Lecture Outline

Quantum Mechanics

The Teaching Company

01.    The Quantum Enigma

Quantum mechanics is an extraordinarily successful physical theory. It accounts for a breathtaking range of phenomena, including the structure of atoms and nuclei, the interaction of light with matter, the behavior of solids and liquids at supercold temperatures, and even the surprising properties of "empty" space. All of these will be examined in the lectures to come. The basic ideas of quantum mechanics are, however, very far removed from our everyday experience. They often come into conflict with our intuition. Even today, physicists and philosophers debate the exact interpretation of the theory. These debates will also form a key part of the course.

02.    The View from 1900

We begin with an account of the development of "classical" physics, the laws of nature as they were understood around the year 1900, just before the quantum revolution. Natural philosophers had long debated whether the physical world was continuous or discrete. The Greek Democritus theorized that everything was made of discrete particles called "atoms." Aristotle thought instead that the world was composed of smooth, continuous substances. A similar debate arose in the 17' century about the nature of light. Newton believed that light was a stream of discrete corpuscles, while Huygens proposed that light was a phenomenon of continuous waves. In the 19th century, experiment and theory combined to yield a resolution of these age-old questions. Matter was made of discrete atoms, while light was shown to be a continuous wave of electromagnetic energy. This synthesis was extremely successful. Nevertheless, a few odd phenomena remained tantalizingly difficult to explain.

03.    Two Revolutionaries—Planck and Einstein

In 1900, Max Planck proposed a startling theory to explain the thermal radiation emitted by a warm object. Despite the continuous-wave nature of light, he argued that light energy could only be emitted or absorbed in discrete amounts called "light quanta." Albert Einstein soon realized that Planck's hypothesis implied that light could behave as a swarm of particles, later christened "photons." Einstein considered the "photoelectric effect," in which electrons are ejected from a metal surface that is exposed to light. Using the photon idea, he was able to explain all of the perplexing details of this effect. Einstein also proposed that the quantum hypothesis applied to matter as well as light. In this way he was able to solve another long-standing puzzle about solids at low temperatures.

04.    Particles of Light, Waves of Matter

Light propagates through space as a continuous wave, but somehow it exchanges its energy in the form of discrete particles.  This weird "wave-particle duality" also applies to matter. In 1924, de Broglie proposed (and experiments soon confirmed) that particles like electrons also have wave properties and can exhibit wave phenomena like interference. Everything, it seems, has both wave and particle properties, and both pictures are needed to explain all the phenomena of nature. The pictures are linked by 2 simple mathematical relations: (1) the Planck-de Broglie equations relating the frequency and wavelength of the wave to the energy and momentum of the particle, and (2) the Born rule relating the intensity of the wave at any point to the probability of finding a particle there. Quantum waves are waves of probability amplitude.

05.    Standing Waves and Stable Atoms

Ernest Rutherford showed that atoms were composed of a dense, positively charged nucleus surrounded by orbiting, negatively charged electrons. But classical physics predicted that these orbiting charges should lose their energy and quickly spiral inward.  Every atom should implode in a microsecond! Bohr suggested that quantum theory could explain the stability of atoms by allowing only certain discrete orbits for the electrons. In effect, the atom can be viewed as a "ladder" of discrete energy levels. Niels Bohr was in this way able to explain the frequencies of light emitted by atoms of different elements. The wave nature of the electron soon made things even clearer. Only certain discrete "standing wave patterns" are possible for an electron moving around the nucleus.  These 3-dimensional patterns exactly account for the atomic energy levels.

06.    Uncertainty

How can we reconcile the particle and wave pictures of an electron? How is it possible for microscopic physics to be so different from our ordinary experience of matter and motion? One of the deepest answers to this question was given by Werner Heisenberg, who showed how wave properties limit the "particleness" of a particle. Because of the phenomenon of wave diffraction, no quantum object can both have a definite location in space and carry a definite momentum. There is also a similar basic relation between energy and time. The trade-off between different particle properties is important for something as small as an electron, but it is too tiny to observe in large-scale objects. Thus Heisenberg's uncertainty principle shows how large-scale, everyday physics can appear to be classical, in spite of quantum mechanics.

