Probability and Schrödinger's Mechanics

The following sections are included:

• Contents

• Preface

• Organisation

In this introductory material I give a rather casual survey of the general approach I shall be using and of some of the attitudes I shall take to philosophy, logic and, particularly, mathematics. In mathematics I shall combine extreme fussiness in some topics with a rather casual lack of rigour in general.

In general, internal, “technological” matters, mathematics is in good hands and needs no gloss from me. But in key areas of interpretation and abstraction the relationship between abstract structures and physical theories are, typically, ignored by mathematicians or deemed to be “arbitrary”.

In the nineteenth century Hegel showed Western Christendom that the God it worshipped was not “out there” but was nothing more than a fantastic image of itself. Modern mathematicians are not yet ready to accept the same conclusion about the logic they rely on; mathematics and logic are not “out there”, they are abstracted from the real world of objects, processes, language and conceptual thought.

Writings about quantum mechanics are hedged about with claims that it changes our whole conception of the applicability of logic and mathematics and revolutionises philosophy. While much of this discussion is patently foolish, it is worth attempting to say a little about some of the assumptions which I have later (mostly silently) made about philosophical, logical and mathematical matters. The ideas simply sketched here will be developed elsewhere. What is said here is trifling as philosophy, it is merely a general orientation to the study of mathematically-articulated science.

The interpretation of the modern (measure-theoretical) theory of probability is at odds with the everyday meaning(s) of the word probability. The material presented stresses that Kolmogorov's theory of probability (a) is the only one which can be used in the context of quantitative theories and (b) that probabilities are the relative measures of sets. Statistical methods are related to probabilities insofar as they are ways of experimentally determining these measures.

After some elementary considerations about the relationship between the “colloquial” and “mathematical” use of some common mathematical terminology, the idea of probability is introduced, not axiomatically at this stage, but descriptively. The relationship between probability and statistics is clarified by use of the simplest and most familiar example; dice. The generation of probability distributions for systems which are entirely deterministic is discussed and their indispensability is emphasised. Some of the difficulties which will occur in the interpretation of Schrödinger's mechanics are briefly visited.

Some of the ideas introduced in the last chapter are placed of a firmer, more formal, footing and a problem in ontology is skirted. The problems associated with time-dependent probability distributions are rehearsed with particular reference to a familiar example. The abstract and concrete objects which are likely to be met in interpreting Schrödinger's mechanics are examined.

Notwithstanding the possibility of setting up axiom systems for quantum mechanics which are independent of classical (Newtonian) mechanics, the concepts and referents of are patently and historically developed from those of Newton's theory as developed by Lagrange, Hamilton and Jacobi.

This part of the work uses elementary (nineteenth century) methods to try to illuminate the relationship between the two mechanical theories.

The Hamilton-Jacobi (H-J) equation represents the culmination of the development of classical (particle) mechanics. In this chapter I concentrate on those points in the interpretation of the Hamilton-Jacobi equation which are absolutely crucial to an appreciation of Schrödinger's mechanics; without which many mysteries appear in the interpretation of quantum theory. In an attempt to be emphatic I may have been repetitive here.

While many of the more elementary paradoxes in the interpretation of quantum theory are generated by misunderstandings about probability, some more subtle ones are due to confusions about angular momentum. Rotational motion, having some seductive and misleading analogies with rectilinear motion, has proved a stumbling block in both classical and quantum mechanics. The most fertile source of “mysteries” in quantum theory — the Einstein-Podolsky-Rosen paradox and the related results due to Bell — are exacerbated by this confusion.

After a discussion of the most famous “thought experiment” based on crystal particle diffraction which purports to show that electrons are both particle and wave, the transition is made from the Hamilton-Jacobi equation of classical mechanics to the underlying dynamical law of Schrödinger's mechanics. The Schrödinger equation and its associated boundary conditions (which generate quantisation) are derived from this law. An attempt is made to distinguish, in Schrödinger's mechanics, between equations (which are expressions of the dynamical law) and identities (which are not) which, historically, has been the cause of some confusion in quantum mechanics.

There are a number of examples of experiments which are presented as supporting the essentially dual nature of what are seen in classical terms as particles. It is suggested both that these experiments make it mandatory to consider individual electrons (for example) as simultaneously having wave and particle properties and the existence in nature of mysterious cooperative properties independently of any physical interactions which the particles might have. In this chapter I take a closer look at the most familiar and ostensibly convincing of these experiments; particle diffraction.

Using material developed in Part 3 a brief outline of the relationship between quantum and classical (particle) mechanics is given, stressing the nature and meaning of the variational principle which is the basis of Schrödinger's mechanics. Much of this material is descriptive and heuristic but the final section (“Summary”) contains the principles which certainly will enable all of Schrödinger's mechanics to be generated as well as the formal structures which may be abstracted from it.

