Primer of Quantum Mechanics

Marvin Chester

• Publisher: Dover Publications
• Year: 2012
• ISBN:
  9780486138206
• Topic: Scienze fisiche
• Subtopic: Fisica

What does quantum mechanics tell us about the key model physical systems of nature? The author of this highly regarded text explores this question in a conceptual manner, fusing mathematical and philosophical elements to present physical imagery that closely parallels the mathematics.
Beginning with an overview that discusses the premise and design for the study, the text proceeds with an examination of the classical quantum bead on a track: its states and representations; its measurement spectra as operator eigenvalues; the harmonic oscillator: bound bead in a symmetric force field; and the bead in a spherical shell. Other topics include spin, matrices, and the structure of quantum mechanics; the simplest atom; indistinguishable particles; and stationary-state perturbation theory.
Geared toward upper-level undergraduate students in physics, this refreshing and instructive text requires the following background: a freshman-year survey course in physics, a first course in classical Newtonian mechanics, and a grasp of mathematics that encompasses integral calculus, vector analysis, differential equations, complex numbers, and Fourier analysis.

Chapter 1

Readers’ Orientation:
Premise, and Design of the Study

1.1 Philosophical Orientation

For what purpose, dear reader, do you study physics?

To use it technologically? Physics can be put to use: so can art and music. But that’s not why you study them.

It isn’t their social relevance that attracts you. The most precious things in life are the irrelevant ones. It is a meager life, indeed, that is consumed only by the relevant, by the problems of mere survival.

You study physics because you find it fascinating. You find poetry in conceptual structures. You find it romantic to understand the workings of nature. You study physics to acquire an intimacy with nature’s way.

Our entire understanding of nature’s way is founded on the subject called quantum mechanics. No fact of nature has ever been discovered that contradicts quantum mechanics. In its existence of over 60 years, quantum theory has experienced only success in describing the physical world. It has survived a stunning multitude of tests on its validity. We must accept it as the soul of nature. Quantum mechanics is nature’s way.

What’s the attraction of studying quantum mechanics? It lies in perceiving the conceptual structure of the subject. I don’t mean by this the mathematical structure. The conceptual structure is more. It confronts the question, When we use the mathematics, what are we saying about the nature of nature? It is in this pursuit—the pursuit of meaning—that both the difficulty and the pleasure lie. The idea is best captured in an aphorism attributed to Niels Bohr, one of the founding fathers of the subject: “Those who are not shocked when they first come across quantum theory cannot possibly have understood it.”

Prepare for a shock, advises Bohr. If you are not shocked, you ought to question whether you have understood. The idea is this: the theory completely confounds our classical common sense about nature.

To understand nature, we must readjust our common sense. That is the central object of this study: to readjust common sense so as to accommodate quantum theory. We make whatever sense we can of quantum mechanics so that after a while it becomes our common sense. That is how we become intimate with nature’s way.

This revision of thinking is not a mere mathematical matter. It is not the mathematical apparatus that shocks us. The mathematics is a beautiful machinery. It is, however, all formality. It can be mastered in shockless serenity. The shock to which Bohr refers is in the interpretation of the mathematics: the physical meaning is shocking.

One of the great savants of quantum mechanics, P. A. M. Dirac, set himself this special task: to mate the conceptual structure with the mathematical. He invented a mathematical notation that directly embeds the philosophy of the subject into the calculational methods. It exhibits what is being said about the nature of nature. Dirac notation exposes the internal logic of quantum mechanics. It displays the sense—the meaning—of the theory in every equation one writes.

The characterization Dirac notation is an infelicitous misnomer for one of the great contributions to modem thought. It is not merely a way of writing; a way of writing expresses a way of thinking. Dirac notation is a way of thinking.

Consider the notation change from roman to arabic numerals. It was one of the most significant advances in the history of mathematical thought. The mechanics of the change in writing was not the issue; the importance lay in the new way of thinking that accompanied the notational change.

So it is with Dirac notation. It expresses the quantum mechanical way of thinking. With it, one can proceed from the philosophy of the subject to its mathematical expression rather than the other way around. That shall be the direction this primer takes in the study of quantum mechanics. The subject will be developed through the philosophy underlying it. The object is to proceed from meaning. Because Dirac notation is exactly suited to this goal, it is used exclusively in the text.

Meaning does not reside in the mathematical symbols. It resides in the cloud of thought enveloping these symbols. It is conveyed in the words; these assign meaning to the symbols.

Here is an example. The most important dictum of quantum mechanics is this one:

WHAT YOU CAN MEASURE IS WHAT YOU CAN KNOW

This precept can be recast as a formal mathematical statement. In Dirac notation it’s particularly simple.

Most of quantum mechanics proceeds from this precept in combination with others. They all can be expressed in both words and in symbols. The weight of thought is in the words; the mathematics and its symbols enables us to probe the consequences of the thought.

One deduces from such dicta the most intriguing (shocking) notions. Here are two of them:

1. There exist particles that are fundamentally indistinguishable from one another: no matter how refined experiments become, none will ever perceive a difference between, say, two electrons. Among a swarm of them, you may be able to recognize an individual fruit fly, but never an individual electron among electrons.

2. Only discrete spectra of atomic energies can be found in nature. There are values of energy that for an orbiting system, no measurement can ever detect! The detectable ones can be calculated.

In Dirac notation each of these notions is a simple mathematical statement. Dirac allows us, while doing the practical calculations, to focus on the meaning of the theory. This brings out its conceptual poetry. It responds to the why of this study. The romance and fascination are preserved while we learn the formal mathematics of the subject.

1.2 Mathematical Orientation

Our study does require a certain level of mathematical sophistication. You need to be proficient in calculus, and you must have been exposed to complex numbers. You must be familiar with the wonderful deduction of Gauss that the unit of imaginary numbers,

, may be used to define the two-dimensional complex plane in which every point corresponds to a complex number.

The complex number may be called z, its real part x, and its imaginary part y, as shown in Figure 1-1. The complex conjugate of z is called z*; it results from changing i to – i everywhere in z. And the magnitude of z, written symbolically as |z|, is never complex, nor even negative: it is the length of the line connecting the point z to the origin—always nonnegative real.

What is important to our study is this most stunning of notions: that the exponential function of i...