This chapter covers the early years of quantum theory, a time of guesswork, inspired by problems presented by the properties of atoms and radiation and their interaction. Later, in the 1920s, this struggle led to the systematic theory known as quantum mechanics, the subject of Chapter 5. Quantum mechanics started with the problem of understanding radiation in thermal equilibrium at a non-zero temperature. It was not possible to make progress in applying quantum ideas to atoms without some understanding of what atoms are. The growth of this understanding began with the discovery of radioactivity.
This chapter covers the early years of quantum theory, a time of guesswork, inspired by problems presented by the properties of atoms and radiation and their interaction. Later, in the 1920s, this struggle led to the systematic theory known as quantum mechanics, the subject of Chapter 5. Quantum mechanics started with the problem of understanding radiation in thermal equilibrium at a non-zero temperature. It was not possible to make progress in applying quantum ideas to atoms without some understanding of what atoms are. The growth of this understanding began with the discovery of radioactivity.
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.
The principles of quantum mechanics are so contrary to ordinary intuition that they can best be motivated by taking a look at their prehistory. In this chapter we will consider the problems confronted by physicists in the first years of the twentieth century that ultimately led to modern quantum mechanics. Photons Physicists in the last decades of the nineteenth century were greatly concerned to understand the nature of black-body radiation — radiation that had come into thermal equilibrium with matter at a given temperature T . The energy ρ(ν, T ) d ν per volume at frequencies between ν and ν + d ν had been measured, chiefly at the University of Berlin, and it was known on thermodynamic grounds that ρ(ν, T ) is a universal function of frequency and temperature, but how could one calculate this function? A simple calculation was given in 1900 by John William Strutt (1842–1919), more usually known as Lord Rayleigh. It was familiar that one can think of the radiation field in a box as a Fourier sum over normal modes. For instance, for a cubical box of width L , whatever boundary condition is satisfied on one face of the box must be satisfied on the opposite face, so the phase of the radiation field must change by an integer multiple of 2π in a distance L .
The origin of the relativity theories was closely bound up with the development of electromagnetic concepts, a development that approached a coherent fieldtheoretical formulation, according to which all actions may vary in a continuous manner. In contrast, quantum theory arose out of the development of atomic concepts, a development that was characterized by the acknowledgment of a fundamental limitation to classical physical ideas when applied to atomic phenomena. This restriction was expressed in the so-called quantum postulate, which attributed to any atomic process an essential discontinuity that was symbolized by Planck's quantum of action. Quantum field theory (QFT) is a later phase of the conceptual developments of quantum theory, and has the old quantum theory and non-relativistic quantum mechanics, essentially the preliminary analyses of the interactions between atoms and radiation, as its predecessors. This chapter will review some features of quantum physics that are relevant to the rise of QFT. The quantization of motion In solving the problem of the equilibrium between matter and radiation, Max Planck (1900) showed that the laws of heat radiation demanded an element of discontinuity in the description of atomic processes. In the statistical behavior of atoms represented, in Planck's description, by linear resonators in their interactions with radiation, only states of vibration should be taken into account whose energy was an integral multiple of a quantum, hv , where h is Planck's constant and v is the frequency of the resonator. Planck himself believed that the discontinuity of energy was only a property of atoms, and was not ready to apply the idea of energy quantization to radiation itself.
By the beginning of the twentieth century, classical physics had accumulated many contradictions that could only be resolved by changing the main concepts of space and time (Einstein’s theory of relativity, Chapter P1) and the laws of motion of particles on atomic scales (quantum mechanics). The first quantum concepts appeared at the very beginning of the century (Planck’s formula for thermal radiation and Einstein’s formula for photoeffect, Chapter P5), but quantum mechanics as a closed science was not constructed until the end of the 1920s.
Quantum Mechanics and its Interpretations Peter E. Hodgson (bio) Quantum Mechanics is an elegant and indeed beautiful formalism, and enables the results of many physical measurements to be calculated with great accuracy. But from the earliest days there has been intense debate about the interpretation of quantum mechanics, and it shows no signs of being settled; if anything, it is becoming more intense over the years. It should therefore be evident that the most that can be done here is to try to show where the real difficul-ties lie, and how the failure to recognize that quantum mechanics is an essentially statistical theory has led to a series of confusing quantum paradoxes. It is just these paradoxes that have often been used to support theological speculations. The Discovery of the Quantum The quantum nature of the physical world was discovered by Max Planck in the early years of the twentieth century. He was studying thermodynamics and his attention was drawn to the frequency distribution of black-body radiation. This radiation is emitted from a small hole in a heated enclosure with rough walls, which ensures [End Page 62] that the radiation reaches statistical equilibrium before emission. The frequency distribution is found to be independent of the material lining the walls of the oven, so Planck realized that it must be a fundamental characteristic of all matter. This frequency distribution had been measured by several experimentalists, and Planck resolved to try to see how it can be derived theoretically. Already Rayleigh and Jeans had used classical radiation theory to derive an expression that fitted the data well at low frequencies but tended to infinity at high energies, which was of course unacceptable. The physicist Wien had derived a formula that fitted the data at high frequencies but failed at low frequencies. Planck used an ingenious argument based on entropy to interpolate between these two formulae and obtained a formula, now known as the Planck distribution, that fitted the data perfectly. His next task was to understand the physics behind his formula. He used the familiar technique of assuming that the energy is emitted in small finite amounts, and then let their size tend to zero, so as to represent a continuous distribution. To his great surprise, the intermediate formula with discrete amounts of energy, called quanta, gave the Planck distribution, but the final result, with infinitely small quanta, did not. All his instincts as a physicist rebelled against this conclusion. Surely energy must be emitted continuously. He tried hard to find a way to avoid this unwelcome conclusion, but without success. He had to admit that it is a feature of the world. Soon it was confirmed by Einstein’s interpretation of the photoelectric effect. This story shows clearly that physicists do not impose their ideas on the world, but recognize and accept what they find, whether or not it agrees with their original ideas. The world is made by God in ways that often we cannot imagine. [End Page 63] The Old Quantum Theory A few years later Rutherford discovered that the atom has a small central nucleus containing most of its mass surrounded by a number of electrons. Bohr used the idea of the quantum to build his model of the atom. He assumed that the electrons go around the nucleus in discrete orbits, and that radiation is emitted when an electron falls from one orbit to a lower one. With this model, he was able to calculate to high accuracy the frequencies of the radiation emitted from an excited hydrogen atom. This was a great achievement, but to get his result he had to assume that the electrons stayed in their orbits, whereas classical theory predicted that they would continuously lose energy and rapidly spiral into the nucleus. Quantum Mechanics and its Interpretations These problems were solved by the development of quantum mechanics in the 1920s by Heisenberg, Schrodinger, Dirac, and others. This enabled what is called the wave function to be calculated for any physical system, and from the wave function the probabilities of occurrence of measurable quantities can be found. It is a beautiful theory, and is the indispensable means of calculation for atomic, nuclear...
In this chapter we review the ideas and discoveries that lead to the development of quantum mechanics. Like other theories in physics, it was motivated by a series of experimental results that were at odds with the prevailing theories of the time. New concepts were introduced to explain those experiments, which in turn led to new experiments and more discoveries. Quantum mechanics took its current shape about 100 years ago and continues to provide a formidable framework for describing not only natural phenomena but also for leading us to new technologies, such as quantum information processing.