Foreword
This is not just another textbook about quantum mechanics as it presents quite a novel, axiomatic path from classical to quantum physics. The novelty begins with the description of classical mechanics, which rests on Eulerâs and Helmholtzâs rather than Newtonâs or Hamiltonâs representations. Special attention is paid to the commons rather than to the differences between classical and quantum mechanics. SchrĂ¶dingerâs 1926 forgotten demands on quantization are taken seriously; therefore, his paradigm âquantization as eigenvalue problemâ is replaced with Einsteinâs idea of âquantization as selection problemâ. The SchrĂ¶dinger equation is derived without any assumptions about the nature of quantum systems, such as interference and superposition, or the existence of a quantum of action, h. The use of the classical expressions for the potential and kinetic energies within quantum physics is justified.
A doubtless benefit of this textbook is its extensive reference to original texts. This includes many details that do not enter contemporary representations of classical mechanics, although these details are essential for understanding quantum physics.
Another benefit consists in that it addresses not only students and scientists, but also teachers and historians; it sheds new light on the history of ideas and notions. The level of mathematics is seldom higher than that of the common (Riemannian) integral. Basic notions and quantities are carefully introduced; steps like âIt can be shown, that textellipsisâŠâ or âItâs sink or swimâ are successfully avoided.
âFrom Newton to Planck and even more. The mix of exacting physics, biography and history exhibits its quite own charm and quickly fascinates the reader â in particular, when nowadays physical equations and the historical literature are so masterly interwoven as the paramount historical figures with their epochal works.â (Carsten Hansen, review on buchkatalog.de)
âI would like to express my thank to the author for this book. It contains an impressive and comprehensible derivation and representation of quantum physics. Due to his approach, I have eventually found an approach that is acceptable for me and not purely formal. The way from classical to quantum physics is impressively understood. Using a profound knowledge of the historical literature, including the original texts, the Schroedinger equation is derived through an extension of Eulerâs and Helmholtzâs representations of classical mechanics to nonclassical systems. It is the concentration on the commons rather than on the differences between classical and quantum systems that makes this approach reasonable. Moreover, this makes the interpretation and meaningfulness of the quantummechanical models and concepts clearly visible.â
Preface
The Preface provides the reader with a very general overview over the goal and the approach of this book, including the new features against previous publications of its author. It emphasizes the need for studying the original texts and stresses, in particular, SchrĂ¶dingerâs discarded requirements to any path from classical to quantum mechanics. Further keys are, (i), Newtonâs forgotten notion of state, (ii), Eulerâs unknown axiomatic of classical mechanics, (iii), Helmholtzâs concept of configurations, (iv), Helmholtzâs less known foundations of energy conservation, (v), Einsteinâs overlooked idea of quantization as selecting the quantum states of a system out of its classical states (discrete versus continuous energy spectrum). Inspired by GĂ¶delâs first incompleteness theorem, the path from classical to quantum mechanics is sought for and found through a question, which can be posed, but not be answered within classical mechanics. The answer to this question provides a novel definition of quantum mechanics. Quantum mechanics is not intuitive, but classical mechanics is not either.
â... so oft ich mich mit den grundlegenden Arbeiten unserer groĂen Meister unmittelbar vertraut machte, hatte ich einen Gewinn an Einsicht und VerstĂ€ndnis zu verzeichnen, der weit ĂŒber das hinaus ging, was aus den sekundĂ€ren Quellen, den LehrbĂŒchern und dergleichen zu entnehmen war.â
âEs ist dies [der Mangel der sonst bereits erfolgreichen Vermittlung wissenschaftlicher Kenntnisse] das Fehlen des historischen Sinnes und der Mangel an Kenntniss jener groĂen Arbeiten, auf welchen das GebĂ€ude der Wissenschaft ruht.â
âWer ĂŒber die handwerkliche Handhabung von Physik hinausgehen will, muss sich zwingend mit der Frage befassen, was physikalisches Denken ausmacht und wie es entstanden ist.â
Why another book about quantum mechanics, and why a physical rather than a philosophical treatise about the relationship between classical (CM) and quantum mechanics (QM)? Because it treats both CM and QM such, that their commons are stressed, while their differences are developed in a natural and smooth manner. This serves not only the unity of physics, but also the understanding of both.
The basic features of CM needed here are exposed along Newtonâs and Helmholtzâs original texts and complemented by Eulerâs forgotten axiomatic of CM . The following transition to QM is guided by,

