Schrödinger's Mechanics

Interpretation

David B Cook

• Publisher: WSPC (EUROPE)
• Year: 2018
• ISBN:
  9781786344922
• Topic: Physical Sciences
• Subtopic: Physics
• Index: Physics

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The interpretation of quantum mechanics has been in dispute for nearly a century with no sign of a resolution. Using a careful examination of the relationship between the final form of classical particle mechanics (the Hamilton–Jacobi Equation) and Schrödinger's mechanics, this book presents a coherent way of addressing the problems and paradoxes that emerge through conventional interpretations.

Schrödinger's Mechanics critiques the popular way of giving physical interpretation to the various terms in perturbation theory and other technologies and places an emphasis on development of the theory and not on an axiomatic approach. When this interpretation is made, the extension of Schrödinger's mechanics in relation to other areas, including spin, relativity and fields, is investigated and new conclusions are reached.

--> Contents:

• Aims
• Basics: The Schrödinger Condition
• The Dimensions of Space
• Momenta, Operators, and a Problem
• Approximation and Interpretation
• Spin in Schrödinger's Theory
• Relativistic Equations
• Fields and Second Quantisation

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--> Readership: Physicists, chemists, philosophers of science, anyone with an interest in physical science. -->
Keywords:Quantum Theory;Interpretation;Wave Mechanics;Quantum PhilosophyReview: Key Features:

• Gives a unique interpretation of the topic

Chapter 1

Aims

This work is divided into two parts: the first part (Chapters 2–5) tries, in the most cogent and positive possible terms, to summarise what Schrödinger’s Mechanics is and its physical interpretation. The remaining chapters are a series of comments, from the point of view of SM, on other developments in quantum theory.

The overall aims of the work, in the first part are as follows:

(1)To place Schrödinger into the position of the culmination of the ‘Great Tradition’ of Analytical Mechanics, that is after Newton, Lagrange, Hamilton and Jacobi. Also adding the pioneering work of Poincar’e and Kolmogorov to create an interpreted, paradox-free theory of the energetics and probability distributions of sub-atomic particles; forming a clear and comprehensible formulation of what I have called Schrödinger’s Mechanics (SM).

(2)To show that other ‘formulations’ of Quantum Mechanics (QM) are either reducible to or abstracted from SM; none of them are capable of generating SM in the same sense that SM can generate them.

(3)The probability distributions in SM, and the state functions from which they are formed, are the same as all other probability distributions. They are mathematical functions of space and exist not in the real world but in our heads, on paper or in our computers. The relationship between Probability and Statistics is emphasised. Confusions between these two and the role of ‘colloquial’ uses of language in discussions of probability and physics can be seen as a source of most difficulties.

When this is done all the ‘traditional problems’ of the interpretation of QM simply fall away. There is no wave–particle duality; no measurement problem; no collapse of the wave function and, most disappointingly for popular science writers, no ‘many-worlds’ or ‘many-universes’ speculation. Heisenberg’s uncertainty ‘principle’ is shown to be a theorem and not a universal excuse for creating what (in a different context) Stephen Jay Gould labeled ‘Just-So Stories’: post hoc justifications of current practice.

The remaining chapters address points of wider interest:

(1)To show the interpretation of SM can be extended to include ‘spin’ entirely within non-relativistic SM by substituting a Geometric Algebra representation of space for the usual symbolic vector system.

(2)To investigate some consequences of imposing the constraints of Special Relativity on the formulation of SM.

(3)To try and draw a sharp distinction between a physical theory and the mathematical technologies used in the approximation methods involved in practical applications; to distinguish interpretation of a theory from attempts to interpret the mathematical terms generated by these technologies.

(4)To distinguish between physical and mathematical ‘fields’ and the consequences of this distinction for the physical interpretation of these entities. To question the conflation of fields and particles.

(5)To stress the limitations of what can be achieved by this approach. Any experimental data which can be conclusively shown to be inexplicable by the methods outlined here call for an extension of the ‘Great Tradition’ beyond SM and will not be considered here. No theory is of unlimited application,^a but any scheme which extends or replaces SM would have to satisfy the strict requirements of the tradition; it would have to be consistent, interpretable and, above all, at least as successful in describing the mechanisms of the real world and generate verifiable physical results. Finally, SM has been at least as successful in interpreting and describing the sub-atomic world (electron physics and chemistry) as Classical Particle Mechanics has been on the human scale. Any acceptable theory of sub-nuclear phenomena would be required to be as successful and not yet another example of semi-empirical ‘Ptolomaic Science’.