07     Complementarity and the Great Debate

During the early days of quantum mechanics, Einstein and Bohr carried on a long-running debate about the new physics. Einstein rejected the indeterminism of quantum mechanics, saying that "God does not play dice with the universe." In a series of clever thought experiments, he tried to show that the theory did not make sense. Bohr defended quantum mechanics with his "principle of complementarity." Observation, Bohr insisted, is not a passive act; it is an active intervention in the world. This intervention necessarily involves a choice, and the choice restricts what we can observe. We may design an experiment to measure precisely either the location of a particle or its momentum, but never both at once.  It is Bohr's principle that unravels the apparent paradoxes of the quantum world.

08.    Paradoxes of Interference

We now begin the task of explaining the laws of quantum physics by considering a very simple type of quantum system: a single photon in an apparatus consisting of a few mirrors and photon detectors. Even in such a simple example, quantum mysteries and paradoxes abound. By considering a series of thought experiments, we can explore the ideas of wave-particle duality, quantum interference, and complementarity. Perhaps the most striking thought experiment is the bomb-testing problem." Imagine a bomb with a trigger so sensitive that a single photon would detonate it. How can we use light to test whether the bomb is in working order—without exploding it?

09.    States, Amplitudes, and Probabilities

The simple photon interferometer example from Lecture Eight is our doorway into the formal math of quantum theory. By learning a few symbols and a handful of rules for manipulating and interpreting them, we can describe the states of quantum particles, show how these states change over time, and predict the results of measurements. One key idea is the principle of superposition, which tells us how to combine 2 quantum states to form a new one. Another concept is the probability amplitude, used to calculate the likelihood of the various outcomes of an experiment.

10,    Particles That Spin

As a planet moves around the Sun, it also may rotate about its own axis. In an analogous way, many quantum particles (such as electrons) both move through space and have an intrinsic "spin."  But a quantum spin is not quite like the spin of a planet. The total amount of spin and its possible spatial orientations are subject to some peculiar quantum rules. Analyzing spin gives us another simple "laboratory" for exploring the basic ideas of quantum mechanics. An appreciation of particle spin will become one of our key tools for understanding the quantum world.

11.    Quantum Twins

Macroscopic objects obey the "snowflake principle": No 2 of them are ever exactly alike. But quantum particles are not snowflakes!  Every electron, for instance, is perfectly identical to every other, with identical mass, charge, spin, and so on. If we swap the positions of 2 electrons, we create a new situation that is physically identical to the original. This leads us to posit 2 types of quantum particle. Bose-Einstein particles, or bosons, are completely symmetric under particle exchange. Fermi-Dirac particles, or fermions, are antisymmetric. Both types of particles exist. For instance, electrons are fermions, and photons are bosons. The difference between the 2 basic particle types is a simple factor of -1 in the quantum rules governing them. As we will see in the next few lectures, this small difference has tremendous consequences.

12.    The Gregarious Particles

Bosons are quantum particles whose states are symmetric under particle exchange. Photons are bosons; so are helium atoms. Under some conditions, even a bound-together pair of electrons can behave like a single boson. Bosons do not mind being in the same quantum state. In fact, they prefer it that way. This tendency of bosons to congregate into the same state leads to some fantastic effects. In a laser, huge numbers of photons are created moving in exactly the same direction with the same energy. And in the low-temperature phenomena of superfluidity and superconductivity, the quantum mechanics of bosons can cause liquids and electric charge to flow forever without any resistance.

13.    Antisymmetric and Antisocial

Fermions are quantum particles whose states are antisymmetric under particle exchange. That is, the exchange of 2 fermions introduces a factor of-1 into the quantum state. Electrons, protons, and neutrons are all fermions. Unlike bosons, fermions are governed by the Pauli exclusion principle: No 2 identical fermions can ever be in the same quantum state. This iron rule determines the structure and chemical properties of atoms with many electrons and influences the stability of atomic nuclei. We see the effects of the Pauli exclusion principle every minute of every day. The reason that matter "occupies space" at all is that its basic constituents—the protons, neutrons, and especially the electrons— are all fermions.

