Travis Norsen - Foundations of Quantum Mechanics

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Springer International Publishing AG 2017

Travis Norsen Foundations of Quantum Mechanics Undergraduate Lecture Notes in Physics

1. Pre-Quantum Theories

Travis Norsen 1

(1)

Department of Physics, Smith College, Northampton, MA, USA

Travis Norsen

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In this introductory chapter we review two theories from classical physics Newtonian mechanics and Maxwellian electrodynamics and use them to introduce a number of concepts (such as determinism, locality, ontology, measurement, and configuration space) that we will explore in the context of quantum mechanics in subsequent chapters.

1.1 Newtonian Mechanics

As a first example of a pre-quantum theory lets consider the picture of the universe formulated by Isaac Newton. The theory, in a nutshell, says that the physical world consists of particles interacting by means of forces which the particles exert on one another and which influence the particles motions. About the particles, Newton wrote:

...it seems probable to me, that God in the Beginning formd Matter in solid, massy, hard, impenetrable, moveable Particles, of such Sizes and Figures, and with such other Properties, and in such Proportion to Space, as most conduced to the End for which he formd them; and that these primitive Particles being Solids, are incomparably harder than any porous Bodies compounded of them; even so very hard, as never to wear or break in pieces.... [A]ll material Things seem to have been composed of the hard and solid Particles above-mentiond, variously associated.... [, pp. 4002]

Newtons endorsement of the idea that observable macroscopic objects are composed of invisibly small, indestructible particles is a kind of bridge between the speculative notion of atomism that had been introduced by Ancient Greek philosophers such as Democritus, and the more scientific atomic theory of matter that grew out of chemistry and physics in the centuries following Newton.

Regarding the forces that these particles exert on one another, Newton wrote that

Bodies act one upon another by the Attractions of Gravity, Magnetism, and Electricity; and these Instances shew the Tenor and Course of Nature, and make it not improbable but that there may be more attractive Powers than these..... [W]e must learn from the Phaenomena of Nature what Bodies attract one another, and what are the Laws and Properties of the Attraction.... The Attractions of Gravity, Magnetism, and Electricity, reach to very sensible distances, and so have been observed by vulgar Eyes, and there may be others which reach to so small distances as hitherto escape Observation.... [, p. 376]

Although he did not have any particular detailed theories about them, Newton thus anticipated the empirical quest to understand the short-range attractions and repulsions between particles that we now think of as responsible for micro-physical, chemical, and even biological processes. But of course Newton did have a rather well-worked-out theoretical account of the long-range gravitational interactions between particles.

According to Newtons law of universal gravitation, the gravitational force exerted on a particle of mass located at position , by another particle of mass located at position , is given by

(1.1)

where is just the distance between the two particles and

(1.2)

is a unit vector pointing along the line from back toward . The gravitational force between two elementary particles, that is, is proportional to the product of the masses of the particles, inversely proportional to the square of the distance between them, and is directed back toward the particle exerting the force. The proportionality constant, G , which we now call Newtons constant, was first measured by Cavendish about a century after Newton.

The total or net force on the particle is then

(1.3)

(Note that here we ignore the existence of other, short-range forces and pretend for simplicity that the particles only interact gravitationally.) And of course it is this net force that influences the particles trajectory through space in accordance with Newtons second law of motion:

(1.4)

Note that Newtons inverse square law, Eq. (), also embodies Newtons third law: for every action theres an equal and opposite reaction. Or more precisely: if j exerts a force on i , then i necessarily also exerts a force on j , and these two forces (that they exert on each other) have equal magnitudes but precisely opposite directions. That is:

(1.5)

It is nice to have some pictures to go along with all the equations, so in Fig. Ive illustrated some of these ideas by showing three particles (which one might think of as two stars forming a binary star system plus an orbiting planet) and the forces they exert on one another.

Fig. 1.1

Three massive bodies and the gravitational forces they exert on one another

Note that the basic laws of Newtonian mechanics (both the expressions for the forces and also Newtons second law which describes how the particles respond to forces) are postulated as applying fundamentally to the elementary, microscopic Particles that Newton spoke of in the first block quote. It is perhaps not terribly surprising, but important and interesting nevertheless, that these same laws (properly understood) also turn out to apply to large macroscopic objects like stars and planets and apples. That is, in Newtonian mechanics, the applicability to macroscopic objects of (for example) the gravitational inverse square law and Newtons second law, are theorems which can be derived from the basic laws (understood as applying to the elementary Particles) rather than postulates. You are invited to consider this point further in some of the end-of-chapter Projects.

It is perhaps worth making more explicit that the long-range gravitational forces exerted on each particle depend, according to Eq. (), are thus non-local , by which we simply mean that they embody what Einstein would describe as a kind of spooky action at a distance. Interestingly, though, Newton himself did not believe that this apparent non-locality should be taken seriously, as accurately capturing the true nature of gravitational interactions. In a famous 1693 letter to Richard Bentley, Newton wrote:

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