Schrödinger’s first part was received January
the 27th 1926
14 Schrödinger cites Louis de Broglie explicitely as an important influence (Schrödinger, 1926, p. 9)
15 This is exactly what happens when you solve the famous wave equation, and gives a beautiful picture of the
use of wave-phenomenon in Schrödinger’s thinking.
15
4.2
4 Schrödinger and Wave Mechanics
Calculation
In this context the solution S is called Hamilton’s Principal Function. It is related to the generalised momentum:
∂S
pi =
(4.2)
∂qi
(Goldstein, 1950, p.274)
What is important is that “when solving the Hamilton-Jacobi equation we are at the same time
obtaining a solution to a mechanical problem.” (Goldstein, 1950, p275). This is due to the fact that
the Hamilton’s Principal Function is the generator of a transformation to constant co-ordinates
and momenta. That is the reason the Hamilton-Jacobi formalism is used in physics and, in this
context, by Schrödinger16 .
Now for the equations. We are only concerned with Hamiltonians which do not involve time
explicitly. In this case the Hamilton-Jacobi equation has the following form:
∂S
=E
(4.3)
H qi ,
∂qi
Where E is a constant. The equation is just a partial differential equation for the function S.
This is Schrödinger’s starting point.
Now Schrödinger wants to find a solution to this equation such that the solution can be
represented as the sum of functions. He wants to have a function separable into the independent
variables qi . Due to this consideration Schrödinger puts “for S a new unknown Ψ such that it
will appear as a product of related functions of the single co-ordinates.” (Schrödinger, 1926, p.1,
emphasis in original).
S = K log Ψ
(4.4)
Where Ψ has no dimension, and Schrödinger introduces K for dimensional purposes, and has those
of action.
What does this form mean? When S needs to be represented as the sum of functions,
Schrödinger chose a new function Ψ such that he can achieve this. With the use of a natural
logarithm of a product of functions this is accomplished fairly easy, so that is where the new function Ψ enters. And that is also thuswhy Ψ appears as a product of functions. When we insert this
new S into equation [4.3] we get the following:
K ∂Ψ
H = qi ,
(4.5)
Ψ ∂qi
Without looking for the real solution of equation [4.5], we can now rewrite this equation in a
“quadratic form equated to zero”. This informally means no more then to write out the Hamiltonian for a system. In his article Schrödinger looks to describe the hydrogen atom. He thus uses
a potential energy of −e2 /r. Using the Hamiltonian of p2i /2m, equation [4.5] can be written in a
three dimensional Cartesian form:17
2 2 2 ! ∂Ψ
∂Ψ
∂Ψ
e2
1 K2
+
+
=
E
+
(4.6a)
2m Ψ2
∂x
∂y
∂z
r
Or;
∂Ψ
∂x
2
+
∂Ψ
∂y
2
+
∂Ψ
∂z
2
−
2m
K2
E+
e2
r
Ψ2 = 0
(4.6b)
∂Ψ
It is easy to recognise this form when one takes in mind the generalised momentum pi = K
Ψ ∂q .
This equation forms Schrödinger’s foundation, from here he continues with the main point of
his paper. This is his postulate which eventually drives him towards his new mechanics. It looks
as follows: For the quadratic form Schrödinger comes up with a condition which must be satisfied
to deduce all the old quantum conditions:
16 When interested in the formal aspects of the Hamilton-Jacobi theory I recommend to you the book on advanced
Classical Mechanics by Herbert Goldstein (1950)
17 The Hamiltonian in question Schrödinger calls “the Hamilton function for Keplerian motion”
16
4.2
4 Schrödinger and Wave Mechanics
Calculation
“We now seek a function Ψ, such that for any arbitrary variation of it the integral
of the said quadratic form, taken over the whole co-ordinate space, is stationary, Ψ
being real, single-valued, finite, and continuously differentiable up to the second order.”
