Measuring magnetic projections

What does the previous chapter have to do with quantum mechanics? I have said that the predictions of quantum mechanics are significantly different from those of classical mechanics only when applied to very small objects. How could we make such a tiny compass needle? In fact we don't need to make one, because nature itself supplies one. It is natural to suppose that an atom acts like a tiny magnetic needle because its orbiting electron mimics a current loop.

In 1922, physicists Otto Stern and Walther Gerlach decided to test this supposition by measuring the magnetic arrow associated with a silver atom. It is clear that they could not do this by watching an individual atom precess in a uniform magnetic field! Instead, they injected a moving silver atom into a non-uniform field and noticed how the resulting force pushed the atom around. The “Stern–Gerlach apparatus” sketched on the next page thus measures the projection of an atom's magnetic arrow on the vertical axis.

What results would you expect from this experiment? Think about this for a moment before reading on.

Classical expectations

I don't know about you, but here is what I would expect: Once the atom enters the non-uniform magnetic field, its magnetic arrow precesses in such a way that its projection on the vertical axis remains constant. While this precession is taking place, there is also a force on the atom, and the magnitude of that force depends upon the value of the projection. If the atom has a large positive projection, it will experience a large upward force and move up sharply.

Capsule history of quantum mechanics

Starting in the seventeenth century, and continuing to the present day, physicists developed a body of ideas that describe much about the world around us: the motion of a cannonball, the orbit of a planet, the working of an engine, the crack of a baseball bat. This body of ideas is called classical mechanics.

In 1905, Albert Einstein realized that these ideas didn't apply to objects moving at high speeds (that is, at speeds near the speed of light) and he developed an alternative body of ideas called relativistic mechanics. Classical mechanics is wrong in principle, but it is a good approximation to relativistic mechanics when applied to objects moving at low speeds.

At about the same time, several experiments led physicists to realize that the classical ideas also didn't apply to very small objects, such as atoms. Over the period 1900–1927 a number of physicists (Planck, Bohr, Einstein, Heisenberg, de Broglie, Schrödinger, and others) developed an alternative quantum mechanics. Classical mechanics is wrong in principle, but it is a good approximation to quantum mechanics when applied to large objects.

What is the nature of quantum mechanics?

I'm not going to spend any time on the history of quantum mechanics, which is convoluted and fascinating. Instead, I will focus on the ideas developed at the end. What sort of ideas required twenty-eight years of development from this stellar group of scientists?

Einstein's theory of relativity is often (and correctly) described as strange and counterintuitive.

We started to investigate quantum mechanics by considering only the quantization of magnetic arrows. In our explorations we found out that the magnetic arrow had some funny properties (for example, it was possible that mx did not have a definite value), but at first it seemed that other properties, such as the position of an atom, behaved in the familiar classical way. Eventually (section 9.3) we found that it was also possible to have an atom without a definite value for its position. In this chapter, we investigate what happens when we apply quantum mechanics to a particle's position.

Ball bouncing from a floor

This chapter will show our framework for quantum mechanics in action, by applying it to the problem of a ball bouncing from a floor. Let us use a very fast ball, such as an electron, so that we can ignore the force of gravity. (We restrict ourselves to an electron that is moving fast on a human scale but slow compared to the speed of light, so that relativistic considerations don't come into play. Also, the magnetic arrow associated with the electron has no effect on the phenomena described in this chapter, so I won't mention it again.)

Imagine a source of balls that could send a ball flying in any direction, for example a hot tungsten filament that boils off electrons. Suppose a ball begins at point P, bounces off the floor, and ends up at point Q. (Points P and Q are equally distant from the floor.

Quantum mechanics is valuable because it opens a discussion about the nature of reality, because it demonstrates the power of reason in revealing the truth even when common sense is an obstacle, and simply because it is good to know how our universe ticks (“knowledge is better than ignorance”). But it is also valuable because a host of practical devices, from lasers to transistors to superconductors, all work because of quantum mechanics.

Most of these applications are beyond the scope of this book. I could tell you in vague terms how a laser works, but I could never convince you that my description was correct — you would have to accept it on my authority, and acceptance on the basis of authority is the very antithesis of scientific thought. However, there is one very recent, very exciting application of quantum mechanics that can be treated in full within the “rigorous but not technical” style of this book, namely the use of quantum mechanics to send coded messages. (You may skip this chapter without interrupting the flow of the book's argument.)

Can you keep a secret?

Sending coded messages is a part of life. Governments and businesses need to transmit secrets that would be deadly in the wrong hands (military plans, formulas for explosives, etc.). But even you have information that you don't want everyone in the world to know: your bank balance, your voting record, your vacation plans. I'm not suggesting that you should be embarrassed about your bank balance, but it's your private information and no eavesdropper has any right to it.

How shall we approach the principles of quantum mechanics? One way is simply to write them down. In fact I have already done that (in the first paragraph of the Preface), but to do so I had to use words and concepts that you don't yet understand. To develop the necessary understanding I will use a particular physical system as a vehicle to propel our exploration of quantum mechanics. Which system? An obvious choice is the motion of a tossed ball. Unfortunately this system, while simple and familiar in classical mechanics, is a complicated one in quantum mechanics. We will eventually get to the quantum mechanics of a tossed ball (in chapter 14, “Quantum mechanics of a bouncing ball”, page 103), but as the vehicle for developing quantum mechanics I will instead use a system that is simple in quantum mechanics but that is, unfortunately, less familiar in daily life. That system is the magnetic needle in a magnetic field. This chapter describes the classical motion of a magnetic needle so that we will be able to see how its classical and quantal behaviors differ.

