PROBABILITY IN THE OLD QUANTUM THEORY Probability was connected to quantum theory right from the start, in 1900, through the derivation of Planck's radiation law. But not much attention has been paid to the concept of probability in quantized radiation, or in the ‘quantum jumps’ from one energy level to another in an atom, and related problems. A whole chapter or book, instead of a section, could be written on the background of quantum theory in statistical mechanics and spectral analysis with this aspect in mind. Probabilistic properties were included in very many of the most important papers dealing with radiation between 1900 and 1925. It became also clear in time that Planck's law could not be derived from classical physics. Einstein admits this around 1908. Later he said, in a work of 1917, that it is a weakness of the theory of photon emission that ‘it leaves to “chance” the time and direction of the elementary processes,’ thereby, according to Pais (1982, p. 412), making explicit ‘that something was amiss with classical causality.’ It remains somewhat open how the origins and shifting interpretations of probability in the old quantum theory affected the acceptance of the probabilistic interpretation of the new quantum mechanics of 1925–1926, and of the indeterminism that found its confirmation in Heisenberg's uncertainty relation in 1927.
PROBABILITY IN THE OLD QUANTUM THEORY Probability was connected to quantum theory right from the start, in 1900, through the derivation of Planck's radiation law. But not much attention has been paid to the concept of probability in quantized radiation, or in the ‘quantum jumps’ from one energy level to another in an atom, and related problems. A whole chapter or book, instead of a section, could be written on the background of quantum theory in statistical mechanics and spectral analysis with this aspect in mind. Probabilistic properties were included in very many of the most important papers dealing with radiation between 1900 and 1925. It became also clear in time that Planck's law could not be derived from classical physics. Einstein admits this around 1908. Later he said, in a work of 1917, that it is a weakness of the theory of photon emission that ‘it leaves to “chance” the time and direction of the elementary processes,’ thereby, according to Pais (1982, p. 412), making explicit ‘that something was amiss with classical causality.’ It remains somewhat open how the origins and shifting interpretations of probability in the old quantum theory affected the acceptance of the probabilistic interpretation of the new quantum mechanics of 1925–1926, and of the indeterminism that found its confirmation in Heisenberg's uncertainty relation in 1927.
In this chapter, we consider the way quantum theory represents deterministic, statistical, and probabilistic aspects of Nature. After a discussion in Section 3.1 of uncertainty in general, we discuss in Sections 3.2-3.4 several formal notions of classical probability, their relation to the probability concept used in applied statistics, and their dependence on the description used. Section 3.5 briefly discusses various examples, where probabilistic features emerge from fully deterministic situations-a theme to be taken up in much more detail in Chapter 11. Section 3.6 then shows how the statistical aspects of the quantum formalism naturally follow from the weak law of large numbers. The notions of c-probability in the classical case and of q-probability in the quantum case are formally defined in Section 3.7. Historically, the concept of classical probability (including its use in stochastic processes) was given an undisputed formal mathematical foundation in 1933 in terms of the measure-theoretic setting of Kolmogorov [164]. Apart from this traditional axiomatic foundation of classical probability theory, there is a less well-known equivalent axiomatic treatment by Whittle [302] in terms of expectations. Here probabilities appear as the expectations of statements, {0, 1}-valued random variables. von Plato [240] discusses the history of the concept of probability. Krüger et al. [169] discuss the history of probability in the various fields of application. Sklar [275] discusses the philosophical problems of the probability concept, with an emphasis on statistical mechanics.
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...
Abstract The theoretical results of quantum mechanics (QM) have been verified by experiments, but the probabilistic Copenhagen interpretation is still controversial, and many counterintuitive phenomena are still difficult to understand. To trace the origin of probability in QM, we construct the amplitudes of a space-time position in a multiparticle quantum objective system and find that the probability in QM originates from the particle number distribution rate in a unit volume near position r at time t in the multiparticle quantum objective system. Based on the origin of probability, the quantization conditions of action and the principle of least action for an elementary particle (referred to as particle) , we find that when a particle moves in space-time, it periodically and alternately exhibit particle and wave state. We have derived the formula for the fringe positions in the double-slit experiment with particles, a new certainty relation and the uncertainty relation, obtain the localized and nonlocalized spatiotemporal range of the particle, the apparent trajectory of the particle motion. Based on this, through rigorous mathematical derivation and analysis, we propose new physical interpretations of the quantum superposition state, wave-particle duality, the double-slit experiment, the Heisenberg uncertainty principle, and the quantum tunnelling effect, and these interpretations are physically logical and not counterintuitive.
In trying to develop a theory of chance, I have so far drawn largely on a metaphysical picture inspired by classical physics, and have used statistical mechanics as my central example of chance arising in physical theories. It has proven difficult to develop an adequate theory of chance. But perhaps things will look different if we turn to quantum mechanics. The theory of quantum mechanics is extremely well confirmed. It is doubtful whether it is strictly true in its current form, due to the well-known conflict between quantum mechanics and the general theory of relativity, but it is an outstandingly good example of a successful physical theory, and it represents a stark break from earlier models of the physical world. Perhaps by tapping into these resources, we can develop an adequate metaphysics of chance. The quantum mechanical world It is widely believed that quantum mechanics is starkly opposed to classical physics, because quantum mechanics claims that the world is governed by fundamentally indeterministic laws. As it happens, this common belief oversimplifies somewhat. Quantum mechanics is a theory that is formulated in relatively mathematical terms, quite removed from concepts of directly observable physical entities. Consequently, there is a great deal of room for interpretation of the meaning of the mathematics. Indeed, there are at least three interpretations of quantum mechanics which are serious candidates for giving an adequate account of how the mathematics relates to reality.
Among the philosophical issues raised by quantum theory, the alleged indeterminism of quantum-physical phenomena, as opposed to the determinism of classical physics, has stood at the center of discussion. There are various other much-debated problems which are directly or indirectly related to this issue or, more generally, the probabilistic character of quantum theory. Less attention has been paid to the questions of the stability of atoms and the specificity of their properties, which quantum theory was primarily designed to resolve. I argue that the indeterminism of quantum theory becomes less intriguing and can be better comprehended when seen in the light of these questions - or briefly, that “stability implies indeterminism, or chance.” Chance, in this respect, appears as an objective and legitimate mode of succession of otherwise stable and discrete well-determined states of certain physical systems.
In this expository paper we discuss the role of probability in statistical physics: it should be that of a general tool guiding induction and must not be restricted only to the empirical interpretation coming from its evaluation by an observed frequency. It is true that in many cases the value of a probability can be sensibly expressed by a suitable frequency, but this does not entails that the latter can be taken as definition of probability. Moreover, in each situation in which one thinks that probabilities are physically real things and believes that something which is only “very probable” can be regarded as being “practically certain”, some “small” probabilities are actually ignored: this attitude of mind may render illegitimate also the probabilistic interpretation of the relevant physical laws. A “law” which is based on a probabilistic interpretation cannot aim at explaining why a fact that we forecast will occur, but rather at explaining why we forecast that this fact will occur. Finally, we emphasize the role of conditional probability, and we show that looking at it from the right perspective allows to avoids also classic paradoxical aspects of quantum mechanics, without the need to resort to the interpretation based on the dualism “particle-wave”.