Einsteinâs 1907 idea of quantization as selection problem,

SchrĂ¶dingerâs 1933 thoughts about the relationship between CM and QM,

SchrĂ¶dingerâs 1926 requirements to any transition from CM to QM.
For the sake of the unity of physics, Hertz had required to represent CM such, that all other branches of physics can be derived from it (âHertzâs programâ). The feasibility of Hertzâs vision will be shown.
Special attention is paid to carefully treating the basic notions. For the notions are the tools of thinking, so that the accuracy of science is by no means better than the accuracy of the notions used. One of the bestknown historical examples is the confusion of energy and force.
Although dealing with basic topics of CM and QM and being selfcontaining, this book does not represent complete introductions into CM respectively QM as there is few account for their experimental bases.
Despite of presenting a nonconventional approach to QM, it presents new results. Among others, for a free particle, a squareintegrable wave function is derived, and the methodologically interesting similarities between its paradigm âquantization as selection problemâ and Mittelstaedtâs reconstruction of QM are considered.
One of the most striking â and often considered to be âthe only mysteriousâ â feature of quantum physics is the âinterference of particlesâ. Its similarities to the interference of classical waves led to the term âwaveparticle dualityâ. This duality is often taken as starting point for introducing quantum physics. On the other hand, the notions âwaveâ and âparticleâ are classical ones, so that the apparent duality arises from the assignment of classical properties to nonclassical objects.
Thus, this book aims at an axiomatic foundation of nonrelativistic QM. For this, it reconsiders the historical development, where it concentrates on aspects, which have been discarded, although being crucial for reaching that goal. It presents an axiomatic derivation of the stationary and timedependent SchrĂ¶dinger equations. âAxiomaticâ means, that features like quantum of action, waveparticle dualism, indistinguishability and uncertainty are deduced rather than postulated.
The possibility of that is usually disbelieved. Admittedly, it became possible only through exploiting largely discarded results, notably, Newtonâs notion of stationary state as well as Helmholtzâs approach to the energy conservation law and concept of configurations. Even more important is Eulerâs forgotten axiomatic of CM. In contrast to Newtonâs axioms, it is not tied to the motion along trajectories. This makes it possible to abandon the trajectories without touching the axiomatic.
Usually, quantum physics is perceived as being counterintuitive. However, one can hardly expect QM to be intuitive, when even common CM is not. For instance,

if Newtonâs 1st axiom (Galileoâs law of relativity ) were intuitive, already Aristotle had stated it ;

if Newtonâs 2nd axiom (the change of the momentum vector is proportional to the vector of the external force applied) were intuitive, it would be taught in school (instead of the oversimplification âforce = mass Ă accelerationâ);

if Newtonâs description of the force of gravity in the Definitions of the Principia were intuitive, the notion of field would have been ascribed to him rather than to Faraday ;