Magnetic needle in a magnetic field

A magnetic needle — like the one found in any woodsman's compass — has a “north pole” and a “south pole”. I will symbolize the magnetic needle by an arrow pointing from its south pole to its north pole.

This book presents the two central concepts of quantum mechanics in such a way that non-technical readers will learn how to work simple yet meaningful problems, as well as grasp the conceptual bizarreness of the quantal world. Those two central concepts are: (1) The outcome of an experiment cannot, in general, be predicted exactly; only the probabilities of the various outcomes can be found. (2) These probabilities arise through the interference of amplitudes.

The book is based on a short course (only fourteen lectures) that I have presented to general-audience students at Oberlin College since 1989, and thus it is suitable for use as a course textbook. But it is also suitable for individual readers looking for intellectual adventure. The technical background needed to understand the book is limited to high school algebra and geometry. More important prerequisites are an open mind, a willingness to question your ingrained notions, and a spirit of exploration. Like any adventure, reading this book is not easy. But you will find it rewarding as well as challenging, and at the end you will possess a genuine understanding of the subject rather than a superficial gloss.

How can one present a technical subject like quantum mechanics to a non-technical audience? There are several possibilities. One is to emphasize the history of the subject and anecdotes about the founders of the field. Another is to describe the cultural climate, social pressures, and typical working conditions of a quantum physicist today.

The previous chapter covered the most important aspects of the Einstein–Podolsky–Rosen conundrum. But some interesting new features have come up since Aspect performed his experiments, and I thought you might enjoy them, so I'll mention two of them here. You may skip this chapter without interrupting the flow of the book's argument.

The results of the Aspect experiment were welcomed by most scientists as a final confirmation of the principles of quantum mechanics, principles that had already been verified magnificently in numerous experiments that were not as clean nor as easy to understand as the test of Bell's theorem. But scientists also looked for possible flaws in the confirmation, and they found one. We have discussed an ideal experiment, in which the source produces a pair of atoms and each tilting analyzer detects one of them. But in Aspect's real experiment, it often happened that after the source launched its atoms only one of the two atoms was detected, and sometimes neither of them were. This is not surprising: perhaps one of the atoms collided with a stray nitrogen molecule and was deflected away from its detector, or perhaps the detector electronics were pausing to reset after detecting one atom when a second atom rushed in. For these reasons, in analyzing his experiment Aspect ignored cases where only one atom was detected. But another possibility is that each atom is generated with an instruction set which could include the instruction “don't detect me”. If this possibility is admitted, then one can invent local deterministic schemes that are consistent with Aspect's experimental results.

According to Charles de Gaulle, Napoleon's military genius lay in his ability “to grasp the situation, to adapt himself to it, and to exploit it to his own advantage”. Most of this book has treated the first two of these steps: learning what quantum mechanics is and how to work with it, whether we like it or not. This appendix moves on to the third step of exploitation.

The applications of quantum mechanics are myriad. Quantum mechanics underlies all chemical and biochemical reactions, the design of drugs and of alloys, and the generation of medical X-rays. It is essential to the laser, to the transistor, and to a sensitive detector of magnetic field called the SQUID (Superconducting QUantum Interference Device). But for the purposes of this book, it is useful to focus on only three of these applications: quantum cryptography, tunneling applications, and quantum computers. The first of these was treated in chapter 13; this appendix describes the second and third. These descriptions are segregated into an appendix because I don't know how to treat them thoroughly at the mathematical level of this book. Consequently, the treatments here are more descriptive and less analytic than the treatments in the chapters.

Many chapters in this book are followed by problems (see page 10) that pertain specifically to that chapter. This appendix contains questions of a more general character. These questions are designed either to consolidate your understanding or to extend your knowledge. The latter sort of question will require further study, such as through reading books listed in the references. But answering the questions will generally require considerable analytic thought and not just parroting a book from the library.

D.1 Is God a deceiver? A central element of René Descartes's philosophy is that we can usually trust our sensual perceptions because God is not a deceiver. The macroscopic world seems to obey the deterministic laws of Newton, yet quantum mechanics maintains that this is just an appearance: the actual laws of physics are probabilistic not deterministic. Does this mean that Descartes was wrong and that God is a deceiver?

D.2 Is quantum mechanics really strange? Throughout this account (beginning with its title) I have emphasized that I find quantum mechanics to be strange. My question here: Is quantum mechanics intrinsically weird, or do I find it weird only because of the way I was brought up? For example, in the Middle Ages most people were brought up believing the earth to be flat. The round earth model must have seemed extraordinarily strange to them when it was first broached. (For example, it must have seemed paradoxical that you could travel always due east and yet eventually arrive back at your starting point.) Yet today even children find nothing unnatural about the round earth because they have heard about it from infancy.

In chapter 9 we concluded that in quantal interference experiments a single atom passes through both branches of an interferometer. In chapter 10.1 we firmed up that everyday-language expression to the technical phrase “there is an amplitude for the atom to go through either branch”. Exactly what do these strange statements mean? How can our minds grow familiar with a real quantal atom, which behaves so unlike a small, hard marble? To prepare for these questions, this chapter examines two variations of the quantal interference experiment. This chapter is not absolutely essential for the logical development of the book, but it dramatically underscores that quantal interference demands a total rethinking of our picture of the atom — no simple trick will suffice.

The Aharonov–Bohm effect

It's possible to build a box called a “corkscrew” from a uniform magnetic field twisted into one turn of a spiral (see figure below). At the left edge of the box the magnetic field points straight out of the page (that is, in the +x direction). Moving towards the right the magnetic field slowly dips until it points straight down, then continues to twist until it points into the page, then straight up, until finally — at the right edge of the box — the magnetic field again points straight out of the page.