Einstein's Dice (Physics 190Q)

detailed syllabus

1. Albert's Discontent

Two controversial ideas lie at the heart of quantum mechanics. The first idea is that the future outcome of an experiment cannot be precisely predicted by any kind of prior knowledge. Only a probabilistic range of results can be anticipated. This was Einstein's biggest objection to quantum mechanics: that the result of a measurement is determined by the throw of a cosmic die. The second idea states that what happens in one part of the universe can instantaneously affect the behavior of a distant part of the universe. This concept led Einstein to suggest that quantum mechanics invoked a "spooky action at a distance", a feature he felt was a serious liability in the theory. Despite our discomfort with these ideas, experimental work since the 1970's has clearly confirmed both of these phenomena.

In order to encompass these odd features of the universe, the theory of quantum mechanics is founded on what seems at times to be contradictory models of the laws of physics. However, judicious use of these models can avoid the apparent contradictions and help to build an intuitive understanding of the world that agrees with quantum mechanical prediction (and experiment).

2. Standing in the dark: the story of light waves

In order to understand the quantum universe, we must develop two types of fundamental models of the physical world: particle models and wave models. We begin a study of wave models by examining the wave theory of light in both historical and scientific context.

Around the turn of the 19^th century, the particle theory of light heretofore advocated by the scientific community was supplanted by a wave theory of light, developed in large part by Thomas Young and Augustin Fresnel. This change from particle model to wave model was driven by the discovery of two phenomena that can only be explained in terms of light waves. The first ("diffraction") describes how light spreads out as it goes through a narrow opening. The second ("interference") describes how, under the right circumstances, two light beams cancel each other out to produce darkness. The same principle can be found in noise-canceling headphones, invented in the 1990's, that use interfering sound waves to produce silence.

Familiar examples of water waves, sound waves and waves on a rope are used to introduce the general features of wave behavior. An introduction to Thomas Young's double slit experiment, in which two sources of light are forced to interfere with each other, further demonstrates the effects of diffraction and interference and shows how they are accommodated in the wave model of light.

Although the need for a wave model of light is obvious today, members of the early 19^th century scientific establishment were reluctant to accept it. The story of Fresnel's humiliation at the hands of the French Academy of Sciences demonstrates the difficulty that beset these scientific luminaries -- though Fresnel was ultimately vindicated by a dramatic experimental result. In the latter half of the 19^th century, James Clerk Maxwell incorporated the theory of light waves into the developing theory of electricity and magnetism. A discussion of this connection between light and electromagnetism completes the description of our modern understanding of light waves and gives final proof for the wave-like nature of light.

3. Particle or wave: Mother Nature's indecision

In the 20^th century, two developments changed the connection we make between the particle and wave models of nature. First, wavelike behavior was observed in many systems (such as beams of electrons, neutrons, atoms and molecules) that are hard to imagine as being made up of waves. The first half of this chapter reviews the key evidence for "matter waves", focussing especially on the serendipitous discovery of electron waves by Clinton Davisson and Lester Germer in 1926. Further examination of single particle interference experiments performed throughout the latter half of the 20^th century and into the 21^st century raises some profound questions about the source of this wavelike behavior. Unlike wave motion in water or air, the wavelike behavior we observe in these experiments is a much more fundamental (and much less intuitive) feature of even the tiniest components of matter.

The second major influence on the wave/particle relationship was the discovery that, following the success of 19^th century scientists to characterize light as a wave, some characteristics of light cannot be explained by a wave model after all. This discovery is discussed in the context of two famous works: Max Planck's characterization of blackbody radiation and Albert Einstein's explanation of the photoelectric effect. The wave/particle relationship we now use as the foundation of quantum mechanics is described with a brief review of the work of Louis de Broglie. Nowadays, we understand there is no such thing as a pure particle, and no such thing as a pure wave. The stuff of the universe exhibits characteristics of both.

4. Games of chance

In the conventional view of quantum mechanics, probability and chance play key roles in the operation of the universe, a concept that was formally developed by Max Born in the 1920's in terms of "probability waves". In order to understand how Born came up with this idea, we explore two lines of scientific inquiry. The first is the search for an underlying medium of light waves at the turn of the 20^th century. The astonishing failure to find such a medium demonstrates that the wave character of light does not itself have a material basis, but manifests itself only as an influence on matter. The second concerns the application of wave characteristics to single particles, as discussed in chapter 3. In those cases, the evident wavelike features provide only probabilistic information about the particles. These two conclusions, that quantum waves have no material basis and that they provide only probabilistic information, together motivate Born's concept of probability waves.

Using modern double-slit experiments as examples, we establish the precise relation between wave amplitudes and the quantum mechanical predictions for particle detection. At the same time, we identify the surprising implications of these experiments for an underlying structure of nature that embodies the two key ideas of chapter 1: the inherent probabilistic nature of the universe and Einstein's "spooky action at a distance".