if Newtonâs representation of CM were intuitive, it had not taken 100 years to accept it, and the confusion between âforceâ and âenergyâ had been settled much earlier, etc.
Moreover, as observed by the pioneers of QM (notably, Bohr 1913, Heisenberg 1925, SchrĂ¶dinger 1926), there is no smooth way from Newtonian or canonical CM to QM. On the contrary, there is a smooth way from Eulerian CM to QM, and this way will be described here.
But what is the motivation for changing from CM to QM? Historically, or empirically, it was the inability of CM to explain certain experimental results. Axiomatically, it is an intrinsic limitation of CM. Such a limitation is suggested by GĂ¶delâs incompleteness theorem , viz, to pose questions, which can be stated, but not be answered within CM.
To be specific, let me recall the following. The set of configurations a classicalmechanical conservative system is able to assume is limited by the energy law. For instance, it makes the motion of an oscillator like a pendulum or spring to be bounded by the two turning points. Therefore, the question reads,
How the mechanics of oscillators without turning points looks like?
This question can be posed, but not be answered within CM. For answering it, Helmholtzâs analysis of the relationships between (momentum) configurations and (kinetic) potential energy is extended to the classically forbidden sets of (momentum) configurations. For an oscillator, these are the sets beyond the turning points. This will lead us to the stationary, timeindependent SchrĂ¶dinger equation. The nonstationary, timedependent SchrĂ¶dinger equation will be obtained through generalizing Eulerâs principles of stationarystate change for classical bodies, first, to classical conservative systems and, then, to quantummechanical systems.
When treating the symmetry of quantum systems, the approach of this book tells immediately, that stationary quantum state quantities assume the symmetries of the corresponding classical quantities. Combining that with a generalization of Helmholtzâs explorations about the relationships between forces and energies leads to a nonclassical, quantum class of interactions, viz, to the gauge invariance of the SchrĂ¶dinger equation and to the (EhrenbergSiday)AharonovBohm effect. In turn, this suggests an approach to the unification of classical mechanics and electromagnetism.
Thus, this book is about the axiomatic derivation of the SchrĂ¶dinger equation and its nonclassical solution rather than about its interpretation. That derivation corroborates SchrĂ¶dingerâs view of Ï(x)^{2} to represent a weight [density] function (1926, 4^{th} Commun., Â§ 7) and reveals x_{ch} Ï(x,t)^{2} V(x,t) and p_{ch} Ï(p,t)^{2} T(p,t) to be the effective potential and kinetic energies, respectively, âseenâ by a quantum particle (x_{ch} and p_{ch} being characteristic extensions). This view is compatible with the quantumlogical interpretation, that QM is just the description of the outcome of measurements of quantum objects by means of quantum methods.
The mathematics used is not more complicated than the Fourier transform . The physical content is accessible even for highschool pupils. For one goal of this book is to free both teachers and pupils from the fear against quantum physics. It should be used as a companion to lectures that provide experimental results having forced the physicists a century ago to search for a novel description of atoms. These experiments provided the facts the novel, nonclassical mechanics had to describe â on the other hand, they gave virtually no hints about the way, how to go axiomatically from CM to QM. An axiomatic way is sought â last but not least â for the sake of the unity of physics.
This book represents a complete revision of my 2006 book From Classical Physics to Quantum Physics. Selection and arrangement of the material as well as many reasonings have been largely changed. The exposition of CM concentrates on the needs for axiomatically deriving QM. The chapters on solidstate theory (except Blochâs theorem and phonons) are omitted as not being related to the axiomatic issues. Accordingly, the citations and quotations have been revisited and updated. The problems have been adapted and extended; an asterisk â*â marks suggestions for own research (and I would be happy to participate in it!). Moreover, symmetry is dealt with in an own part, in order to highlight the benefits of this approach and to scope with new results. The ideas on field quantization in Â§ 9.2.5 of the 2006 book are revised and exposed in much more detail in a new part, too. The index has been extended accordingly.
