In order to discover what effect the quantum nature of radiation has on the detection of signals at optical frequencies, the detectability of coherent and, in the threshold limit, incoherent signals is calculated on the basis of an idealized quantum-mechanical model of a receiver. The detectabilities reduce at low frequencies to those found by ordinary signal-detection theory; at high frequencies they depend on Planck's constant. For incoherent signals of high frequency the threshold detector is equivalent to a system that focuses the radiation on a photoelectric surface of unit quantum efficiency and optimally processes the emitted electrons.
In order to discover what effect the quantum nature of radiation has on the detection of signals at optical frequencies, the detectability of coherent and, in the threshold limit, incoherent signals is calculated on the basis of an idealized quantum-mechanical model of a receiver. The detectabilities reduce at low frequencies to those found by ordinary signal-detection theory; at high frequencies they depend on Planck's constant. For incoherent signals of high frequency the threshold detector is equivalent to a system that focuses the radiation on a photoelectric surface of unit quantum efficiency and optimally processes the emitted electrons.
The detection of an incoherent object, light from which is received in the presence of thermal radiant energy, is treated by quantum detection theory under the assumption that the product of the observation time and the bandwidth of the object light is large. The threshold detector measures a quantum-mechanical operator that is a quadratic functional of the field at the aperture of the receiver. The statistics of the outcome of this measurement are approximately Poisson, and the detectability of the object depends—as under a background limitation—on the number of effectively independent spatio-temporal modes of the field. The detectabilities of objects with a rectangular and a Lorentz spectrum are compared under the assumption that the light from each possesses first-order coherence at the aperture.
It was found possible to develop a mathematical theory of communication which includes quantum effects, and as is also important, reduces to the mathematics of classical communication theory when quantum effects can be neglected. Because photoelectric detection is used universally at optical frequencies, and because it lends itself to a simpler theoretical description, earlier publications have considered the statistics of photo detection - how they differ for coherent and incoherent signals, and the consequences of these differences for information transmission. The work was concentrated in two major areas: (a) statistics of photodetection, and (b) quantum statistical analysis of communication.
The detection process of photons is studied at the limit of a vanishing Planck's constant. The detectors are conceived as particular cases of systems composed of atomic entities, called sources, acting in some given regions of space-time. When these regions are small and when their number is large with respect to the number of atoms of a real detector, we speak of a many-source system. The observable detection probabilities at given times are defined in general in an S-matrix formalism. It is shown that, in the case of a many-source system, these probabilities satisfy the same relations as those valid for a classical statistical system. When Planck's constant goes to zero, the energy density of the radiation remaining finite, an approximate expression of the amplitudes is constructed, which preserves the unitarity of the S matrix, and therefore the conservation of probability. Consequently, no recourse to the attenuation of the radiation field is necessary to ensure a coherent set of probabilities. Due to a factorization property of the amplitudes, it is then shown that the behavior of a real detector is nearly the same as that of a many-source system, namely, a classical behavior. \textcopyright{} 1996 The American Physical Society.
This chapter deals with the important experimental studies on black-body radiation, Wein’s law, Rayleigh–Jeans law, Planck’s distribution law, etc. The black-body radiation studies reveal that the maximum wavelength of radiation 1 is inversely proportional to temperature T . The expression for Wein’s law has been deduced on the basis of thermodynamic considerations. The law gives a clue to understand that it holds true only at low temperatures and short wavelengths. The Rayleigh–Jeans law is based on the general theorem of elasticity and is in excellent quantitative agreement with experimental observations for large wavelengths and high temperatures, leading to ultraviolet ‘catastrophe’. Actually, Max Planck enunciated the correct form of the temperature–radiation law based on some assumptions and finally proposed Planck’s quantum theory. This chapter also explains the photoelectric effect and important conclusions drawn on the basis of experimental facts. Einstein applied the quantum theory to explain the photoelectric effect and to define threshold frequency. On the basis of plot of frequency of radiation versus voltage applied, a linear graph was obtained, which enabled finding the value of Planck’s constant h. The Compton effect has also been written in a lucid manner involving the mathematical formulation from which the Compton shift can easily be obtained. This chapter also details the atomic hydrogen spectrum including Bohr’s model of an atom, energy of electron revolving around the nucleus, radius of an orbit and expression for velocity of electron. It also mentions the drawbacks of Bohr’s theory. References, problems, and questions on this concept are provided at the end of the chapter.
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...