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By Robert J. Gould
Princeton University PressCopyright © 2005 Princeton University Press
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Chapter OneSome Fundamental Principles
1.1 UNITS AND CHARACTERISTIC LENGTHS, TIMES, ENERGIES, ETC.
In the measurement of quantities by laboratory instruments, both the c.g.s. and m.k.s. units are convenient. However, for the description of particle and atomic processes, the c.g.s. system is preferable in that equations and formulas are sometimes simpler in form; for this reason, the c.g.s. system will be employed throughout this book. At the same time, it is often useful to express quantities in dimensionless units in terms of certain "fundamental" values defined in terms of the fundamental physical constants. Different fundamental quantities-for example, a characteristic length-can be formed from different combinations of physical constants, and the particular choice appropriate for the description of some process is dictated by the nature of the process.
Concerning the physical constants themselves, the most fundamental one is perhaps the "velocity of light" (c). The constant is of more general significance than the name given to it, since it is the characteristic parameter of spacetime, and its value is relevant to all dynamical processes in physics. Our fundamental theory of spacetime is special relativity, and we shallreview certain basic features of the theory in the following section. Considerations of some general consequences of special relativity are extremely powerful, in particular, as a guide in formulating the fundamental equations of physics.
After c, the most fundamental physical constant is probably Planck's constant ([??]). Loosely put, this constant might be designated as a "quantization parameter," but this is probably not a good description. Another try at description might be to call it the "fundamental indeterminacy parameter," but it is questionable whether the "uncertainty relations" deserve the title of principle, since they follow from the superposition principle (which really is a principle). Given that discrete particle motion is to be described in terms of an associated wave or propagation vector k and frequency [omega], Planck's constant is then the proportionality factor between k and the particle momentum:
p = [??]k. (1.1)
The uncertainty relations for an individual particle follow from this relation and the superposition principle. If momentum is to be regarded as a particle dynamical property and the wave propagation vector a kinematical variable, we might designate [??] more descriptively as a parameter of particle dynamics. However, we shall, as usual, refer to [??] simply as Planck's constant like everyone else.
The third most fundamental physical constant may be the "electronic charge" (e), since it seems to be a fundamental unit common to the various charged elementary particles. That is, although there is a spectrum of masses for the particles, except for the fractionally charged "quarks," the particle charges are multiples of e.
From the three physical constants c, [??], and e, it is not possible to construct a fundamental length by various combinations of products. From e and [??] it is possible to form a characteristic velocity [[upsilon].sub.0] = [e.sup.2]/[??], (1.2)
and this velocity is of significance for particle processes. Combining the fundamental physical constants, a dimensionless number [alpha] = [e.sup.2]/[??]c [approximately equal to] 1/137 (1.3)
can be formed that is of great importance, especially for electromagnetic processes. This number is called the "fine structure constant" because of its role in determining the magnitude of the small relativistic level shifts in atomic hydrogen; it can also be regarded as a dimensionless coupling constant for electromagnetic processes. Because of its small value, these processes can be calculated well by perturbation theory.
The masses of the various elementary particles play a major role in particle processes. The electron (and positron) mass (m), being the smallest of all, is of great importance because the particle is easily perturbed by an electromagnetic field. In particular, a variety of radiative (photon-producing) processes are associated with the electron and its interactions. A description of these processes is the principal task of this book. Almost all of our knowledge about the world outside our solar system comes from the analysis of the spectral distribution of radiation from distant sources. Our understanding of the details of the microscopic photon-producing processes allows us to interpret these source spectra and learn something of the nature of the sources. Fortunately, the electromagnetic processes are very well understood, and they can be calculated to high accuracy by perturbation theory.
The nucleon mass (M)-say, the mass of the proton, which is stable-is significant in that it is much larger (about 1836m) than that of the electron. Along with their corresponding antiparticles, the electron and proton are the only stable "particles." In fact, there is now good evidence, from inelastic scattering of very high energy electrons off protons, that the latter are not "elementary" or "fundamental" particles. Instead, protons are thought to be composites, built from quarks, and they have, as a consequence, structure. For example, protons have a characteristic size and charge distribution that can be measured. Pions (and also kaons) are also quark composites, and the pion is especially important as the least massive of the strongly interacting species. In the older theory of strong interactions, the pion was treated as a fundamental particle and its mass ([m.sub.[pi]]) determined the characteristic range of the interaction. These ideas are still useful in understanding certain particle and nuclear processes.
