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CRC Press eBooks

Quantum Theory

A. C. Fischer-Cripps

Published Jun 20, 2019• 0 citations
Publisher

Abstract

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.

DOI
10.1201/9780429243844-5

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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.
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.
Toward the end of the nineteenth century, it became clear that some experiments cannot be explained on the basis of known physics, now referred to as classical physics. In 1900, Max Planck was able to correctly describe the distribution of blackbody radiation energy as a function of wavelength λ or frequency ν = c/λ. He had to accept the existence of energy packets (quanta) with energy E = hν. A few years later, Albert Einstein found that this unexpected relation between energy and frequency is also necessary in order to understand photoemission from metallic surfaces. A decade later, Niels Bohr arrived at the correct equation for the electronic spectrum of the hydrogen atom by applying “quantum conditions” on the motion of the electron around the nucleus and the Planck’s relation E = hν on the energy difference between the energy levels (1916). A decade later, Heisenberg, Born, and Jordan introduced “quantum mechanics” and de Broglie and Schrödinger (1925-1926) introduced “wave mechanics.”
Most textbooks say that quantum mechanics came into being when Max Planck, in a desperate attempt to construct a theory that explained experimental data, explored the radical idea of energy being radiated or absorbed by all bodies in discrete packets (or quanta). The energy content of a quantum of radiation is proportional to the frequency of the radiation emitted or absorbed, as in , and the proportionality constant bears the name of Planck. Planck himself was rather uncomfortable with the idea of energy packets, and spent a considerable amount of time in later years trying to “restore” the classical picture of continuous energy exchange between bodies and surroundings. However, in 1905 Einstein was able to use the idea of the quantum to explain the mechanism of the photoelectric effect, which cannot be explained in any other way! This established the reality of the quantum and so, it may be said that Einstein, rather than Planck, is the father of quantum mechanics. Within a few years, Neils Bohr constructed his model of the hydrogen atom in which the electron can only occupy certain special orbits around the nucleus, and any jump between these orbits is accompanied by energy being absorbed or emitted in the form of quanta. The nearly exact agreement of Bohr’s predictions based on the model and observed hydrogen atomic spectra was among the first impressive accomplishments of quantum mechanics.
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.
525. IN preceding chapters we saw how the classical system of mechanics failed to account for certain phenomena of physics, and it became clear that these could only be explained in terms of a new system of dynamics which must be supposed to supersede the old classical system when we have to deal with phenomena conditioned by the fine-scale structure of matter. The first instance of the failure of the classical mechanics was provided by the phenomenon of black-body radiation. On the assumption that black-body radiation represents a state of thermodynamical equilibrium, we found that the observed distribution of radiant energy in the spectrum of an ideal black body could be explained on the supposition that the energy of each part of the radiation-producing mechanism fell into “quanta.” Corresponding to a simple harmonic vibration of frequency v , there was supposed to be an energyquantum of amount hv , where h is Planck's universal constant of which the value has already been given. Calling this energy ∈ , it was found that the energy of vibration could be 0, ∈, 2∈, 3∈,… or any integral multiple of ∈ , but could not contain fractional parts of ∈ . The energy, after retaining one of these values for a certain time, had to be supposed to jump abruptly to some other value, which in turn was retained until another jump occurred.