At the beginning of the 20th Century new branches of physics were developed to describe two extreme realms of reality: the very small and the very fast. The special theory of relativity was developed around 1905 and describes physical systems traveling at near the speed of light. Likewise, quantum mechanics was developed starting around 1900 to describe systems at the atomic level. Both theories radically transformed our understanding of science and philosophy, and now affect our everyday life.
At the beginning of the 20th Century new branches of physics were developed to describe two extreme realms of reality: the very small and the very fast. The special theory of relativity was developed around 1905 and describes physical systems traveling at near the speed of light. Likewise, quantum mechanics was developed starting around 1900 to describe systems at the atomic level. Both theories radically transformed our understanding of science and philosophy, and now affect our everyday life.
The beginning of the 20th century is marked by tremendous new scientific developments and specifically by the birth of quantum mechanics. It starts with Max Planck and Max Einstein, who introduce and apply the concept of quanta to characterize the energy of light. Quantum computers can considerably decrease the time it takes to complete calculations and solve extremely complex problems. Until the 18th century, very little is understood about light. According to the philosophers Euclid and Plato, light is produced by our eyes. Light is clearly capable of traveling through an interstellar void, while a wave needs a medium for material propagation to exist. However, because of Isaac Newton’s reputation, the corpuscular theory persists until the 19th century. When light is beamed on the top layer, it frees and pushes an electron that moves towards the bottom layer, creating a current. This current is then used to power engines for electric cars.
The early twentieth century was a revolutionary time in the history of physics. People often think of Einstein's special theory of relativity of 1905, which changed our conceptions of time and space. But among physicists, quantum mechanics is usually regarded as an even more radical change in our thinking about the physical world. Quantum mechanics, which was developed between 1900 and 1926, began as a theory of atoms and light but has now become the framework in terms of which all basic physical theories are expected to be cast. We need quantum ideas not only to understand atoms, molecules, and elementary particles, but also to understand the electronic properties of solids and even certain astronomical phenomena such as the stability of white dwarf stars. The theory was radical in part because it introduced probabilistic behavior as a fundamental aspect of the world, but even more because it seems to allow mutually exclusive situations to exist simultaneously in a “quantum superposition.” We will see later that the possibility of quantum superposition is largely responsible for a quantum computer's distinctive advantage over an ordinary computer. The present chapter is devoted to introducing the basic principles of quantum mechanics. There are essentially four components of the mathematical structure of quantum mechanics. We need to know how to represent (i) states, (ii) measurements, (iii) reversible transformations, and (iv) composite systems. We will develop the first three in stages, starting with a very simple case – linear polarization of photons – and working toward the most general case.
Science has developed through a variety of investigations more or less over the time scale of human existence. On this scale, quantum mechanics is an extremely young field, existing essentially for only just over a century. Even our understanding of classical mechanics has existed for a comparatively long period—roughly having been formalized with Newton's equations published in his Principia Mathematica , in April 1686 (1687). Up until the very end of the nineteenth century, mechanics in the form of Newton's laws were comfortable, being understood and accepted by all who dealt with the motion of reasonable bodies. With the turn of the new century, however, mechanics would be shaken to its roots. The change began already on December 14, 1900, when Planck (1900) presented his new theory of black-body radiation to the German Physical Society, and this change would affect mechanics in physically small structures (and lead to quantum mechanics). Mechanics of large bodies would similarly be changed in 1905 with the arrival of Albert Einstein's (1905a) relativity theory. In this book, we will deal with only the first of these two great breakthroughs.
Unlike special relativity, quantum mechanics was not the inspired product of one mind. It developed, in fits and starts, over a period of 25 years (1900–1925), and many cooks stirred the broth. Even now, a century later, quantum mechanics raises profound conceptual questions. Every competent physicist can “do” quantum mechanics, and the predictions it makes are in spectacularly good agreement with experimental results; there can be little doubt that quantum mechanics is “right.” On the other hand, as Richard Feynman once remarked, “nobody understands quantum mechanics.” I'll tell the story in two parts. First the history (how the essential pieces of the puzzle were assembled), and then the implications (what the finished picture has to say about our world). Photons Like relativity, quantum mechanics began with the study of light. In both cases, this historical association is misleading. Relativity, really, has nothing to do with light – to be sure, it involves that magic speed limit, c , and of course the most familiar thing that travels at speed c is light. So light is a convenient vehicle for introducing the subject. But it is perfectly possible to imagine a universe in which there is no such thing as light, and yet relativity might still be valid. Quantum mechanics, too, starts with light – in this case the “quantum” of light (the photon). But, as it turns out, this is just one rather specialized example of much more general principles – in fact, a rather bad example, because photons are by their nature relativistic (they travel at the speed of light), and it would be more natural to begin with the quantum mechanics of slow-moving objects, in analogy with Newtonian mechanics.
The first decades of the twentieth century were marked by two revolutionary scientific accomplishments, the theory of relativity and quantum mechanics, with repercussions still felt today. Relativity theory and quantum mechanics became the two most important branches of “Modern Physics” that emerged as an alternative to “Classical Physics” (a term often used interchangeably with that of “Newtonian” or “Galilean” physics). No field of science (from astronomy to the life sciences), no field of knowledge (from philosophy to sociology), and no artistic practice (from architecture to the fine arts) was left untouched by these investigations into the nature of our physical universe.
In the 20th century, two new terms were added to the scientific lexicon: relativistic mechanics and quantum mechanics. The challenge to the concept of the absoluteness of space and time started at the end of the 19th century and gave birth to Einstein's special relativity theory in 1905. This unified space and time to a single entity, space–time, and we have been introduced to many new concepts such as time dilation, space contraction, mass–energy relation, and the space–time continuum in relativity theory. We have also learned that the highest possible velocity in this universe is that of light and that nothing in the universe can move faster than light. Einstein's general theory of relativity took into consideration gravitation and established the chronogeometric concept of gravity. Quantum mechanics developed by Bohr, de Broglie, Heisenberg, Schrödinger, and others gave us knowledge in the atomic and subatomic domain and led to the wave–particle duality of de Broglie, the uncertainty principle of Heisenberg, the wave mechanics of Schrödinger, and so forth. We have learned that three distinct chords are at the heart of quantum mechanics—granularity, relational, and indeterminism. Although quantum mechanics is a weird subject, as we will discuss, the modern physics is totally based on it.