Twentieth century physics began exactly in the year 1900 when the German physicist Max Planck proposed a possible solution to a problem that had been haunting physicists for years. The problem concerned the light emitted by any object that is heated to a certain temperature, and also the softer infrared radiation emitted, with less intensity, by colder objects. It had become well accepted that all this radiation had an electromagnetic origin, and the laws of Nature for these electromagnetic waves were already known. Also known were the laws for heat and cold, so-called ‘thermodynamics’. Or so it seemed. But if we use thermodynamic laws to compute the intensity of the radiation in question, the outcome does not make any sense at all. Such a calculation would tell us that an infinite amount of radiation should be emitted in the very far ultraviolet. Of course, this is not what happens. What does happen is that the intensity of the radiation shows a peak at a certain characteristic wavelength, and diminishes both at wavelengths longer and shorter than this. This characteristic wavelength is inversely proportional to the absolute temperature of the radiating object (absolute temperature is defined by a temperature scale that begins at 273°C below zero or 459.6 OF below zero). At 1000°C or 1800°F an object is ‘red hot’; at this point the object is radiating in the visible light region.
Twentieth century physics began exactly in the year 1900 when the German physicist Max Planck proposed a possible solution to a problem that had been haunting physicists for years. The problem concerned the light emitted by any object that is heated to a certain temperature, and also the softer infrared radiation emitted, with less intensity, by colder objects. It had become well accepted that all this radiation had an electromagnetic origin, and the laws of Nature for these electromagnetic waves were already known. Also known were the laws for heat and cold, so-called ‘thermodynamics’. Or so it seemed. But if we use thermodynamic laws to compute the intensity of the radiation in question, the outcome does not make any sense at all. Such a calculation would tell us that an infinite amount of radiation should be emitted in the very far ultraviolet. Of course, this is not what happens. What does happen is that the intensity of the radiation shows a peak at a certain characteristic wavelength, and diminishes both at wavelengths longer and shorter than this. This characteristic wavelength is inversely proportional to the absolute temperature of the radiating object (absolute temperature is defined by a temperature scale that begins at 273°C below zero or 459.6 OF below zero). At 1000°C or 1800°F an object is ‘red hot’; at this point the object is radiating in the visible light region.
It was exactly one hundred years ago on 14 December 1900 that the great physicist Max Planck (1858–1947) presented to the German Physical Society his revolutionary concept of the quantum of action, using which he successfully explained the spectrum of radiation emitted by a black body. Planck was aided in this momentous discovery by, first, his insistence on empirical veracity and second, the laws of thermodynamics. This history is narrated in several places1, I shall just briefly recall the main features.
The quantum revolution began in 1900 with a novel solution to what might seem to be one of the minor puzzles of theoretical physics, the problem of black-body radiation: the problem of the interaction of those two sorts of entity which figure in the dualist ontology of late classical physics. The trouble for classical physics was that the facts about a black body's capacity to absorb and emit radiation defied thermodynamic explanation. The best that classical physics could do was to get the facts wrong and to generate paradox. So what was the problem of black-body radiation and why was it considered to be so puzzling? First, what is a black body? Some bodies absorb and emit radiation more readily than others. A mirror is a poor absorber of radiation in the form of light, a piece of coal is a good one. A good absorber of radiation in the form of light is ‘blacker’ than one that is not. One can extend the metaphor to radiation outside the visible range. Clearly, a body that is ‘blacker’ than another for one range of radiation frequencies need not be ‘blacker’ for all frequencies. A body's ‘blackness’ will, in general, depend on its chemical structure. But we can imagine a body whose blackness is maximal for all wavelengths. Such a body, if it exists, has a blackness which is independent of its chemical structure. We call it a black body. Two interesting questions are: first, are there bodies in Nature which are uniformly and maximally black for all wavelengths? and second, how does such a black body distribute the energy that it radiates as a function of the wavelength of the radiation it emits?
From prehistoric times has come recognition that sunlight and firelight provide warmth, and that such illumination casts shadows. Expressed in more contemporary terms one would say that light travels in rays, and that this radiation has the potential to provide heat energy to absorbing material. From the time of Newton it has been known that light sources emit radiation comprising a distribution of colors. During the nineteenth century it was recognized that radiation had characteristics of transverse waves (with wavelength associated to color) but until the late twentieth century, when lasers became laboratory tools, it was hardly necessary to delve into the equations of electromagnetic theory to treat such experiments as were then possible; interest lay primarily with thermodynamic considerations of energy flow or with measurements of the dark or bright lines seen in the distribution of light that, after passing through a slit, was dispersed by a prism or grating into constituent colors. Although laser light sources are essential for the types of atomic excitation considered in this monograph, the legacy from thermal radiation still influences many interpretations of the interaction between radiation and matter, and it is therefore useful to summarize some of those concepts. The mathematics needed for describing laser radiation, or polarized light in general, has much in common with the mathematics of quantum theory discussed starting in Sec. 3.5 and specialized to two-level atoms in Sec. 5.6. In both cases one deals with two complex-valued functions – independent electric field amplitudes or probability amplitudes – whose absolute squares are measurable.
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 .
At the beginning of the twentieth century, it became clear that microscopic particles do not obey Newton's laws of classical mechanics. The idea of light quanta was at odds with the established wisdom that light is a wave phenomenon, of course, but no other explanation of the photoelectric effect seemed possible. The next step in the development of quantum mechanics was taken by Niels Bohr, who introduced postulates about the discreteness of the energy of the bound electrons in an atom similar to the discreteness of the energy of light quanta. Modern quantum mechanics was founded by Werner Heisenberg, Max Born, Erwin Schroedinger, Paul Adrian Maurice Dirac, Wolfgang Pauli, Enrico Fermi, and others. In quantum mechanics, it is in general fundamentally impossible to determine the exact outcome of an experiment; one can only determine the probability of any given outcome.
This chapter presents the basic physical ideas that underpin the quantum theory. Historians place the beginnings of the quantum theory at 1900. That year Max Planck published his famous formula for the distribution of energy as a function of frequency in black-body radiation. Curiously, although radiation theory gave birth to quantum mechanics, the interaction of matter and electromagnetic radiation is a rather advanced topic in quantum mechanics and it took almost fifty years for an acceptable formulation to be achieved. One reason for this is that photons are quanta of the electromagnetic field, and the quantum theory of fields requires a relativistic treatment to be entirely successful. The corpuscular nature of light was only half of the input that gave birth to quantum mechanics. The other half concerns matter. Given that the discreteness of photons manifests itself during the emission and absorption process, it is not surprising to find that quantum oddities afflict the behaviour of material particles too.