(Schrödinger, 1926, p.2)
As quoted above, Schrödinger thus was looking for [4.6b] to be stationary when integrated over the
whole of co-ordinate space. But for what purpose would one state this? So reflecting, he takes the
integral over all space of H − E, because this is essentially what is expressed by [4.6b], and states
this integral should be stationary. You could say Schrödinger comes up with a new and stronger
condition for the function Ψ, replacing the the quadratic form of his Hamilton-Jacobi equation
[4.6b]. According to Schrödinger another condition needs to be satisfied to describe phenomena
correctly in the quantum regime. But why it should exactly be this condition stays unclear.
For now we could say this is an incomprehensible step in Schrödinger’s deduction of a quantum
mechanics, a step for which in his first article a explicit drive is missing. So in this section I will
just work on with this statement. In the second part Schrödinger’s interpretation of this step will
be discussed. First we look at the results of these conditions.
Schrödinger is stating that the equation must be stationary. This corresponds with equating
the variation of this equation to zero. From the discussed step thus the following appears:
"
2 2 2
#
ZZZ
∂Ψ
∂Ψ
2m
e2
∂Ψ
+
+
− 2 E+
Ψ2 = 0
(4.7)
δJ = δ
dxdydz
∂x
∂y
∂z
K
r
all space
Now we have an expression that is the mathematical equivalent of the mentioned quote. This
is a variational problem which can be solved fairly easy18 . We will now briefly dive into this
mathematical procedure.
First the variation can be distributed over the terms in the integral:
"
2 2 2
#
ZZZ
∂Ψ
∂Ψ
∂Ψ
2m
e2
dxdydz δ
δJ =
+
+
− 2 E+
Ψ2 = 0 (4.8a)
∂x
∂y
∂z
K
r
all space
ZZZ
=
#
" 2
2
2
∂Ψ
∂Ψ
2m
e2
∂Ψ
+δ
+δ
− 2 E+
δΨ2 = 0(4.8b)
dxdydz δ
∂x
∂y
∂z
K
r
all space
To insert the variation into the squared of the partial derivatives, one must proceed in the following
way:
2
∂Ψ
∂Ψ
∂Ψ
∂Ψ
∂
δ
= 2
δ
=2
δΨ
(4.9)
∂qi
∂qi
∂qi
∂qi ∂qi
∂
∂Ψ
∂2Ψ
= 2
δΨ − 2 2 δΨ
∂qi ∂qi
∂qi
2
2
1
∂Ψ
∂
∂Ψ
∂ Ψ
δ
=
δΨ −
δΨ
(4.10)
2
∂qi
∂qi ∂qi
∂qi2
Using this procedure in [4.8b] and computing the variation of Ψ2 gives:
"
ZZZ
∂ ∂Ψ
∂2Ψ
∂ ∂Ψ
∂2Ψ
δJ = 2
dxdydz
δΨ −
δΨ
+
δΨ
−
δΨ
+
∂x ∂x
∂x2
∂y ∂y
∂y 2
all space
(4.11)
∂
∂z
#
∂Ψ
∂2Ψ
2m
e2
δΨ −
δΨ − 2 E +
ΨδΨ = 0
∂z
∂z 2
K
r
18 In his paper at this point Schrödinger only mentions that one can find the solution “in the usual way”, no
derivation is given and the following then also is totally my own construction. This type of calculus was quite
omnipresent at that time.
17
4.2
4 Schrödinger and Wave Mechanics
Calculation
Note the factor of 2 in front of the integral, which besides in equation [4.10] also appeared in the
variation of Ψ2 ; δΨ2 = 2ΨδΨ. From here on we can collect terms to get a nice expression we can
further manipulate.