Major changes concern also the account for Mittelstaedtâs recent book Rational Reconstructions of Modern Physics and the representation of Newtonâs axioms, of the notion of state, of the selection problems, of the representation of the energy of a QM system, and of the recursion relations. The Fourier transform between the wave functions in position and momentum representations is now established by means of a single (new) argument. For free spinless particles in whole space, R^3, squareintegrable wave functions are derived. Chapter 1 about the conservation and changes of stationary quantum states has been completely rewritten. Chapter 5 âAequat causa effectumâ presents not previously published results about a common feature of the symmetries of classical and quantum systems. Section 4.2 with the completion of SchrĂ¶dingers transition from QM to CM by means of coherent states is new, too. There is a new Chapter 2 about the energy law as foundation of CM. It discusses Planckâs 1909 treatment in view of Eulerâs forgotten results in the Anleitung and Carlsons 2016 approach. Subsection 2.3.2 about energy and extension in momentum space is improved by means of the hodograph. The treatment of tunneling has been extended to Bohmian formulae. In connection with that, there is a new Section 3 about the benefits of our approach for treating conservations of weight (probability) and energy.
Sections not being necessary for understanding the essentials of this book are marked with an asterisk (*).
First of all, and again, I feel most indebted to my friend and colleague Dr. Dieter Suisky. He has discovered the power of Eulerâs axiomatic of CM for tackling QM and special relativity, and he has extracted from Einsteinâs 1907 pioneering paper on the specific heat of solids the paradigm âquantization as selection problemâ. I am also still indebted to all the colleagues mentioned in my foregoing book. The discussions with Prof. Sigurd Schrader and his coworkers at the Technical University of Applied Sciences Wildau as well as with my friend and former colleague Dr. Michael Erdmann have encouraged me to change substantially the order as well as the manner of exposition of the material to the present form.
Moreover, I thank Dr. Christian Baumgarten for useful hints, Prof. Shawn Carlson for the discussions about his âPrinciple of Dynamic State Equilibriumâ prior publication, Prof. Maurice A. de Gosson for elucidating various aspects of quantization, Dr. Pekko Kuopanportti for hints on the energy density, Prof. Peter Mittelstaedt for explanations to his book Rational Reconstructions of Modern Physics and Prof. Michael MĂŒllerPreuĂkerâ for his hints about gauge theory. The discussions with Prof. GĂŒnther Nimtz have very much clarified my picture of the timedependent âtunnel effectâ. I feel indebted to Prof. Harry Paul for earlier discussions and for sending me his contribution about the blackbody radiation, Prof. Lev Petrovich Pitaevskii for insightful remarks on the energy density, Prof. Joseph Rosen for his comments on the relationship between symmetry, cause and action, and Prof. Robert H. Swendsen for discussions on (in)distinguishability and for providing texts prior publication. Dr. Eugene V. Stefanovichâs work has influenced this book much more than being visible, and thus I feel highly indebted to him for his numerous additional explanations. Prof. Norbert Straumann has pointed to the Hilbert space formalism as a reason for the fact that SchrĂ¶dingerâs own criticism of his derivation of wave mechanics has eventually been discarded. Prof. Johannes Richter has provided valuable explanations about localization in ideal crystals (flatband systems).
My friend and schoolmate Dr. Wolfgang Peters posed many stimulating philosophically driven questions and gifted me Torrettiâs book. My friends and former colleagues Dr. Andreas FĂ¶rster and Andreas Rothenberg have made several valuable proposals for the text.
Moreover, numerous suggestions and hints to references I have obtained from the newsgroup sci.physics.foundations. The teaching at the Institute of Physics, Mathematics and Informatics at the Abai University Almaty, Kazakhstan, was a source of rethinking several issues.
I am also indebted to anonymous referees for pointing to various weaknesses in the draft. I hope to have eliminated (most of) the mixing of introducing and advanced paragraphs.
Fig. 1. Science should make fun (source: ericweisstein.com).
Despite of the varying quality of its articles, Wikipedia has become a most valuable reference to biographical and bibliographical data. This makes every euro I have donated to it to be worth it. Thus, I feel highly indebted to all the enthusiasts and projects making the wisdom of the past and present available in the web.
This book has been typeset using TeXstudio and pdfLaTeX. I thank the German TeX user group Dante e.V. (http://www.dante.de) as well as the many enthusiasts on the www TeX help pages for their support.
Peter Enders
Senzig, August 2018
CONFLICT OF INTEREST
The author declares no conflict of interest, financial or otherwise.
ACKNOWLEDGEMENTS
My beloved parents, Dr. Lieselott Enders and Dr. Gerhart Enders, gifted me a universal humanistic education and the freedom of independent thinking. My faithful wife, Galina Nurtasinowa, has shared the up and downs of writing, again.