Principles Of Newtonian And Quantum Mechanics, The: The Need For Planck's Constant, H (Second Edition)

The Need for Planck's Constant, h

Maurice A de Gosson

• Publisher: WSPC
• Year: 2016
• eBook ISBN: 9789813200982
• Edition: 2
• Topic: Physical Sciences
• Subtopic: Mathematical & Computational Physics

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The second edition of this book deals, as the first, with the foundations of classical physics from the "symplectic" point of view, and of quantum mechanics from the "metaplectic" point of view. We have revised and augmented the topics studied in the first edition in the light of new results, and added several new sections. The Bohmian interpretation of quantum mechanics is discussed in detail. Phase space quantization is achieved using the "principle of the symplectic camel", which is a deep topological property of Hamiltonian flows. We introduce the notion of "quantum blob", which can be viewed as the fundamental phase space unit. The mathematical tools developed in this book are the theory of the symplectic and metaplectic group, the Maslov index in a rigorous form, and the Leray index of a pair of Lagrangian planes. The concept of the "metatron" is introduced, in connection with the Bohmian theory of motion. The short-time behavior of the propagator is studied and applied to the quantum Zeno effect.

--> --> Contents: From Kepler to Schrödinger... and Beyond;Newtonian Mechanics;The Symplectic Group;Action and Phase;Semi-Classical Mechanics;Metaplectic Group, Maslov Index and Quantization;Schrödinger's Equation and the Metatron; Appendices: Symplectic Linear Algebra;The Lie-Trotter Formula for Flows;The Heisenberg Groups;The Bundle of s-Densities;The Lagrangian Grassmannian;
Readership: Reserchers and graduate students in mathematical physics and physics.Newtonian Mechanics, Quantum Mechanics, Bohmian Theory, Kepler, Schrödinger, Semi-Classical Mechanics, Metatron0

Chapter 1

From Kepler to Schrödinger... and Beyond

Summary. The mathematical structure underlying Newtonian mechanics is symplectic geometry, which contains a classical form of Heisenberg’s uncertainty principle. Quantum mechanics is based on de Broglie’s theory of matter waves, whose evolution is governed by Schrödinger’s equation. The latter emerges mathematically from classical mechanics using the metaplectic representation of the symplectic group.

The purpose of this introductory Chapter is to present the basics of both classical and quantum physics “in a nutshell”. Much of the material will be further discussed and developed in the forthcoming Chapters.

The three first sections of this Chapter are devoted to a review of the essentials of Newtonian mechanics, in its Hamiltonian formulation. This will allow us to introduce the reader to one of the recurrent themes of this book, which is the “symplectization” of mechanics. The remainder of the Chapter is devoted to a review of quantum mechanics, with an emphasis on its Bohmian formulation. We also briefly discuss two topics which will be developed in this book: the metaplectic representation of the symplectic group, and the non-squeezing result of Gromov, which leads to a topological form of Heisenberg’s inequalities.

It is indeed a discouraging (and perilous!) task to try give a bibliography for the topics reviewed in this Chapter, because of the immensity of the available literature. I have therefore decided to only list a few selected references; no doubt that some readers will felicitate me for my good taste, and that the majority probably will curse me for my omissions –and my ignorance!

The reader will note that I have added some historical data. However, this book is not an obituary: only the dates of birth of the mentioned scholars are indicated. These scientists, who have shown us the way, are eternal because they live for us today, and will live for us in time to come, in their great findings, their papers and books.

1.1Classical Mechanics

I will triumph over mankind by the honest confession that I have stolen the golden vases of the Egyptians to build up a tabernacle for my God far away from the confines of Egypt. If you forgive me, I rejoice; if you are angry, I can bear it; the dice is cast, the book is written either for my contemporaries, or for posterity. I care not which; I can wait a hundred years for a Reader when God has waited six thousand years for a witness (Johannes Kepler).

Johannes Kepler (b.1571) had to wait for less than hundred years for recognition: in 1687, Sir Isaac Newton (b.1643) published Philosophiae Naturalis Principia Mathematica. Newton’s work had of course forerunners, as has every work in Science, and he acknowledged this in his famous sentence:

“If I have been able to see further, it was because I stood on the shoulders of Giants.”

These Giants were Kepler, on one side, and Nicolas Copernicus (b.1473) and Galileo Galilei (b.1564) on the other side. While Galilei studied motions on Earth (reputedly by dropping objects from the Leaning Tower of Pisa), Kepler used the earlier extremely accurate –naked eyed!– observations of his master, the astronomer Tycho Brahe (b.1546), to derive his celebrated laws on planetary motion. It is almost certain that Kepler’s work actually had a great influence on Newton’s theory; what actually prevented Kepler from discovering the mathematical laws of gravitation was his ignorance of the operation of differentiation, which was invented by Newton himself, and probably simultaneously, by Gottfried Wilhelm Leibniz (b.1646). It is however noteworthy that Kepler knew how to “integrate”, as is witnessed in his work Astronomia Nova (1609): one can say (with hindsight!) that the calculations Kepler did to establish his Area Law involved a numerical technique that is reminiscent of integration (see Schempp [136] for an interesting account of the “Keplerian strategy”).

1.1.1Newton’s Laws and Mach’s Principle

Newton’s Principia (a paradigm of the exact Sciences, often considered as being the best scientific work ever written) contained the results of Newton’s investigations and thoughts about Celestial Mechanics, and culminated in the statement of the laws of gravitation. Newton has often been dubbed the “first physicist”; the Principia were in fact the act of birth of Classical Mechanics. As Newton himself put it:

“The laws which we have explained abundantly serve to account for all the motions of the celestial bodies, and of our sea.”

We begin by recalling Newton’s laws, almost as Newton himself stated them:

Newton’s First law: a body remains in rest –or in uniform motion– as long as no external forces act to change that state.

This is popularly known as “Newton’s law of inertia”. A reference frame where it holds is called an inertial frame. Newton’s First Law may seem “obvious” to us today, but it was really a novelty at Newton’s time where one still believed that motion ceased with the cause of motion! Newton’s First Law moreover contains in germ a deep question about the identification between “inertial” and “gravitational” mass.

Newton’s Second law: the change in momentum of a body is proportional to the force that acts on the body, and takes place in the direction of that external force.

This is perhaps the most famous of Newton’s laws. It was rephrased by Kirchhoff in the well-known (and somewhat unfortunate!) form “Force equals mass times acceleration”.

Newton’s Third law: if a given body acts on a second body with a force, then the latter will act on the first with a force equal in magnitude, but opposite in direction.

This is of course the familiar law of “action and reaction”: when you exert a push on a rigid wall, it “pushes you back” with...

Table of contents

• Cover Page
• Title
• Copyright
• Dedication
• Contents
• Preface to the First Edition
• Preface to the Second Edition
• Foreword by Basil Hiley
• 1 From Kepler to Schrödinger... and Beyond
• 2 Newtonian Mechanics
• 3 The Symplectic Group
• 4 Action and Phase
• 5 Semi-Classical Mechanics
• 6 Metaplectic Group, Maslov Index, and Quantization
• 7 Schrödinger’s Equation and the Metatron
• A Symplectic Linear Algebra
• B The Lie-Trotter Formula for Flows
• C The Heisenberg Groups
• D The Bundle of s-Densities
• E The Lagrangian Grassmannian
• Bibliography
• Index