Abstract We explore the introduction of Planck’s constant via the study of black body radiation and the need for quantized energies, including the use of statistical mechanics methods to address this problem. This naturally leads to the Sackur Tetrode equation and the connections between entropy and quantum theory. The formalism of quantum mechanics is explored, focusing on the dimensionality of the otherwise formal mathematics involved. Bound states modelling important physical systems (such as the infinite well, harmonic oscillator, linear potential, and the quantum Coulomb problem) are explored in the context of dimensional analysis. The dimensionless phases inherent in quantum wavefunctions are discussed in the context of Planck’s constant, including discussions of some of the classic tests of quantum theory. Systems of units related to aspects of quantum mechanics, including Hartree and Planckian ones, are explored.
Abstract We explore the introduction of Planck’s constant via the study of black body radiation and the need for quantized energies, including the use of statistical mechanics methods to address this problem. This naturally leads to the Sackur Tetrode equation and the connections between entropy and quantum theory. The formalism of quantum mechanics is explored, focusing on the dimensionality of the otherwise formal mathematics involved. Bound states modelling important physical systems (such as the infinite well, harmonic oscillator, linear potential, and the quantum Coulomb problem) are explored in the context of dimensional analysis. The dimensionless phases inherent in quantum wavefunctions are discussed in the context of Planck’s constant, including discussions of some of the classic tests of quantum theory. Systems of units related to aspects of quantum mechanics, including Hartree and Planckian ones, are explored.
One of the first inklings of a problem in physics at the end of the 19th century was the nature of the radiation emission spectrum from a black body. This could not be explained in terms of established and irrefutable methods of statistical mechanics. It was Planck’s idea to divide up the energy spectrum of a series of harmonic oscillators each with a different state of energy but whose energy steps depended upon the frequency of the oscillation that marked the beginning of quantum physics. As a result, Planck introduced a fundamental constant of nature, Planck’s constant.
The development of quantum mechanics in the early twentieth century marked one of the most profound conceptual revolutions in physics. Beginning with Planck's quantization of radiation and Einstein's interpretation of the photoelectric effect, followed by Bohr's atomic model, and later formalized by Heisenberg and Schrödinger, the theory provided a new description of nature—one in which probability amplitudes replaced deterministic trajectories. Within this framework, microscopic systems such as atoms, electrons, and photons are characterized by wavefunctions and operators rather than by points in phase space.
The experimental foundations of quantum theory are presented in some detail in this chapter: on the one hand, the investigation of black-body radiation, which helps in developing an interdisciplinary view of physics, besides having historical interest; on the other hand, the energy and linear momentum of photons, atomic spectra, discrete energy levels, wave-like properties of electrons, interference phenomena and uncertainty relations. Black-body radiation Black-body radiation is not just a topic of historical interest. From a pedagogical point of view, it helps in developing an interdisciplinary view of physics, since it involves, among the other, branches of physics such as electrodynamics and thermodynamics, as well as a new constant of nature, the Planck constant, which is peculiar to quantum theory and quantum statistics. Moreover, looking at modern developments, the radiation that pervades the whole universe (Gamow 1946, Penzias and Wilson 1965, Smoot et al . 1992, Spergel et al . 2003) is a black-body radiation, and the expected emission of particles from black holes (space–time regions where gravity is so strong that no light ray can escape to infinity, and all nearby matter gets eaten up) is also (approximately) a black-body radiation (Hawking 1974, 1975). In this section, relying in part on Born (1969), we are aiming to derive the law of heat radiation, following Planck's method. We think of a box for which the walls are heated to a definite temperature T .
Classical statistical mechanics tries to derive the macroscopic properties of matter starting from the mechanical laws that rule the behaviour of single particles. To describe equilibrium states, only observables accounting for correlations among states are considered, and only systems consisting of a large number of particles are taken into account. The observables are described by continuous functions on phase space, and the states are represented by linear assignments of a number to each observable. Within the framework of the canonical ensemble, one deals with mechanical systems in thermal equilibrium with a thermal reservoir, and the equilibrium state is reached as a result of the interaction with the external world. The external world may be really external, or equally well, the unrecognized internal degrees of freedom, like in the calculation of viscosity or thermal conductivity. If really only the external world is the cause we may expect some surface dependence while an internal unrecognized degree of freedom would have a volume effect, unless the interactions are long range. The microcanonical ensemble is instead introduced to study isolated mechanical systems, and the equilibrium is viewed as a temporal average, rather than as a limit. Attention is then focused on partition functions, the theorem of equipartition of energy and an elementary theory of specific heats. In the second part, the Planck derivation of the law of black-body radiation is analysed, presenting in chronological order the Kirchhoff theorem, the Stefan law, the Wien displacement law, the Rayleigh–Jeans formula and the Planck hypothesis. Further topics discussed are the Einstein and Debye quantum models for specific heats of solids. These topics prepare the ground for the introduction of quantum statistical mechanics. The third part is, in fact, devoted to the analysis of identical particles in quantum mechanics.
Introduction The quantized harmonic oscillator has proven to be perhaps the most important system in continuous time (CT) quantum mechanics (QM). It was the atomic oscillators assumed to line the containing walls of a black-body radiation field that Planck referred to when he postulated the quantization of energy in 1900 (Planck, 1900, 1901), thereby triggering the development of QM. Subsequently, the quantized harmonic oscillator resurfaced in many contexts, most notably in quantum optics and relativistic quantum field theory, including string theory. There are two reasons for the importance of the quantized oscillator: (i) it can be solved completely; and (ii) it has ladder operators that generate Planck's quantized energy levels in particle QM and particle-like states in relativistic quantum field theory. Temporal discretization does not destroy the properties of the oscillator. Rather, it enhances them. We shall show that the parabolic barrier encountered in the classical theory of the discrete time (DT) oscillator, which was discussed in the previous chapter, provides a natural upper bound for important physical quantities such as energy and momentum. In other words, the quantized DT oscillator has a natural cutoff. This may turn out to be of fundamental importance in the regularization (renormalization) of relativistic quantum field theories, where there exist immense problems related to the fact that the CT oscillator has no upper bound to its energy spectrum.
The study of quantum statistical mechanics begins with a review of the basic principles of quantum mechanics. Schrödinger’s equation is introduced and Eigenstates (or stationary states) are defined. Model probability for quantum statistics is assumed to have a uniform distribution in phases. Wave functions for many-body systems are defined. The density matrix is introduced. The Planck entropy and the microcanonical ensemble are defined. The differences between classical and quantum statistical mechanics are all based on the differing concepts of a microscopic ‘state’. While the classical microscopic state (specified by a point in phase space) determines the exact position and momentum of every particle, the quantum mechanical state determines neither; quantum states can only provide probability distributions for observable quantities.