The masses of the elementary particles determine the various fundamental or characteristic lengths, all of which are inversely proportional to the mass value. There are different kinds of lengths, each having a different physical meaning and playing a different role in determining the characteristic magnitude of importance of various processes. Along with [??] and e, the electron mass determines the characteristic atomic size [a.sub.0] = [[??].sup.2]/m[e.sup.2]. (1.4)
This is the Bohr radius, and it is one of the triumphs of quantum mechanics that the atomic radius (~ [a.sub.0] ~ [10.sup.-8] cm) is explained by physical principles. Classical physics had no explanation for the characteristic size of atoms as determined in the last century. The basic physical meaning of the characteristic length [a.sub.0] can be indicated through considerations of atomic binding. The classical total energy of an electron of momentum p in the neighborhood of a proton is
[E.sub.cl] = [p.sup.2]/2m - [e.sup.2]/r. (1.5)
In a quantum-mechanical description, the spectra of position and momentum values are such that there is a spread in each, determined by the uncertainty relation. Setting pr ~ [??] as a constraint condition added to Equation (1.5), we see that [E.sub.cl] is minimized at a value
[([E.sub.cl]).sub.min] ~ -[e.sup.2]/2[a.sub.0] [equivalent to] -Ry (1.6)
for the r-value
[r.sub.min] ~ [a.sub.0]. (1.7) This little analysis shows, very simply, why atoms have a ground state or state of minimum energy. In a classical model with [??] [right arrow] 0 the electron "orbit" size could be infinitesimally small and the energy would be infinitely negative.
A characteristic length that does not involve is the "classical electron radius" [r.sub.0]. If the electron mass is attributed to its electrostatic self-energy (~ [e.sup.2]/[r.sub.0] ~ m[c.sup.2]), the result is
[r.sub.0] = [e.sup.2]/m[c.sup.2]. (1.8)
This is a very small distance (~ 3 x [10.sup.-13] cm), and the quantity really has no physical meaning, because the classical self-energy considerations are not valid. However, the combination [e.sup.2]/m[c.sup.2] appears often to various powers in expressions for parameters for electromagnetic quantities. Thus, it is still designated [r.sub.0] and called by its original name. The erroneous nature of the classical model for electromagnetic self-energy is clear through considerations that introduce another characteristic length. If we attempt to localize an electron to a very small distance, of necessity we introduce a spectrum of momentum states extending to high values. For p ~ mc, the energy values become large enough to produce [e.sup.[+ or -]] pairs, which affect and limit the localization. The uncertainty relation then suggests a minimum localization distance [r.sub.loc] ~ [??]/mc [equivalent to] [LAMBDA]. (1.9)
Again for historical reasons, the quantity [LAMBDA] is called the electron Compton wavelength. It appears often as a factor in formulas for cross sections for electromagnetic processes and, in general, in many equations describing phenomena involving electrons.
The three lengths [a.sub.0], [r.sub.0], and [LAMBDA] are related through a linear equation with the fine-structure constant as a proportionality factor:
[r.sub.0] = [alpha][LAMBDA] = [[alpha].sup.2] [a.sub.0]. (1.10)
Although the three lengths are connected by means of the factor [alpha], only [a.sub.0] and [LAMBDA] have a useful physical meaning, and most formulas given throughout this work will not be expressed in terms of [r.sub.0].
It might be noted that each of the lengths [a.sub.0], [LAMBDA], and [r.sub.0] is inversely proportional to the electron mass. For some problems it is convenient to consider corresponding lengths involving masses of other particles. While [a.sub.0] determines the characteristic (electron) atomic unit of length, and [E.sub.0] = [e.sup.2]/[a.sub.0](= 2Ry) the atomic unit of energy, the electron mass can be replaced by the nucleon (proton) mass M to introduce a "nucleon atomic unit" of distance
[a.sub.M] = (m/M)[a.sub.0] (1.11)
and a characteristic "nucleon Rydberg energy"
[Ry.sub.M] = (M/m)Ry. (1.12)
These units are convenient, for example, in the treatment of proton-proton scattering; in that problem, in which the nuclear and Coulomb forces contribute, the Coulomb force plays the major role (except at very high energy).