"
ZZZ
∂ ∂Ψ
∂ ∂Ψ
∂ ∂Ψ
δJ = 2
dxdydz
δΨ +
δΨ +
δΨ −
∂x ∂x
∂y ∂y
∂z ∂z
all space
(4.12)
#
2
∂2Ψ
∂2Ψ
2m
e2
∂ Ψ
δΨ +
δΨ +
δΨ − 2 E +
ΨδΨ = 0
∂x2
∂y 2
∂z 2
K
r
It is now evident that we can rewrite the newly formed terms inside the integral. First notice
the partials of x, y and z which together with the partial derivatives of Ψ form a divergence of a
gradient. Secondly notice the formation of a Laplacian operator working on Ψ times the variation
of Ψ. All together this forms:
"
#
ZZZ
2
e
2m
~ · δΨ∇Ψ
~
Ψ =0
(4.13a)
− δΨ∇2 Ψ − 2 δΨ E +
dxdydz ∇
δJ = 2
K
r
all space
or
1
δJ =
2
ZZZ
all space
ZZZ
e2
2m
~ · δΨ∇Ψ
~
Ψ = 0 (4.13b)
dxdydz ∇
−
dxdydzδΨ ∇2 Ψ + 2 E +
K
r
all space
In this form we can easily apply the divergence theorem on the first integral to get a final expression
which gives insight in what kind of conditions the variational principle of Schrödinger lead to.
I
Z Z Z
1
2m
e2
2
~
δJ = d~s · δΨ∇Ψ −
dxdydzδΨ ∇ Ψ + 2 E +
Ψ2 = 0
(4.14)
2
K
r
Where ds is an infinitesimal element of the infinite closed surface over which the integral is taken.
~ equates
The dot product of the gradient with the normal vector of the integration surface d~s · ∇Ψ
to a single differential since of course the dot product of perpendicular vectors results in zero.
Z
Z Z Z
1
∂Ψ
2m
e2
2
δJ = dsδΨ
−
dxdydzδΨ ∇ Ψ + 2 E +
Ψ =0
(4.15)
2
∂n
K
r
Here ends our mathematical venture, and what a fine result we have. Now a reflection is needed.
From this expression Schrödinger famously concludes a condition which at that time drastically
changed the field of physics; Schrödinger’s Wave Equation. Since the terms in previous expression
are independent they both must equate to zero, and therefore we can formulate the following
conditions:
2m
e2
Ψ=0
(4.16a)
∇2 Ψ + 2 E +
K
r
I
∂Ψ
dsδΨ
=0
(4.16b)
∂n
First note that equation [4.16b] can be ignored since it only says something which is already
known; Ψ should become zero at infinity (remember the integration is taken over an infinite
closed surface). The important result is thus reached with [4.16a]. From this point in his article
Schrödinger deduces the energy-levels of the hydrogen atom by solving this equation, and gets a
result which perfectly matches with Bohr’s calculation. Today it is known to every undergraduate
of Physics how to solve this equation, the ‘Schrödinger time-independent wave equation’ for the
hydrogen atom. I will hence not go into the calculation of the energy relation as Schrödinger does
in his article. It is not this that leads to a new mechanics.
18
4.2
4 Schrödinger and Wave Mechanics
Calculation
More important is to reflect upon the way Schrödinger took to get to the point of a governing
wave equation. Starting with the standard Hamilton-Jacobi differential equation, we now have
a condition for a new function Ψ from which the energy-levels of the hydrogen atom can be
perfectly calculated. The essential thing of this procedure, which is a start of a mechanics of
quantum theory, Schrödinger says to be “that the postulations of “whole numbers” no longer
enters into the quantum rules mysteriously, but that we have traced the matter a step further
back, and found the “integralness” to have its origin in the finiteness and single-valuedness of
a certain space function.”(idem, p9, quotation marks in original). By this he means that the
quantum-numbers, and thus the discreteness of observables, are no postulates but really follow
from a more fundamental condition. These quantum numbers come into play in solving [4.16a]
and implementing Schrödinger’s conditions for Ψ (e.g., single-valuedness).