It might be thought that the work presented here is ‘old-fashioned’ and, indeed, so it is, but insofar as all the more modern interpretations of the various quantum theories are ultimately based on the interpretation of SM, they depend for their validity on this interpretation.

Throughout the work the ‘derivations’ given are simply outlines of how more rigorous methods might be used.

^a There will never be a ‘Theory of everything’.

Chapter 2

Basics: The Schrödinger Condition

“Did all of those old guys swing like that?”¹

2.1 Method

Much of 20^th century theoretical physics can be characterised by,

Schrödinger (Maxwell, Dirac) Equation, how many ways can I write thee? Let me count the ways.

Mathematicians are notorious for ignoring the physical interpretation of their excursions into theoretical physics — sometimes brazenly so — and tend, like the Red Queen, to be capable of ‘believing six impossible things before breakfast’.² The ‘standard — Copenhagen — interpretation’ of quantum theory is often taken for granted by mathematicians and philosophers, ignoring the fact that not all good physicists are good philosophers and it is common for philosophers to know very little physics. All too often no distinction is made between equations (which are an expression of a law of nature) and identities (definitions). There is, for example, more interest in the study of the Schrödinger (Maxwell, Dirac) equation than in the study of the physics of the real world. In fact, there is a long ‘tradition’ among mathematically inclined physicists dating from classical times that it should be possible to derive substantive scientific results from mathematics or logic alone.³

In this essay, I take the opposing view: that the physical interpretation of mathematics must be given greater importance than its structure, generality or even elegance (Boltzmann is reported to have remarked that elegance should be left to tailors). This means, naturally, that some of what is written here may be controversial, indeed may, at times, seem a little pedestrian. But, I use as my excuse an essay by that great historian E. P. Thompson where he compares himself to that English bird the Great Bustard; stumbling about in the undergrowth of the real world trying to get airbourne in a pathetic contrast to some ‘theorists’ of history, who attempt to soar like eagles, way above the brute facts.⁴

There are several clear descriptions of the necessary conditions for a (physical) theory to be considered scientific; below is an extract from Lucio Russo’s superbly written book⁵:

(1)Their statements are not about concrete objects, but about specific theoretical entities. For example, Euclidean geometry makes statements about angles or segments, and thermodynamics about the temperature or entropy of a system, but in nature there is no angle, segment, temperature or entropy.

(2)The theory has a rigorously deductive structure. It consists of a few fundamental statements (called axioms, postulates, or principles) about its own theoretical entities, and it gives a unified and universally accepted means for deducing from them an infinite number of consequences. In other words, the theory provides general methods for solving an unlimited number of problems. Such problems, posable within the scope of the theory, are in reality ‘exercises’, in the sense that there is general agreement among specialists on the methods of solving them and of checking the correctness of the solutions. The fundamental methods are proofs and calculation. The ‘truth’ of scientific statements is therefore guaranteed in this sense.

(3)Applications to the real world are based on correspondence rules between the entities of the theory and concrete objects. Unlike the internal assertions of the theory, the correspondence rules carry no absolute guarantee. The fundamental method for checking their validity, which is to say the applicability of the theory, is the experimental method. In any case, the range of validity of the correspondence rules is always limited.

Any theory with these three characteristics will be called a scientific theory.

Mario Bunge⁶ has, perhaps, gone furthest with this approach.

What is missing from the list cannot be expressed so succinctly but is equally important. Perhaps most importantly, these criteria can easily be satisfied by purely phenomenological, even empirical theories; there is no doubt, for example, that these criteria are satisfied by Ptolomy’s theory of the solar system.

Secondly, these criteria are notably, and I suppose intentionally, static; they give no indication of the development of the ...

Table of contents

• Cover Page
• Title
• Copyright
• Contents
• Preface
• About the Author
• 1. Aims
• 2. Basics: The Schrödinger Condition
• 3. The Dimensions of Space
• 4. Momenta, Operators, and a Problem
• 5. Approximation and Interpretation
• 6. Spin in Schrödinger’s Theory
• 7. Relativistic Equations
• 8. Fields and Second Quantisation
• Epilogue
• Suggested Reading
• Index