At last we are able to generate the Schrödinger equation which is at the heart of all applications of Schrödinger's mechanics. The solutions of this equation for particular systems enables the probability distributions for abstract systems to be computed and, as it turns out, the other product of the Schrödinger Condition (the boundary conditions on these distributions) generates the most surprising feature of Schrödinger's mechanics, the existence of discrete, quantised, values of some dynamical variables; in particular the existence of quantised energies for particles in some environments. The interpretation of the solutions of the Schrödinger equation is, of course, fixed in advance by the considerations of the previous chapter; but there are some unfamiliar consequences of this interpretation.

Some problems of interpretation are addressed here. The nature of some of the eigenvalue problems which are not associated with the Schrödinger's equation are investigated and found to be identities independent of the law of motion in Schrödinger's mechanics. Difficulties in understanding the nature of the momentum probability distribution for real functions ψ also have to be faced.

The most widely used and most familiar applications of Schrödinger's mechanics are in those cases where the potential energy function is independent of time and is a conservative field so that, both classically and in quantum theory, the energy of the system is conserved. The Schrödinger equation for these important cases takes a characteristic form; that of an eigenvalue problem. In this case it is possible to abstract the algebraic structure of a Hilbert space from the solutions of the Schrödinger equation. This algebraic structure transforms much of the mathematics from problems in analysis (which are hard) to problems in algebra (which are easier) while doing some violence to the interpretation of the theory. Quantum theory is not deepened or developed by the use of powerful abstract methods, merely sophisticated. It is via the interpretation of the symbolism that a theory is deepened, not by an improvement of the tools of articulation.

Schrödinger's mechanics, like many other physical theories, cannot be satisfactorily interpreted and understood without a careful scrutiny of the way it works when applied to particular physical systems. However instructive it is to think that conclusions may be drawn about the real world from thought experiments on non-existent objects, it is even more instructive to see how the theory actually applies to the real world. The applications discussed in this part are chosen because they are exactly soluble in Schrödinger's mechanics but, more importantly, because each one has important, even decisive, things to say about the interpretation of the theory.

The Kepler problem (two particles in mutual attraction) is not only a classic problem in its own right with extensive application throughout physics and chemistry, but its theoretical ramifications bring into sharp focus some ideas which are crucial to the use and understanding of Schrödinger's mechanics. In this chapter we look at the several ways in which the associated Schrödinger's equation may be solved and the interpretation of the solutions. There are some surprises in the general properties of the solutions and some quantities which emerge on the way to those solutions. I have risked boring the reader by stressing fundamental points here.

In some ways the quantum theory of simple harmonic motion (SHM) is a more basic application of Schrödinger's mechanics than the Kepler problem. It certainly provides, in addition to its obvious applications in the theory of mechanical vibrators, much of the theory and conceptual structure of various types of “field”. This chapter provides the simple solutions of the Schrödinger equation for SHM and the development and interpretation of the more abstract applications of the associated formalism. We shall see that some interpretations of the solutions of the SHM problem are responsible for much obfuscation and discussion about non-existent particles.

Perturbation theory, as an approximation method for tackling intractable Schrödinger equations, might seem out of place in a work concentrating on the interpretation of Schrödinger's mechanics. But, in fact, this approximation method is the principle (perhaps the only) tool of modern field theories and, as such, has proved to be the source of many of the concepts used by modern theorists. These concepts have tended to be taken over uncritically by philosophers and detached from their original source. It is worth looking at this method and, perhaps, comparing it and its conceptual structure with some earlier approximation techniques.

The most familiar and therefore, perhaps, the most widely ignored examples of explanations of phenomena in terms of hidden variables are in the use of various semi-empirical Hamiltonians. The archetypes are the spin Hamiltonians used in the theory of the chemical bond and various magnetic phenomena and the vector model of the atom, but modern particle physics abounds with examples.

Most, if not all, the paradoxes associated with quantum mechanics are associated with the interpretation of the mathematical theory of probability. In Part 2 these matters were addressed. In this part some of the more infamous of these paradoxes are examined individually.

It should be clear that the question of the (statistical) verification of the probabilistic predictions of Schrödinger's mechanics is just a particular application of the statistical method in general. Measurements are made on random concrete objects which are instances of the relevant abstract object and summed to form approximate numerical quadratures to be compared to the theoretical measures of the appropriate model subsets. However, the “problem of measurement” and “theories of measurement” are totems in quantum theory and so I do obeisance to them here.

Here is a miscellany of matters which have caused difficulties of interpretation; some are soluble, some are not.

We have seen how it is possible to extract some of the formal structures abstracted from Schrödinger's mechanics and use them in a semi-empirical way to provide impressive technologies for the rationalisation of a wide range of phenomena for which no interpreted theory is yet available. It is interesting but perhaps not very profitable to compare this situation with previous experiences of this kind in the history of science and to comment on some current developments.

The following sections are included:

• Index