But now remember what Schrödinger did at the start of his paper. He made a step in which
he states an incomprehensible condition for a newly introduced function Ψ. We could be sceptical
and conclude Schrödinger with this procedure only replaces the old postulates with a new one in
the form of this condition, from which the old one can be deduced on a rather elegant way. He
delays the inevitable.
Such interpretations are of course unnecessary and totally unwanted in a historical reflection
like this one. But it is in this sense interesting, and historically relevant, because of the second
paper Schrödinger published on this topic (which ended up in the same volume of the Annalen
der Physik of 1926). In this paper he himself takes on this point of view and rectifies the steps
taken in the paper we just discussed.
4.2.2
Part II
In Schrödinger’s second paper, properly named “Part II”, he proceeds on a manner totally unlike
his first part. “Part I” was a very chaotic paper, one in which there was no significant structural
approach towards the problem he wanted to solve. Conditions are thrown around and many
solutions are almost postulated without any derivation given at all. It gave rise to a feeling of
a sudden idea which he came up with and worked out as fast he could. This is stressed again
by the fact a correction by Schrödinger himself on his first article was submitted five days after
he submitted his second part.19 This chaotic feeling does not at all appear in his second part.
In Part II Schrödinger writes again three paragraphs on an new mechanics for quantum theory,
but now starts with a conceptual idea. He wants to use an old analogy, found by Hamilton, to
conceptually make clear where this quantum mechanics is coming from and how the equations that
must govern this mechanics arise. He starts with a reflection on his “unintelligble” transformation
of equation [4.4] in the first part and the “equally incomprehensible transition from the equating
to zero of a certain expression to the postulation that the space integral of the said expression shall
be stationary” (Schrödinger, 1926, p. 13), a daring but almost necessary statement since his first
part missed an important drive.
In the first paragraph of part II he explains this old analogy where after he concludes one
could expand this analogy further. This he does in the second paragraph and comes up with a
construction of a new mechanics. In the third paragraph he applies the vocabulary of the analogy
on some examples. We shall not reproduce all of his work in this paper, and focus on the fascinating
analogy Schrödinger used. The derivations will be brief, but it is the motivation which interests
us and which is also the main point of his second communication.
To understand his motivation we first must distinguish between two forms of optics, undulatory
optics and geometrical optics. Undulatory optics, or wave optics, is the optics generally is worked
with in physics. Also called physical optics, undulatory optics deals with phenomena of dispersion,
diffraction, interference and polarization, it is the optics of the propagation of waves and waves
fronts. In geometrical optics wave phenomena are approximated by rays and ray phenomena. One
here speaks of rays which determine the propagation of light, which can quite correctly describe
reflection and refraction in optics. Fermat’s principle is a most famous description in geometrical
19 The
first part was submitted 27 January, the second communication 23 February and his correction on the first
part the 28th of February.
19
4.2
4 Schrödinger and Wave Mechanics
Calculation
optics. Formally geometrical optics is the short wavelengths limit of undulatory optics, i.e. with
very short wavelengths wave propagation can be approximated by rays. Hamilton found out that
he could describe both geometrical optics and mechanics with the same formalism, the HamiltonJacobi formalism; “With the help of his characteristic function he discovered that the actual motion
of a point mass in a field of force is governed by the same formal law as the propagation of the rays
of light.” (Jammer, 1966, p. 236). This analogy is what Schrödinger expounds in paragraph one.
