Lectures on Quantum Mechanics

Volume 1: Basic Matters

Berthold-Georg Englert

• Publisher: WSPC
• Year: 2006
• ISBN:
  9789814365529
• Topic: Naturwissenschaften
• Subtopic: Quantentheorie

Readership: Undergraduates in physics; also in chemistry, mathematics, and engineering; physics lecturers; Perturbed Evolution for graduate students in physics as well.

Chapter 1

A Brutal Fact of Life

1.1 Causality and determinism

Before their first encounter with the quantum phenomena that govern the realm of atomic physics and sub-atomic physics, students receive a training in classical physics, where Isaac Newton’s mechanics of massive bodies and James C. Maxwell’s electromagnetism — the physical theory of the electromagnetic field and its relation to the electric charges — give convincingly accurate accounts of the observed phenomena. Indeed, almost all experiences of physical phenomena that we are conscious of without the help of refined instruments fit perfectly into the conceptual and technical framework of these classical theories. It is instructive to recall two characteristic features that are equally possessed by Newton’s mechanics and Maxwell’s electromagnetism: Causality and Determinism.

Causality is inference in time: Once you know the state of affairs — physicists prefer to speak more precisely of the “state of the system” — you can predict the state of affairs at any later time, and often also retrodict the state of affairs at earlier times. Having determined the relative positions of the sun, earth and moon and their relative velocities, we can calculate highly precisely when the next lunar eclipse will happen (extreme precision on short time scales and satisfactory precision for long time scales also require good knowledge of the positions and velocities of the other planets and their satellites, but that is a side issue here) or when the last one occurred. Quite similarly, present knowledge of the strength and direction of the electric and magnetic fields together with knowledge about the motion of the electric charges enables us to calculate reliably the electromagnetic field configuration in the future, or the past.

Causality, as we shall see, is also a property of quantal evolution: Given the state of the system now, we can infer the state of the system later (but, typically, not earlier). Such as there are Newton’s equation of motion in mechanics, and Maxwell’s set of equations for the electromagnetic field, there are also equations of motion in quantum mechanics: Erwin Schrödinger’s equation, which is more in the spirit of Maxwell’s equations, and Werner Heisenberg’s equation, which is more in Newton’s tradition.

We say that the classical theories are deterministic because the state of the system uniquely determines all phenomena. When the positions and velocities of all objects are known in Newton’s mechanics, also the results of all possible measurements are predictable, there is no room for any uncertainty in principle. Likewise, once the electromagnetic field is completely specified and the positions and velocities of all charges are known in Maxwell’s electromagnetic theory, all possible electromagnetic phenomena are fully predictable.

Let us look at a somewhat familiar situation that illustrates this point and will enable us to establish the difference in situation that we encounter in quantum physics. You have all seen reflections of yourself in the glass of a shopping window, while at the same time having a good view of the goodies for sale. This is a result of the property of the glass sheet that it partly transmits light and partly reflects it. In a laboratory version we could have 50% probability each for transmission and reflection:

A light source emits pulses of light, which are split in two by such a half-transparent mirror, half of the intensity being transmitted, the other half reflected. Given the properties of pulses emitted by the source and the material properties of the glass, we can predict completely how much of the intensity is reflected, how much is transmitted, how the pulse shape is changed, and so forth—all these being implications of Maxwell’s equations.

But, we know that there is a different class of phenomena that reveal a certain graininess of light: the pulses consist of individual lumps of energy — “light quanta”, or “photons”. (We are a bit sloppy with the terminology here, at a more refined level, photons and light quanta are not the same, but that is irrelevant presently.) We become aware of the photons if we dim the light source by so much that there is only a single photon per pulse. We also register the reflected and transmitted light by single-photon counters:

What will be the fate of the next photon to come? Since it cannot split in two, either the photon is transmitted as a whole, or it is reflected as a whole, so that eventually one of the counters will register the photon. A single photon, so to say, makes one detector click: either we register a click of detector

or of detector

, but not of both.

What is important here is that we cannot predict which detector will click for the next photon, all we know is the history of the clicks of the photons that have already arrived. Perhaps a sequence such as

was the case for the last ten photons. In a long sequence, reporting the detector clicks of very many photons, there will be about the same number of

clicks and

clicks, because it remains true that half of the intensity is reflected and half transmitted. On the single-photon level, this becomes a probabilistic fact: Each photon has a 50% chance of being reflected and an equal chance of being transmitted. And this is all we can say about the future fate of a photon approaching the glass sheet.

So, when repeating the experiment with another set of ten photons, we do not reproduce the above sequence of detector clicks, but rather get another one, perhaps

And a third set of ten would give yet another sequence, all 2¹⁰ possible sequences occurring with the same frequency if we repeat the experiment very often.

Thus, although we know exactly all the properties of the incoming photon, we cannot predict which detector will click. We can only make statistical predictions that answer questions such as “How likely are four

s and six

s in the next sequence of ten?”

1-1 Answer this question.

What we face here, in a simple but typical situation, is the lack of determinism of quantum phenomena. Complete knowledge of the state of affairs does not enable us to predict the outcomes of all measurements that could be performed on the system. In other words, the state does not determine the phenomena. There is a fundamental element of chance: The laws of nature that connect our knowledge about the state of the system with the observed phenomena are probabilistic, not deterministic.

1.2 Bell’s inequality: No hidden determinism

Now, that raises the question about the origin of this probabilistic nature. Does the lack of determinism result from a lack of knowledge? Or, put differently, could we know more than we do, and then have determinism reinstalled? The answer is No. Even if we know everything that can possibly be known about the photon, we cannot predict its fate.

It is not simple to make this point for the example discussed above with simple photons incident on a half-transparent mirror. In fact, one can construct contrived formalisms in which the photons are equipped with internal clockworks of some sort that determine in a hidden fashion where each photon will go. But in more complicated situations, even the most ingenious deterministic mechanism cannot reproduce the observed facts in all respects. The following argument is a variant of the one given by John S. Bell in the 1960s.

Consider the more general scenario in which a photon-pair source always emits two photons, one going to the left, the other going to the right:

Each photon is detected by one of two detectors eventually—with measurement results +1 or −1 — and the devices allow for a number of parameter settings. We denote by symbol a the collection of parameters on the left, and by b those on the right. Details do not matter, all we need is that different settings are possible, that is: there is a choice between different measurements on both sides. The only restriction we insist upon is that there are only two possible outcomes for each setting, the abstract generalization of “transmission” and “reflection” in the single-photon plus glass sheet example above.

For any given setting, the experimental data is of this kind:

The products in the last row distinguish the pairs with the same...

Table of contents

• Cover Page
• Title Page
• Copyright Page
• Dedication Page
• Table of Contents
• Preface
• 1. A Brutal Fact of Life
• 2. Kinematics: How Quantum Systems are Described
• 3. Dynamics: How Quantum Systems Evolve
• 4. Motion along the x Axis
• 5. Elementary Examples
• Index
• Back Cover