Another important characteristic distance is the particle Compton wavelength associated with the least massive of the strongly interacting particles (i.e., the pion). The quantity [[LAMBDA].sub.[pi]] = [??]/[m.sub.[pi]]c (1.13)
determines the range of the strong interaction and the magnitude of characteristic cross sections associated purely with this interaction. The cross section is
[[sigma].sub.s] ~ [[LAMBDA].sup.2.sub.[pi]] 20 mb, (1.14)
where 1 mb = [10.sup.-3] b, the barn (b), defined as [10.sup.-24][cm.sup.2], being a cross section unit common in nuclear and strong-interaction physics (1 barn is a large cross section for nuclear processes: "as big as a barn").
The choice of units for the description of some particular phenomenon is dictated not just by considerations of characteristic numerical values of relevant quantities. Depending on the type of units chosen, the equations describing a process take on slightly different form. When formulated in terms of the "most natural" units, the equations are more transparent in exhibiting the nature of the physics involved. In problems of atomic structure or in the description of the scattering of non-relativistic electrons by atomic systems or by a pure Coulomb field, the so-called atomic or "hartree" units are natural. In these units e, [??], and m are each set equal to unity, and lengths are in units of the Bohr radius [a.sub.0], cross sections are in units of [a.sup.2.sub.0], and energies are in units of [e.sub.2]/[a.sub.0] = 2Ry. The atomic units are, however, not as convenient in problems involving relativistic particles; then the more useful choice is [??] = c = 1 for which [e.sup.2] = [alpha] is fixed by the dimensionless fine-structure constant [Equation (1.3)]. These units are particularly useful in describing electromagnetic phenomena. Further, if the process involves electrons or positrons, the rest energy m([c.sup.2]) is a natural characteristic energy. Throughout this book, certain important results will often be expressed in forms that exhibit dimensions clearly by collecting products of factors that are dimensionless ratios. For example, if a cross section for some electromagnetic process at energy E is expressed in terms of a factor [[LAMBDA].sup.2],a function of E/[mc.sup.2], and a factor [[alpha].sup.n], we immediately identify n as the "order" of the process. Higher order electromagnetic processes have cross sections down by powers of [alpha]. Equations expressed in this manner are preferable to those in which numerical values of physical constants are substituted in.
1.2 RELATIVISTIC COVARIANCE AND RELATIVISTIC INVARIANTS
Ideas of covariance are extremely powerful as a guide in formulating basic physical laws and in the derivation of results in mathematical descriptions of certain physical processes. Considerations of covariance can even provide a path to the discovery of new fundamental laws and then to the development of these new areas of physics. In the description of physical processes, it is often possible to simplify derivations by imposing conditions of relativistic covariance as a trick to arrive at formulas of general validity. We shall often make use of this kind of device.
1.2.1 Spacetime Transformation
The basic laws of physics are generally expressed as differential equations with space and time coordinates as independent variables. The spacetime coordinates refer, in some cases, to "events" such as the position (or possible position) of a particle or of a particle process. Further, the properties of spacetime are described in terms of its "structure" or its transformation properties, and this is the theory of special relativity. For spacetime reference frames K and K' whose spatial coordinate axes are moving with constant relative velocity, the relationship between the coordinates of events in the two frames is the Lorentz transformation
[x'.sub.µ] = [summation over ([v])[a.sub.µv][X].sub.v]. (1.15) Here, [x.sub.v], with v = 0, 1, 2, 3, represents the time (v = 0) and space (v = 1, 2, 3) coordinates. Because of the fundamental isotropy of space, it is convenient to choose Cartesian coordinates ([x.sub.1], [x.sub.2], [x.sub.3] = x, y, z) for the spatial coordinate description. These are thus "natural" or "preferred" coordinates for formulating the basic equations of physics. It is, in one sense, convenient to choose an imaginary component [x.sub.0] = ict for the time variable. This is because the fundamental property of space-time can then be described by, in addition to the property (1.15), the equation
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], (1.16)
in which d[x.sub.µ] are the differential coordinate separations between two spacetime events.
Because of the choice of an imaginary time component, it has not been necessary to introduce a "metric" or metric tensor. The spacetime is essentially four-dimensional cartesian, and the metric tensor ([g.sub.µv]) is identical to the Kronecker [delta]-function [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]. (1.17)
Excerpted from Electromagnetic Processes by Robert J. Gould Copyright © 2005 by Princeton University Press. Excerpted by permission.
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