David Park

Relation between the parabolic and spherical eigenfunctions of hydrogen

It is shown that the transformation coefficients relating the eigenfunctions of the Kepler problem in parabolic and spherical coordinates respectively are the normalized Clebsch-Gordan coefficients.

Light and Number: Ordering Principles in the World of an Autistic Child.

This study describes part of a complex system of ideas in which a 12-year-old autistic girl unites concepts of number with elements of sunlight and weather, and seeks to explain some of her reasoning. Certain prime numbers are endowed with positive or negative affect, which persists even when they are multiplied to form composite numbers. The numbers are associated with 29 kinds of weather, real and fictitious, which also have strong affect attached to them. It is conjectured that the system is an ingenious and laborious attempt by the child to compensate for her failure to endow events and emotions with ordinary social and emotional meaning by the invention of an entirely personal kind of meaning in whose terms at least some elements of daily experience can be ordered and understood.

Rays of Light

Following the historical order of ideas is usually a good way of introducing people to physics, but the historical order is never the logical order. If it were, there would be no room for discovery, since each new development would merely follow deductively from the preceding ones. New discoveries usually involve radically new ideas not implicit in what went before, and deductive connections, when they can be made at all, must be made backwards. You cannot derive the formulas of relativistic mechanics from those of Newton. You can derive Newtonian formulas from relativistic ones by taking a low-velocity limit, but even this is not a derivation of Newtonian physics, since that physics takes place against a background of absolute space and time which Einstein abandoned. This example contains a lesson: We may be able to derive the equations of an earlier theory from those of a later one, but to recapture the theory itself requires imagination and scholarship.

The Myth of the Passage of Time

The main purpose of this paper is to point out that the technique, familiar to physicists, of adopting various representations of physical phenomena for various purposes is convenient (and, I believe, necessary) in order to clarify some of the aspects of time that have been regarded as mysterious. Two representations are distinguished: the one that physics always uses and I have called atemporal, and the temporal one implicit in ordinary language and metaphor and most of the philosophical speculation with which I am familiar. The myth mentioned in the title is the idea that this representation somehow is time itself. The representation are explained by the use of pictures, which are to be distinguished from the representations they illustrate, and an attempt is made to clarify some old questions by their use.

Theory of Perturbations

The number of calculations that can be carried out exactly, either in classical or in quantum dynamics, is comparatively small and the number of useful results so obtained is even smaller. The greater part of physics, engineering, and dynamical astronomy involves approximations, and the worst difficulties arise when the variables of a problem are not separable. What one hopes to do is to find a corresponding problem in which not only are the variables separable but the resulting differential equations can be solved exactly, and then move from the solved problem to the unsolved one by a systematic procedure of approximation, usually involving expansion in series. The procedures in quantum and in classical mechanics can be carried out in somewhat analogous fashions, but there is no use in pursuing the comparison very far since what is observable differs in the two theories. An astronomer, for example, wishes to calculate positions in the sky and has little interest in knowing energies, whereas in atomic physics an electron’s position is not an observable while its energy is.

The Motion of a Rigid Body

In the narrow world of the exact sciences there are several kinds of rigid body. There are nuclei, generally not spherical and precessing in the atomic field or some applied field. There are molecules, whose shapes may be very complex, and there are stars and planets interacting through gravitation. In the closely adjacent field of engineering, and especially space engineering, there is an immense variety of oscillating and rotating devices, some of which, such as the gyroscopic stabilizers used on space craft, are extremely sophisticated.

The Hamilton-Jacobi Theory

The principal object of classical dynamics is to find where everything is at time t; that is, to find a set of q n (t). The principal object of quantum mechanics is to find a wave function ψ(q, t). From this you cannot calculate where everything is in the sense of classical physics, but you can calculate all there is to know. Lagrangian and Hamiltonian dynamics find q(t) and q(t) or p(t) by means of ordinary differential equations. In ψ(q, t), q is in no sense a function of t, since ψ has a value for every q and every t. The equation it solves is a partial differential equation. The differences between the two mathematical descriptions are so wide that they seem to belong to different universes of ideas, but in Ehrenfest’s theorems (Sect. 1.3), for example, familiar ordinary differential equations came out of the partial differential equation for ψ(q, t) that are the same as the equations of Newtonian dynamics. It is now our task to show that out of the dynamics of Lagrange and Hamilton comes a partial differential equation that is the same as the eikonal equation (1.31), which was derived from wave optics. In Chap. 7 we will see how to reconstruct quantum mechanics, under suitable assumptions, if the eikonal equation is known. These arguments will help to explain the mathematical relation between the two theories, and a study of the necessary assumptions will do much to illuminate the subtler question of their physical relation.

N-Particle Systems

Newton’s first and second laws really refer only to particles, for they make no reference to properties like size, elasticity, and internal angular momentum. The objects described by these laws have only mass and position. The properties of extended objects must be included in the theory by a process of summation or integration. In physics generally there are two situations of special interest: where the bodies are few in number, say up to 4 or 5, and where there are about 1023 of them. The situations are very different, starting with the kind of question one wishes to ask — macroscopic quantities like pressure, temperature and density have no counterparts in a few-body system. If one is thinking about a rigid solid one must decide what rigidity means; if the solid is elastic there are assumptions to be made concerning the nature of its elasticity. In Sect. 4.6 an elementary calculation in hydrodynamics will require four assumptions in addition to the assumption that a small element of the fluid obeys Newton’s second law. Except for Sects. 4.5 and 4.6 this chapter will be concerned with the simplest cases, in which the N particles can be considered one at a time, each one obeying Newton’s laws. Rigid and elastic bodies will be discussed in Chaps. 8 and 9.

Action and Phase

It is relatively simple to derive formulas of classical mechanics from those of quantum mechanics. One translates statements about probabilities into statements about measurable quantities, and one finds ways to eliminate ħ, either by considering situations in which phase changes much more rapidly than other parameters (Sect. 1.3) or by taking averages in a suitable way (Sect. 2.1).

Orbits of Particles

If a system has only a very few degrees of freedom its equations of motion are often obvious. One writes down Newton’s laws and sets about solving them. If they are not easy to solve it is because some equations are difficult. In complicated systems, even formulating the equations presents some problems, and most of this book is devoted to more advanced and general ways of writing down such equations and constructing their exact and approximate solutions. This chapter exhibits some relatively simple systems chosen so as to illustrate various forms of dynamical behavior. The first few examples need only Newton’s laws and a little mathematics, but for the later ones it will be convenient to introduce new variables and write equations of motion in terms of generalized coordinates. This will be done using Lagrange’s technique, which leads not only to general and useful equations of motion but also to the powerful methods of solution that follow.