In a somewhat more modern interpretation the analogy, and the first paragraph of Schrödinger’s
second part, may to be summarized as follows.20
R
We start with Hamilton’s variational principle; δ L dt = 0. From this relation one can derive
Maupertuis’ principle of least action for systems with a constant energy:
Z
δ 2T dt = 0
(4.17a)
Or:
δ
Z p
2m(E − V ) ds = 0
(4.17b)
With this we have a formulation of the equations of motion for a physical system, not as differential
equations but as an integral equation. This we can compare with Fermat’s idea on rays of light,
the basis of geometrical optics. Fermat’s principle of least time can formally also be expressed as
a variation principle:
Z
n
δ
ds = 0
(4.18)
c
With n the refractive index. It immediately becomes clear that there is a similarity between [4.17b]
and [4.18]. One is a variation for mechanical problems, the other for problems in geometrical optics
and still they have much in common. If we in optics formulate the phase velocity, u = c/n, we
see that the expression C[2m(E − V )]−1/2 plays in mechanics the same role as this phase velocity!
This can be represented as follows:
u=
c
⇐⇒ C[2m(E − V )]−1/2
n
(4.19)
Where C denotes a constant. This is an analogy which was the result of the theoretical research by
Sir William Rowan Hamilton, a result which was published between 1828 and 1837. “Hamilton’s
optical-mechanical analogy”, as we will call it, can be extended further. First, one can say the
action function S defines an action surface, S(qi , t)=constant for which we have p = ∇S and
∂S/∂t = L − pv = −E. Second, in the optic region we can express the phase in terms of the
frequency and the wave vector k, φ(ω, t) = −ωt + kr. From we can construct the following
relations: k = ∇φ and ∂φ/∂t = −ω. This suffices to see a complete formal analogy between the
surface of constant action of a mechanical system and the surface of constant phase in optics. The
wave vector corresponds to the momentum and the frequency to the energy of the particle.
Schrödinger ends the paragraph with a discussion on what to do with this analogy. Maybe one
cannot draw the analogy any further, and should a real parallel between optics and mechanics not
be made. He continues:
”But even the first attempt at the development of the analogy to the wave theory leads
to such striking results, that a quite different suspicion arises: we know to-day, in fact,
that our classical mechanics fails for very small dimensions of the path and for very
great curvatures. Perhaps this failure is in strict analogy with the failure of geometrical
optics, i.e. “the optics of infinitely small wave lengths”, that becomes evident as soon
as the obstacles or apertures are no longer great compared with the real, finite, wave
length. Perhaps our classical mechanics is the complete analogy of geometrical optics
and as such is wrong and not in agreement with reality; it fails whenever the radii of
curvature and dimensions of the path are no longer great compared with a certain wave
20 In
this summary I will briefly follow Max Jammer’s formulation (Jammer, 1966, pp 236,237).
20
4.3
4 Schrödinger and Wave Mechanics
Contextual
length, to which in, q-space, a real meaning is attached. Then it becomes a question of
searching for an undulatory mechanics, and the most obvious way is the working out
of the Hamiltonian analogy on the lines of undulatory optics.”
(Schrödinger, 1926, p. 18. Emphasis in original.)
Further considerations presented in Schrödinger’s second part will not be discussed. In his
second paragraph Schrödinger does so, but these are compared with the main argument elaborated
above not noteworthy. The most important consideration forms out of this quite naturally: “...we
must treat the matter strictly on the wave theory, i.e. we must proceed from the wave equation
and not from the fundamental equations of mechanics, in order to form a picture of the manifold
of the possible processes. These latter equations are just as useless for the elucidation of the
micro-structure of mechanical processes as geometrical optics is for explaining the phenomena of
diffraction.” (idem., p. 25). Analogous to undulatory optics there must also exist a undulatory
mechanics of which classical mechanics is a limiting case.
We have thus seen his wave equation appear in part I, and in Part II a most extraordinary
justification of the use of the wave equation came forth. It strikes really deep into a physical
understanding of the world. It is a theory in which different phenomena can be explained from
the same principles, a venture which modern string theory wishes to accomplish on a larger scale
even today.
4.3
Contextual
Schrödinger eventual submitted four communications in which his findings were more and more
elaborated. In the third communication he goes into in perturbation theory, with applications to
the stark effect of the Balmer lines very elaborately. And in the fourth a series of independent
subjects is treated on all of which his new mechanics is applicable. In the end Schrödinger manages
to elaborate an old existing formal system, lost and forgotten, and give in new life.