5. Superposition and collapse

The distinction between everyday uncertainty, resulting from a lack of specific knowledge (like the uncertainty in the role of a die), and the fundamental uncertainty inherent in quantum mechanics derives from the nature of "superposition". Superposition describes how waves combine together, sometimes enhancing each other, sometimes canceling each other out. When applied to quantum probability waves, the result is surprisingly different from the combination of probabilities in ordinary games of chance. In quantum mechanics, it may be highly likely that a photon can pass through a barrier and arrive at a certain location when one of two particular holes is opened in the barrier. However, if both holes are opened simultaneously, it is impossible for the photon to reach its destination.

Quantum wave "collapse" extends the concept of quantum probability by describing how a specific experimental result is obtained from a range of possibilities. Examples using polarized light illustrate the counterintuitive nature of quantum superposition and collapse, and underscore the non-realist interpretation of quantum mechanics. Further examples involving "quantum Zeno" experiments, developed in the 1990's, demonstrate how repeated collapse can be used to prevent a system from changing. Examples of this effect mimic the proverbial "watched pot", which refuses to boil as long as someone is keeping an eye on it. A study of "interaction-free-measurement" experiments, also devised in the 1990's, shows how to use the features of superposition and collapse to detect the presence of an object without interacting with it in any way, not even scattering a single photon off it. This technique can been used to take an X-ray-like image of an object without the use of X rays.

6. Complementarity and uncertainty

Experiments with polarized light illustrate another principle of quantum mechanics called "complementarity". This concept was introduced by Werner Heisenberg in his "uncertainty principle" and was debated by Albert Einstein and Niels Bohr for many decades. The uncertainty principle proposes that certain combinations of measurements, such as measurements of polarization in different directions, are complementary; precision in one measurement produces uncertainty in the other. Einstein and Bohr struggled to understand this idea in terms of idealized thought experiments. In doing so, they inadvertently misled generations of scientists into thinking that the uncertainty principle was a simple consequence of measurement. This chapter uses more fundamental examples of complementarity, such as the masses and lifetimes of elementary particles, to convince the reader that uncertainty is not just a limitation in measurement apparatus, but is inherent in the structure of nature.

To better understand this issue we explore how the laws of quantum uncertainty can be used in many practical applications. Our discussion of polarization measurements leads to a scheme for secret communication, known as "quantum cryptography", based on an absolutely unbreakable code carried by photon polarizations. This encryption scheme is guaranteed to be secure no matter what code-breaking computer power is brought to bear on the problem. Quantum uncertainty further explains how objects can pass through barriers that are supposed to be impenetrable according to classical physics. This phenomenon, known as "quantum tunneling", is the principle behind scanning tunneling microscopes, which have been used to make images of atomic-sized structures. It has also led, in special circumstances, to cases of apparent faster-than-light travel. Finally, we discuss how, in accordance with the uncertainty principle, matter and energy can temporarily be created out of nothing. This is realized in the production of virtual particles (out of nothing!) that mediate the known fundamental forces: gravity, electromagnetism, and nuclear forces. Apparently, even a perfect vacuum is never completely empty. Extending this idea to its logical extreme, we explore the notion that the entire universe might be one big quantum fluctuation.

7. The Einstein Podolsky Rosen paradox

So far, we have developed arguments in favor of quantum mechanics and against a local realist model of the universe using the concepts of superposition and collapse. In this chapter we make these arguments more precise by examining one of the most famous and controversial experiments in the history of quantum mechanics. Einstein, Podolsky and Rosen initially formulated the "EPR paradox" as a thought experiment to argue that the theory of quantum mechanics is incomplete. It uses the notions of superposition and collapse to demonstrate how, according to quantum mechanical interpretation, the behavior of one physical system instantaneously affects another distant system. It's as if two gloves chosen at random from two large piles of gloves on opposite sides of the Earth always turned out to be a matched pair. Einstein called this "spooky action at a distance" and considered it a fatal flaw in the quantum mechanical interpretation.

To the scientific community at large, much of this debate seemed to be pure semantics with no consequence for the physical world. However, in the 1960's, an Irish physicist by the name of John Bell recognized that not only did quantum mechanics explain the results of EPR experiments differently from local realist theories, in certain circumstances it made verifiably different predictions for the outcome of the experiments. As a result of Bell's revelation, the philosophical debate was transformed into an experimental test. Since then, extensive laboratory tests have dramatically verified the correctness of quantum mechanics, and the failure of other theories, beyond a shadow of a doubt. It seems that quantum gloves always come in pairs.

8. An entangled web

The EPR experiment is an instance of the more general phenomenon of "entanglement", which describes the non-local behavior of the universe. Entanglement is a special case of superposition in which two quantum mechanical systems are correlated, even though they may be spatially separated. To understand this concept, imagine that the first system is allowed to be in either of two states A or B; the system is a superposition of the two possibilities. Similarly, the second system might be in a superposition of states C and D. Now suppose the two systems are set up so that the first system is in state A if and only if the second system is in state C, and the first is in state B if and only if the second system is in state D. In other words, the combined system can either have State1=A and State2=C or State1=B and State2=D. In this case, each individual system remains ambiguous -- the first system is either in A or B, the second in C or D -- but if we were to make a measurement on the first system to determine which state it is in, we would automatically know the state of the second system. The two systems are completely correlated.

