Classical Mechanics and Quantum Mechanics: An Historic-Axiomatic Approach


Peter Enders

eISBN: 978-1-68108-449-7, 2018
ISBN: 978-1-68108-450-3

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This unique textbook presents a novel, axiomatic pedagogical path from classical to quantum physics. Readers are introduced to the des...[view complete introduction]


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 Euler’s and Helmholtz’s rather than Newton’s or Hamilton’s representations. Special attention is paid to the commons rather than to the differences between classical and quantum mechanics. Schrödinger’s 1926 forgotten demands on quantization are taken seriously; therefore, his paradigm ‘quantization as eigenvalue problem’ is replaced with Einstein’s idea of ‘quantization as selection problem’. The Schrödinger 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 “It can be shown, that textellipsis…” or ‘It’s sink or swim’ are successfully avoided.

“From Newton to Planck and even more. The mix of exacting physics, biography and history exhibits its quite own charm and quickly fascinates the reader – in particular, when nowadays physical equations and the historical literature are so masterly interwoven as the paramount historical figures with their epochal works.” (Carsten Hansen, review on

“I 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 Euler’s and Helmholtz’s representations of classical mechanics to non-classical 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 quantum-mechanical models and concepts clearly visible.”


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, Schrödinger’s discarded requirements to any path from classical to quantum mechanics. Further keys are, (i), Newton’s forgotten notion of state, (ii), Euler’s unknown axiomatic of classical mechanics, (iii), Helmholtz’s concept of configurations, (iv), Helmholtz’s less known foundations of energy conservation, (v), Einstein’s overlooked idea of quantization as selecting the quantum states of a system out of its classical states (discrete versus continuous energy spectrum). Inspired by Gödel’s 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.

“... so oft ich mich mit den grundlegenden Arbeiten unserer großen Meister unmittelbar vertraut machte, hatte ich einen Gewinn an Einsicht und Verständnis zu verzeichnen, der weit über das hinaus ging, was aus den sekundären Quellen, den Lehrbüchern und dergleichen zu entnehmen war.”

“Es ist dies [der Mangel der sonst bereits erfolgreichen Vermittlung wissenschaftlicher Kenntnisse] das Fehlen des historischen Sinnes und der Mangel an Kenntniss jener großen Arbeiten, auf welchen das Gebäude der Wissenschaft ruht.”

“Wer über die handwerkliche Handhabung von Physik hinausgehen will, muss sich zwingend mit der Frage befassen, was physikalisches Denken ausmacht und wie es entstanden ist.”

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 Newton’s and Helmholtz’s original texts and complemented by Euler’s forgotten axiomatic of CM . The following transition to QM is guided by,

  • Einstein’s 1907 idea of quantization as selection problem,
  • Schrödinger’s 1933 thoughts about the relationship between CM and QM,
  • Schrödinger’s 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 (‘Hertz’s program’). The feasibility of Hertz’s 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 best-known historical examples is the confusion of energy and force.

Although dealing with basic topics of CM and QM and being self-containing, this book does not represent complete introductions into CM respectively QM as there is few account for their experimental bases.

Despite of presenting a non-conventional approach to QM, it presents new results. Among others, for a free particle, a square-integrable wave function is derived, and the methodologically interesting similarities between its paradigm ‘quantization as selection problem’ and Mittelstaedt’s reconstruction of QM are considered.

One of the most striking – and often considered to be “the only mysterious” – feature of quantum physics is the ‘interference of particles’. Its similarities to the interference of classical waves led to the term ‘wave-particle duality’. This duality is often taken as starting point for introducing quantum physics. On the other hand, the notions ‘wave’ and ‘particle’ are classical ones, so that the apparent duality arises from the assignment of classical properties to non-classical objects.

Thus, this book aims at an axiomatic foundation of non-relativistic 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 time-dependent Schrödinger equations. “Axiomatic” means, that features like quantum of action, wave-particle 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, Newton’s notion of stationary state as well as Helmholtz’s approach to the energy conservation law and concept of configurations. Even more important is Euler’s forgotten axiomatic of CM. In contrast to Newton’s 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 counter-intuitive. However, one can hardly expect QM to be intuitive, when even common CM is not. For instance,

  • if Newton’s 1st axiom (Galileo’s law of relativity ) were intuitive, already Aristotle had stated it ;
  • if Newton’s 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 over-simplification ‘force = mass × acceleration’);
  • if Newton’s 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 Newton’s representation of CM were intuitive, it had not taken 100 years to accept it, and the confusion between ‘force’ and ‘energy’ had been settled much earlier, etc.

Moreover, as observed by the pioneers of QM (notably, Bohr 1913, Heisenberg 1925, Schrödinger 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 Gödel’s 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 classical-mechanical 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, Helmholtz’s 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, time-independent Schrödinger equation. The non-stationary, time-dependent Schrödinger equation will be obtained through generalizing Euler’s principles of stationary-state change for classical bodies, first, to classical conservative systems and, then, to quantum-mechanical 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 Helmholtz’s explorations about the relationships between forces and energies leads to a non-classical, quantum class of interactions, viz, to the gauge invariance of the Schrödinger equation and to the (Ehrenberg-Siday-)Aharonov-Bohm 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 Schrödinger equation and its non-classical solution rather than about its interpretation. That derivation corroborates Schrödinger’s view of |ψ(x)|2 to represent a weight [density] function (1926, 4th Commun., § 7) and reveals xch |ψ(x,t)|2 V(x,t) and pch |ϕ(p,t)|2 T(p,t) to be the effective potential and kinetic energies, respectively, ‘seen’ by a quantum particle (xch and pch being characteristic extensions). This view is compatible with the quantum-logical 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 high-school 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, non-classical mechanics had to describe – 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 – last but not least – 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 solid-state theory (except Bloch’s 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 ‘*’ 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 § 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 Mittelstaedt’s recent book Rational Reconstructions of Modern Physics and the representation of Newton’s 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, square-integrable wave functions are derived. Chapter 1 about the conservation and changes of stationary quantum states has been completely rewritten. Chapter 5 ‘Aequat causa effectum’ presents not previously published results about a common feature of the symmetries of classical and quantum systems. Section 4.2 with the completion of Schrödingers 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 Planck’s 1909 treatment in view of Euler’s 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 Euler’s axiomatic of CM for tackling QM and special relativity, and he has extracted from Einstein’s 1907 pioneering paper on the specific heat of solids the paradigm ‘quantization as selection problem’. I am also still indebted to all the colleagues mentioned in my foregoing book. The discussions with Prof. Sigurd Schrader and his co-workers 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 “Principle of Dynamic State Equilibrium” 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 Müller-Preußker† for his hints about gauge theory. The discussions with Prof. Günther Nimtz have very much clarified my picture of the time-dependent ‘tunnel effect’. I feel indebted to Prof. Harry Paul for earlier discussions and for sending me his contribution about the black-body 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. Stefanovich’s 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 Schrödinger’s own criticism of his derivation of wave mechanics has eventually been discarded. Prof. Johannes Richter has provided valuable explanations about localization in ideal crystals (flat-band systems).

My friend and schoolmate Dr. Wolfgang Peters posed many stimulating philosophically driven questions and gifted me Torretti’s book. My friends and former colleagues Dr. Andreas Förster 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:

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. ( as well as the many enthusiasts on the www TeX help pages for their support.

Peter Enders
Senzig, August 2018


The author declares no conflict of interest, financial or otherwise.


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.

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