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Progress in mathematical physics

The Quantized Hamiltonian for a Charged Particle in an Electromagnetic Field

Floyd L. Williams

Published Jan 1, 2003• 0 citations
Publisher

Abstract

Electric fields and magnetic fields exert a manifest influence, or effect, on the energy levels of an atom. These effects, revealed for example by the splitting of spectral lines, are called Stark and Zeeman effects, respectively. The Stark effect (first observed by Johannes Stark in 1913) plays an important role, for example, in theoretical attempts to understand the formation of molecules from atoms. Our interest will be in the Zeeman effect—especially as we consider the notion of “spin,” and spin wave functions in the next chapter, particularly in Section 9.3 there. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

DOI
10.1007/978-1-4612-0009-3_9

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Progress in mathematical physics•Jan 1, 2003•0 citations
Electric fields and magnetic fields exert a manifest influence, or effect, on the energy levels of an atom. These effects, revealed for example by the splitting of spectral lines, are called Stark and Zeeman effects, respectively. The Stark effect (first observed by Johannes Stark in 1913) plays an important role, for example, in theoretical attempts to understand the formation of molecules from atoms. Our interest will be in the Zeeman effect—especially as we consider the notion of “spin,” and spin wave functions in the next chapter, particularly in Section 9.3 there. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
In the introductory remarks to Chapter 5 mention was made of the 1925 proposal of Uhlenbeck and Goudsmit that to an electron there should be assigned an intrinsic angular momentum or spin S = (S x , S y , S z )—this in addition to its quantum angular momentum (or orbital angular momentum) M = (M x , M y , M z ) defined in (5.4.4). This is not to be taken to mean that an electron actually spins about an axis in some physical way. Experiments (in particular the noted Stern-Gerlach experiment) have provided adequate justification for the proposed concept of spin, a proposal also made by R. Bichowsky and H. Urey. Dirac has shown that the notion of spin is necessary if quantum mechanics and Einstein relativity are to be consistent theories. Chemical properties of atoms, and their ability to combine and form molecules, the splitting of spectral lines by a magnetic field applied to an atom (i.e., the Zeeman effect, which we discuss later), all depend on this important notion.
Abstract Bohr’s theory of the atom did a fine job of explaining the absorption and em1ss10n spectra of one-electron atoms in terms of the quantum rules. Schrodinger’s wave mechanics did better in the sense that the rules of quantization were given a firm mathematical basis. However, problems still abounded. Experimental spectra revealed quite a number of problems for which wave mechanics (and matrix mechanics) appeared to have no solutions. In particular, some atomic emission lines predicted by the theory were seen to be split in the presence of a magnetic field into several quite distinct lines (a phenomenon known as the Zeeman effect). Neither wave mechanics nor matrix mechanics could explain this phenomenon without some further, largely ad hoc, assumptions. Worse still, when the spectra of atoms with two or more electrons were studied, certain predicted lines were found to be missing.
Since the finiteness of the speed of light, and other considerations of special relativity, play such an important role in arguments concerning the interpretation of quantum mechanics, a good understanding of quantum field theory is a prerequisite. In this chapter, a brief summary is presented of what Quantum Field Theory is, and what one can conclude from it. The fundamental fields are bosonic fields, accounting for fundamental particles with spin 0 and 1, and fermionic fields describing fundamental particles with spin 12\frac{1}{2}21​ . Within the framework of Quantum Field Theory, there are no conflicts between quantum mechanics and locality, causality, and positivity of the energy, but there seems to be no ‘underlying real world’. We do note the importance of correlations, even in the vacuum state. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This concise and accessible book provides a detailed introduction to the fundamental principles of atomic physics at an undergraduate level. Concepts are explained in an intuitive way and the book assumes only a basic knowledge of quantum mechanics and electromagnetism. With a compact format specifically designed for students, the first part of the book covers the key principles of the subject, including the quantum theory of the hydrogen atom, radiative transitions, the shell model of multi-electron atoms, spin-orbit coupling, and the effects of external fields. The second part provides an introduction to the four key applications of atomic physics: lasers, cold atoms, solid-state spectroscopy and astrophysics. This highly pedagogical text includes worked examples and end of chapter problems to allow students to test their knowledge, as well as numerous diagrams of key concepts, making it perfect for undergraduate students looking for a succinct primer on the concepts and applications of atomic physics.
The basic equation of quantum mechanics is the Schrödinger equation which expresses the wave function Ψ of a quantum system as an eigenfunction of a quantized Hamiltonian operator H : HΨ = λΨ where the (real) eigenvalue λ is the quantum energy of the system in the state Ψ see equations (1.3.2), (1.3.4). Embodied already in this equation is the basic quantum mechanical principle that quantum energies cannot take on arbitrary values but are quantized : they are given by a discrete set of eigenvalues of a suitable second-order differential operator. This mathematical phenomenon of the discreteness of eigenvalues explains, for example, the observed discreteness of absorption and emission atomic spectral lines; compare remarks in Sections 1.2 and 1.3 of Chapter 1. The Schrödinger theory, and the equivalent theory of Heisenberg, Born and Jordan, represents a distinct advancement of the Bohr theory. Some early basic papers on quantum mechanics are compiled in the book [82], which includes a historic introduction by B. van der Waerden. Also see [6, 8, 9, 24, 75, 76]. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.