14.    The Most Important Minus Sign in the World

The difference between bosons and fermions shapes the whole physical world. But where does this crucial -1 factor come from? A clue is provided by the "spin-statistics rule." Particles with spin 0 or 1 are always bosons, while particles with spin ½  are always fermions. But how is spin related to particle exchange?

Surprisingly, rotating an object by 360° (1 full turn) is not exactly the same thing as not rotating the object at all. We can illustrate this by a simple demonstration involving a twisted ribbon attached to 2 pencils. For spin -½ particles, a full rotation introduces a "twist" factor of -1. These particles must be rotated by 2 full turns to restore their original "untwisted" state. The twist factor for spin - ½ particles turns out to be the origin of the particle-exchange factor for identical fermions—the most important minus sign in the universe.

15.    Entanglement

When 2 particles are part of the same quantum system, they may be entangled with each other. This means that the states of the particles are connected in a somewhat mysterious way. We can illustrate this by considering the "total spin 0" state of 2 spin-½ particles. If the same spin measurement is made on the 2 particles, they always give opposite results. A few years after the original Bohr-Einstein debate, Einstein used entanglement to argue that quantum mechanics had to be incomplete—that is, that there must be physically real things that cannot be described by quantum mechanics. Bohr replied that 2 entangled quantum particles, even if they are far apart in space, must be regarded as a single quantum system, so Einstein's argument did not hold water. Who was right?

16.    Bell and Beyond

Thirty years after Einstein's argument, an astonishing discovery by John Bell turned it completely upside down. Common sense suggests that each particle carries its own local "instructions" about how to behave in an experiment. Bell showed that, for entangled quantum particles, this common-sense view cannot be reconciled with quantum mechanics. Furthermore, the experiments show that quantum mechanics is right! Bell's breakthrough has tremendous philosophical implications. It presents us with a logical dilemma. Either we must give up the idea that particles have definite (but unknown) spin, position, momentum, etc., before they are measured, or we must imagine that the particles in the universe are all connected by a web of instantaneous communication links. Either way, our everyday intuition breaks down when we enter the quantum realm.

17.    All the Myriad Ways

How does an electron get from point A to point B? Richard Feynman gave a surprising answer. He said that the electron travels from A to B along every conceivable path. Each path contributes equally to the total quantum amplitude, whose square gives the probability that the electron gets from A to B. This is quantum interference taken to its ultimate extreme! Feynman's wild idea provides a powerful way of analyzing how things happen in the quantum world. By drawing a series of more and more complex "cartoons," we can give a more and more precise account of how electrons interact with light and move from place to place. Some of these cartoons show particles moving forward and backward in time, while other particles appear from nowhere and disappear again. But these wacky alternate histories make real contributions and have observable effects on the behavior of electrons and photons.

18.    Much Ado about Nothing

A classical pendulum can swing back and forth with any energy. We can even imagine it hanging at rest, with no energy at all. Not so a quantum pendulum! Even in the lowest-energy "ground state," a quantum pendulum still has an irreducible vibration, or "zero-point" motion. Zero-point energy of atoms is the reason why liquid helium will not freeze even at absolute zero. Zero-point energy is also present in the electromagnetic field. In the vacuum state, with no photons at all, there remains an irreducible quantum vibration of the field. The quantum vacuum is a complex, seething, rapidly fluctuating medium.  Hendrik Casimir discovered that the energy of the quantum vacuum can actually be observed as a tiny attraction between 2 metal plates. More dramatically, vacuum energy may be the source of the "dark energy" that causes our universe to expand at an ever-accelerating rate.

19.    Quantum Cloning

We will now begin to explore a contemporary topic in quantum physics, the subject of quantum information. The idea is that we can use the strange rules of the quantum world to store, retrieve, transmit, and process bits of data. The idea of information turns out to be quite useful for understanding the quantum world. Nowadays, we are very familiar with the classical idea of information. We know how the same information can be represented in different physical forms, can be sent from one place to another, can be copied, and so on. Quantum information is much the same, with one profound difference: Quantum information cannot be perfectly copied. This is the lesson of the "no-cloning" theorem of quantum mechanics, a principle with many amazing implications.