It seems this day however the work of Schrödinger can, in a tragic way, be compared with
the work of his predecessor, Sir Hamilton. Hamilton’s famous mechanics was only elaborated
disjointed from his optics, Hamilton’s optics reappeared later in the theory of the eikonal21 , again
disjointed. His proposition of a synthesis of a formalism for mechanics and geometrical optics was
ignored. When we today look in our highly synthetic and didactic graduate books on Quantum
Mechanics one can find no trace of Schrödinger’s initial drive towards a wave equation of micromechanics. If this is a good or a bad thing for physics one must decide for oneself, but I think it
could be said a beautiful work of physical apprehension, useful or not, is to be forgotten.
In his approach of quantum theory Schrödinger did not do away with the old physics. He saw
in the old quantum conceptions no need for a new beginning but the need for a sequel, a sequel of
the old theories of optics and mechanics. No new firmaments needed to be laid down, it was all
part of a bigger whole. One can with this observation notice fundamental differences in the style
of Heisenberg compared with that of Schrödinger. But it also appeared that both formalism, how
different their origins may be, formally have much in common. Consideration of this type will be
discussed in the following section.
21 The eikonal equation is a partial differential equation useful in problems of wave propagation. It provides a
link between undulatory optics and geometric optics.
21
5.0 Discussion
5 Discussion
5
Discussion
“Ever-newer waters flow on those who step into the same rivers.”
Heraclitus
In previous sections we have seen the intriguing mathematical and physical steps both Heisenberg
and Schrödinger undertook to create a new formalism for quantum theory. A formalism from
which all old quantum conditions naturally appear. Using concepts which differ radically from
each other, both physicists managed to formulate a theory which eventually could explain the
quantum nature of our experience. Notice it is very extraordinary that these two theories were
formulated parallel to each other, independently. We are speaking of the parallel development of
two totally different explanations to describe the very same range of experience.
It is most interesting to reflect upon this fact. I will therefore now consider some topics for
which this development I think has, or could have, profound implications.
5.1
Style and Stylistic Differences
The development in question exemplifies a fundamental aspect of scientific endeavour; the concept
of style. I like to say Heisenberg and Schrödinger came up with different theories because they both
had a very different background in physics, institutions but also in life as a whole. A difference
which ends up in a different style of thinking.
Heisenberg formulated a real new mathematical calculus to cope with the existing problems.
A calculus where special computational rules were involved and quantities turned out to be noncommutative. Where the latter aspect was something rarely accounted before. The basis for
the theory was Bohr’s work on spectral lines, where discreteness was introduced in postulation
and formed the foundation of quantum theory. Therefore this element of discontinuity was also
deeply embedded in the Heisenberg’s mechanics. Looking chiefly at the mathematical properties
of quantum-theoretical quantities made it also a real algebraic approach. This together made the
theory of Heisenberg conceptually very hard to understand, and resisted any possible graphical
interpretation. Finally we can also state that matrix mechanics is, in the end, based upon the
concept of a particle (or corpuscle), a discrete entity who governs the micro world in his interactions
and possible transitions (Jammer, 1966, p. 271). He had an abstract mathematical style of
approach. Why does the theory of Heisenberg have all these characteristics? Is this a physical
truth, or rather a consequence of other elements?
Totally unlike Heisenberg, Schrödinger worked with a mathematical apparatus which had great
familiarity among physicists; differential equations. Further characterized by a more analytical
approach with the help of the discussed analogy, Schrödinger manages to get a conceptually very
clear theory. This analogy also stresses his dependence on the element of continuity instead of
discontinuity, in principle working with an extension of the classical laws of motion. And again,
the analogy naturally gives also the concept of wave a prominent place in Schrödinger’s mechanics,
depending on only the wave equation as both classical mechanics and geometrical optics depend on
the Hamilton-Jacobi equation (Jammer, 1966, pp. 271,272). Schrödinger had a more Anschaulich
physical style. We again see specific elements of this theory which are used by Schrödinger, why
are these and not others incorporated in the theory?