In this section, the ideas behind entanglement are developed using a number of examples from new technologies, including clock synchronization, tracking by Global Positioning Satellites, and photolithography. Finally, and most dramatically, we use the principles of entanglement to explain "quantum teleportation", in which a quantum mechanical state is transferred intact to a distant location. Although this feat may not seem revolutionary in a world where next day shipping is a common convenience, at the quantum level this is a remarkable accomplishment. The uncertainty principle specifies that a quantum system cannot be precisely measured, much less copied. Transporting a quantum state precisely intact is therefore a technological wonder, which can only be done utilizing a loophole in quantum theory. Although not exactly the technology envisioned in Star Trek transporters, quantum teleportation is remarkable in its own right, and is expected to provide the basis for new opportunities in information transfer.

9. Of cats and coherence

To this day, the concept of quantum wave collapse remains controversial among quantum philosophers, for good reason.

Here we return to the issue of wave collapse, and build on the notion of entanglement to introduce the modern idea of "decoherence".

To set the context for further discussion we explore the problems associated with quantum collapse in a number of thought experiments, including a famous one due to Irwin Schroedinger in which a cat in a box is contrived to be in a superposition of being both alive and dead. As soon as the box is opened, the superposition is destroyed and the state of the cat collapses to either one possibility or the other. This case is carefully constructed to emphasize the odd features of quantum collapse that seem unremarkable in the description of invisible microscopic systems, but appear ridiculous for macroscopic felines.

The traditional interpretation of quantum mechanics demands an ill-defined distinction between the quantum system under study (e.g., a cat) and the measurement apparatus (the person who opens the box) in order to produce a distinct unambiguous experimental result (alive or dead). The interaction of the two causes a collapse of the cat's wave function, not the observer's, even though one would expect that both observer and observed are governed by the same universal laws of quantum mechanics. This unnatural asymmetry between observer and observed is something that quantum philosophers have struggled with for decades.

To try to understand the subtleties of quantum collapse more precisely, we explain how the collapse of large systems affects probability distributions, producing what is called a probability "mixture". Studies of "quantum erasure" experiments developed in ?? clarify the nature of observation and measurement, demonstrating that states of quantum superposition are not so much destroyed by measurement as rearranged into different states of quantum entanglement. Building upon these ideas, we introduce the recent (and controversial) idea of decoherence as an alternative to wave collapse. In this view, measurement is a source of entanglement, rather than collapse, but a large number of random entanglements typically involved in a macroscopic measurement mimic the effect of a probability mixture.

10. Discreteness and confinement

"Discreteness" can be said to be the "quantum" in quantum mechanics; it is at the heart of the theory. Many types of measurements result in one of a discrete range of possibilities. For example, particles in a double slit experiment do not land at any possible location on the viewing screen, but rather in one of a set of discrete fringes. In this discussion we tie together several previous ideas to explain discreteness as a result of confining a quantum wave to a finite region. In turn, we use discreteness to explain the energy spectrum of electrons confined to an atom, the residual energy of materials chilled to absolute zero temperature, and the behavior of nanoscale devices such as quantum dots.

In the mid 80's scientists learned how to put this concept to use to make artificial atoms, called quantum dots, which promise unprecedented potential in technologies of tunable lasers, flat panel displays, biological and chemical sensors, fast optical switching, demultiplexing, self replication, and high density information storage.

11. Spin statistics and the spin mystery

Mother Nature is choosy in what she lets interfere. Vertically polarized photons will interfere with other vertically polarized photons, but not with horizontally polarized photons, and certainly not with other types of particles. Like only interferes with like. Moreover, fundamental particles, such as photons or electrons, interfere differently depending on the "spin" of the particle, a characteristic akin to polarization. Particles with integer spin (called bosons) and particles with half-integer spin (called fermions) have opposite interference effects. In cases where bosons interfere constructively, fermions will interfere destructively, and vice versa. To illustrate the profound impact of this effect on science, bosonic interference is used to explain the operation of lasers, superconductors, superfluids, and Bose-Einstein condensates, while fermionic interference is used to explain the energies of atoms, the operation of semiconductors and the structure of white dwarf stars.

12. Quantum computing and information

The theory of quantum mechanics has given rise to a new perspective on information, which incorporates the concepts of superposition and entanglement. To reflect this change in perspective, information technology has developed a new language for quantum information, incorporating the concepts of "qbits" and "ebits", and new algorithms that make direct use of this information. We review some of the most remarkable developments in quantum information technology in recent years, including factoring of large integers (such as those used in cryptographic keys), and high-speed searches of large databases.

Parallel to the progress in information technology, quantum computing has also made great strides. The new quantum computing industry incorporates most of the principles discussed in this course. A future generation of quantum computers will use quantum superpositions to implement quantum algorithms, quantum entanglement and teleportation to transfer information, quantum error correction in order to overcome the effects of decoherence, and secure communication channels based on quantum cryptography. Several examples of physical systems with potential quantum computing capability are examined, including microelectronic systems, strings of atoms, ions in magnetic traps, and nuclear magnetic resonance of molecules in liquids, bio materials.