20.    Quantum Cryptography

Sometimes we want to keep information private, to protect it from being intercepted by an eavesdropper. This is the task of cryptography, the science of secret codes. We can encrypt our messages so that no one can read them without a special "key," but how do we distribute the secret key without it being intercepted? In the classical world, where any information may be copied, perfect security is impossible. But quantum mechanics provides a method for doing this known as "quantum key distribution." Alice and Bob can use this method to establish a secret key that is guaranteed by the no-cloning theorem to be unknown to any eavesdropper Eve.  And this is more than a pie-in-the-sky thought experiment. A handful of banks and government agencies already use quantum key distribution systems to ensure the security of their most secret data.

21.    Bits, Qubits, and Ebits

What are the laws governing quantum information? Charles Bennett has proposed some fundamental rules governing the relations among different sorts of information. He envisions 3 basic kinds of information resource: bits, qubits, and ebits. Thus Alice may send a 1-bit ordinary message to Bob. Alternatively, Alice may send a 1-qubit quantum message to Bob, perhaps encoded in the spin of a particle. Finally, Alice and Bob may share a pair of entangled particles, such as 2 spins in a total spin 0 state. Bennett's rules show how one sort of resource may be used to do the job of another. For instance, 1 qubit can be used to send a 1-bit message, but not vice versa. The most astonishing of Bennett's rules is "quantum teleportation," in which entanglement can be used to send quantum information "instantaneously" from Alice to Bob.

22.    Quantum Computers

As computers become more powerful, and as their components become more and more microscopic, we can envision a future when their basic operations are performed by individual electrons or photons. What then? A quantum computer would be a device that makes full use of the laws of quantum mechanics to perform its computations. In theory, certain very hard mathematical problems would be much easier to solve on a quantum computer.  But any would-be builder of a quantum computer faces a daunting challenge. The machine must be extremely well isolated from the outside world, or else external interference will cause the quantum process to lose its coherence. However, the different parts of the computer must rapidly interact with each other, or else the computation cannot be done. So far, despite some progress, this essential dilemma remains unsolved. A true quantum computer remains as yet only a fascinating possibility.

23.    Many Worlds or One?

More than 80 years after the birth of quantum mechanics, physicists and philosophers still debate its meaning. What is the fundamental nature of the quantum world? For everyday purposes, most physicists use the so-called Copenhagen interpretation of quantum mechanics. The microscopic realm only acquires meaning through macroscopic observations and records, subject to the principles of complementarity and uncertainty. In the hidden-variables interpretation, however, the world is actually deterministic. Apparent quantum indeterminacy is merely due to our ignorance of factors that we cannot directly observe. These unseen factors are highly nonlocal, cooperating instantaneously over vast distances. The most provocative approach is the many-worlds interpretation. Here we regard the entire universe as a single enormous quantum system. In a measurement process, the universal state "splits" into many branches, 1 for each possible outcome. The many-worlds interpretation paints a picture of a vast, multivalued universe encompassing every possibility of the quantum world.

24.    The Great Smoky Dragon

Recall our old single-photon example. Quantum interference between 2 paths only occurs if no record of any sort is made anywhere in the universe of which path the photon chose. "Quantum mechanics is what happens when no one is looking." John Wheeler describes this as the Great Smoky Dragon. The dragon's tail shows where the quantum particle enters our experiment. Its head appears at the other end, biting one of the detectors. But in between, the exact shape of the dragon's body is forever shrouded in smoke. The Great Smoky Dragon dwells in every part of the quantum world.  We can never say whether 2 photons have been exchanged.

We can never assign a definite state to 1 of a pair of entangled particles. We can never tell which path an electron follows from A to B. We can never exactly duplicate quantum information. This inescapable elusiveness is the deep principle behind the quantum enigma.