Looking at these consideration one could say that we are dealing with two radically different
thought styles 22 . Due to the different background, in physics but also in general, of both physicists
they have a different directed perception of the world they investigate. This background could be
defined as them being a member of a specific thought-collective, “a community of persons mutually
exchanging ideas or maintaining intellectual interaction” (Fleck, 1935, p. 39). In such collective
22 A term which is introduced by Ludwik Fleck in his magnum opus “Genesis of a Scientific Fact” (1935). The
philosophical notes on style in this paper are largely influenced by his work on the philosophy and sociology of
science.
22
5.2
5 Discussion
Scientific Knowledge
one shares the meaning of different concepts, and they are together in control of the meaning
assigned to concepts. Scientists possess a style and are member of one or more collectives23 , where
of course both style and collective are intimately related to each other. With this conceptual
toolbox we can thus more elegantly describe the above development.
Heisenberg’s work on dispersion theory was a key influence on the development of his thought
style. As an apprentice of Born and having collaborated with Kramers and Bohr (van der Waerden,
1967, p. 17) he became a member of their thought collective, and shared the problems they were
working on and the way they treated those problems. His perception became directed in this
particular way. Schrödinger’s early work on continuous media and the applications of eigenvalue
problems made that Schrödinger had a special readiness for directed perception, directed towards
a continuous interpretation of quantum theory governed by differential equations. He could see
the wave mechanics where Heisenberg saw matrix mechanics. They both saw something different,
and assigned a different meaning to it because of their different readiness of perception.
It is thus with studying such a strange development, where two different theories were formed
at the same time, to cover the same range of experience, one can clearly see that science is an
enterprise in which different styles exist which direct toward a different perception of the field and
it problems. And that real “true” descriptions of our experience do not come forth, just because
we have a different experience coupled to styles and collectives. Even physics is no rigid structure
where one has a well determined and true concept. It is an ever-changing system of interacting
thought-styles and collectives.
Really interesting is the fact that Erwin Schrödinger also wrote a paper on the relation between
his and Heisenberg’s mechanics. He concludes in his paper a formal equivalence of both theories.
We will not consider the formal equivalence itself, but it does show in physics unity is needed
in the concepts they use. The thought collective of the field of physicists must be as smooth as
possible, but fluctuations of course always exist; fluctuations in the collective but also in the style of
individuals, of which the paper’s subject is great example. It is here that in quantum mechanics,
after this paper about the formal equivalence, the styles of Heisenberg and Schrödinger have
merged. To clarify: In physics we all use, for example, the same Maxwell-equations. But this does
not mean that this is just a simple fact, these equations were also constructed by physicists which
each a different styles of thought, and also has dynamic history. And even now these equations are
determined they are still not rigid and mean different things, concerning modern developments,
but also concerning individuals.
5.2
Scientific Knowledge
There are two more point which I would like to make concerning a discussion on what the parallel
development of wave and matrix mechanics can learn us about science. The first point has to
do with how a theory like quantum mechanics, and then particularly the discussed formalisms, is
formed and discovered. I think namely that it gives a beautiful example of how should see this
concept of discovery. We have seen a very complicated process in which many steps occurred in
finding a formalism for quantum theory. This process was guided by the different style of the
physicists. And it is this process I want to emphasize. In the process of working their way towards
a fully fledged theory from out their own style, they created knowledge of how one genuinely
could describe this world. Could one thus say it is here were it really is discovered that Quantum
Mechanics exists? I think this is a bad interpretation. A discovery is not a matter of ‘uncovering
reality’, uncovering the facts. We have seen people actively participate and create this mechanics,
using their own style to construct it. Facts are not passively discovered, they are actively created.
How they are created depend on the different styles and collectives in question.
The second point concerns a reflection upon our own place in physics. In today’s physical endeavours and teachings the process of the discussed development is of course no longer
present. The whole process in which quantum theory has developed, from Planck to Heisenberg
23 One is of course a member of a thought collective implicitly, just like most are not aware of their own thought
style.
23
5.2
5 Discussion
Scientific Knowledge
and Schrödinger, will no longer be represented with all details of construction and many steps of
the process will never be looked at again. I think that eventually the actual course of the development becomes rationalized and schematized. We no longer see the moment which physicists were
constructing the now determined facts of quantum mechanics. The long path which came before
the construction of the formalism, where there was no defined quantum theory and strange steps
were taken to acquire workable equations, is rationalized and one again begins thinking in the
static accumulation of facts, false and true. Preventing this ‘whiggish’ type of history of science,
I think reflections of such types presented here could act to reflect upon the scientific enterprise
in which one places themselves. And I would even say that reflections of the kind presented in
this paper are essential for a conceptual understanding of a physical theory, looking back at the
process and trying to understand in a less rationalized way. But even more to learn how science
works. We can learn that we also possess a style of thought and we also are member of thought
collectives which directs our perception of science.
24
6.0 Conclusion
6 Conclusion
6
Conclusion
In this paper the development of a formalism for quantum theory by Erwin Schrödinger and
Werner Heisenberg has been discussed. It became clear that from completely different conditions
and tenets, both physicists independently and parallel to each other, created a mechanics which
could describe the exact same range of experience and account for the concept of line spectra.
Heisenberg’s work builds upon the old quantum mechanics. He looked what the characteristics
were of quantum-theoretical observables and tried to compute a generalised quantum-theoretical
quantity. For this he used an analogous form of the classic Fourier expansion, which he extended
to the region of quantum theory. With this new type of expansion he tried to compute a quantumtheoretical x(t)2 for which he deduced a new type of multiplication rule. This was later recognised
as the multiplication of matrices, and observables in quantum theory turned into infinite dimensional matrices. Schrödinger approached quantum theory in a totally different fashion. It appeared
in the second part of his paper Schrödinger got inspired by an old analogy worked out by Hamilton. Since geometrical optics was a small wavelength limiting case of wave optics, and geometrical
optics had through the Hamilton-Jacobi equations a formal analogy with classical mechanics, this
new quantum mechanics of which classical mechanics was a limiting case should be governed by
the same formalism as wave optics; the wave equation. But in his first paper Schrödinger used
an “unintelligible” variational principle to come up with the same equation, and no analogy was
used to justify his steps.
The different formalisations of both physicist I tried elegantly to describe by their difference
in style. I tried to the determine different styles and collectives in which these physicists held
place. Both their style and the collective they communicated with directed their perception in
specific ways, ways in which they could see both a different quantum theory. Realising this, such
an historical reflection as presented above could become an important tool for reflection upon the
scientists own discipline and their place in it. One sees the flux of scientific enterprise where there
are no rigid conceptions available, and not the static discoveries in history which today seem so
natural. It could also be discussed regarding the above historical reflection, that such reflections
are in some way essential for the conceptual understanding of modern theories
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7.0 Acknowledgements
7 Acknowledgements
7
Acknowledgements
The author would like to thank Prof. Dr. A.J. Kox for his professional guidance into the world of
the History of Physics. Under his supervision it became clear how important it sometimes is to
reflect upon the history of your own discipline, to not only gain consciousness of your position in
the field but also to learn a lot about your discipline and even science itself.
The author would also like to thank Prof. Dr. E.P. Verlinde for a short informal discussion on
his view of the origin of Heisenberg’s matrix mechanics.
26
8.0 Bibliography
8 Bibliography
8
Bibliography
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