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Mathematics for Quantum Mechanics
An Introductory Survey of Operators, Eigenvalues, and Linear Vector Spaces
By John David Jackson
Dover Publications, Inc.Copyright © 1990 John David Jackson
All rights reserved.
The purpose of these notes is to set forth the essentials of the mathematics of quantum mechanics with only enough mathematical rigor to avoid mistakes in the physical applications.
In various parts of quantum theory it is appropriate to use mathematical methods that at first sight are quite different and unconnected. Thus in dealing with potential barriers or the hydrogen atom, the techniques of ordinary or partial differential equations in coordinate space are employed, whereas for a problem such as the harmonic oscillator, the use of abstract linear operator methods leads to an elegant solution. The chief aims of the present discussion are to show the underlying unity of all the methods and to build up enough familiarity with each of them so that in the subsequent treatment of quantum mechanics as a subject of physics the best method can be applied to each problem without apology and without the need to explain new mathematics.
Quantum theory was originally developed with two different mathematical techniques —Schrödinger wave mechanics (differential equations) and Heisenberg matrix mechanics. The equivalence of the two approaches was soon demonstrated, and the mathematical methods were generalized by Dirac, who showed that the techniques of Heisenberg and Schrödinger were special representations of the formalism of linear operators in an abstract vector space.
The concept of discreteness is central in quantum theory. Physically measurable quantities (called "observables") are often found to take on only certain definite values, independent of external conditions (e.g., preparation of light source, detailed design of deflecting magnet, etc.). Important examples of discrete observables are energy (Ritz combination principle and Rydberg formula, Franck-Hertz experiment) and angular momentum (Stern-Gerlach experiment). In mathematical language the discrete, allowed values of an observable are called eigenvalues (sometimes called characteristic or proper values). The physicist is often interested in predicting and correlating the eigenvalues for a given physical system. Provided he has an appropriate mathematical description of the physical system, he wants to solve "the eigenvalue problem." Thus the mathematical eigenvalue problem is an important aspect of quantum mechanics. This problem can be phrased in terms of differential equations, in terms of matrices, or in terms of linear vector spaces. We shall consider the various techniques and explore their essential unity below. Not all of quantum mechanics concerns itself with discrete eigenvalues, of course. Hence some of the mathematical discussion will not bear directly on the eigenvalue problem. Furthermore, a number of topics, such as perturbation theory and variational methods, will be omitted from these notes, to be dealt with separately.CHAPTER 2
Eigenvalue Problems in Classical Physics
Eigenvalue problems dominate quantum mechanics, but they were well known and understood in classical physics. They occur for mechanical systems with a finite number of degrees of freedom (discrete systems), or for mechanical or electromagnetic systems with an infinite number of degrees of freedom (continuous systems). We shall discuss briefly a few examples to recall the mathematical methods used, and to see how and why eigenvalues arise.
2-1 VIBRATING STRING
A string of uniform mass density ρ tension T, fastened at the points x = 0 and x = a, can move in the xy plane with a displacement measured perpendicular to the x axis equal to u(x,t). The equation of motion for small oscillations of the string is a linear, second-order, partial differential equation of the form
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)
If we define a quantity with dimensions of velocity v = (T/ρ)1/2, the equation can be written
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)
Using the separation-of-variables technique, we assume a solution
U(x, t) = y(x)z(t) (3)
and find that y(x) and z(t) must satisfy the separate ordinary differential equations
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)
where k2 is some as yet undetermined constant. We note that y(x) and z(vt) satisfy the same equation whose solution is sines and cosines.
So far there has been no mention of an eigenvalue problem. That comes with the imposition of the boundary conditions. These are
1. Spatial: u(O,t) = u(a,t) = 0, for all t.
2. Temporal: [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. First we consider the spatial boundary condition. The solution for y(x) is
y(x) = A sin kx + B cos kx (5)
To satisfy boundary condition 1 it is necessary that y(0) = y(a) = 0. Hence B = 0 and, if A [not equal] 0, then ka = nπ. The unknown parameter k has thus been found to have a countably infinite set of discrete eigenvalues,
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)
The most general solution consistent with the spatial boundary conditions is, therefore,
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)
Note that linear superposition has been assumed valid, consistent with the linearity of the original differential equation. The coefficients An and Bnare determined by the temporal boundary conditions at t = 0. Since this is a problem in Fourier series inversion which will be discussed below (Sec. 3-1), we shall not bother with it here.
The essential point is the discrete nature of the parameter k as a result of the spatial boundary conditions. A direct consequence of this discreteness is the discreteness of the frequencies of vibration of the string. If the fundamental frequency is defined as ω = πV/a, the allowed frequencies of vibration are
ωn = nω (n = 1,2,3, ...) (8)
The mechanical motions associated with these frequencies [y = sin(ωn x/v)] are called the normal modes or eigenmodes of oscillation. An arbitrary small-amplitude motion of the string can be built up by linear superposition, according to (7).
Problem: Show how eigenfrequencies of oscillation occur for the small oscillations of a uniform, flexible, rectangular membrane with sides of length a and b. Find the most general solution for the displacement, and show that the eigenfrequencies can be written
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
where m and n are positive integers.
2-2 VIBRATING CIRCULAR MEMBRANE
The problem of the eigenfrequencies of a circular membrane, besides involving an additional independent variable, demands explicit consideration of another physical condition on the solution besides the satisfying of boundary conditions. That physical condition is the single-valuedness of the displacement. We shall see that this requirement, as much as the boundary conditions, determines the eigenvalues of the problem.
The small oscillations of an elastic membrane are described by the partial differential equation
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)
where u(x,y,t) is the displacement and v is a velocity characteristic of the membrane. Because the boundary condition of zero displacement is to be applied along a circle, it is obviously convenient to use polar coordinates (p, φ instead of (x,y). The two-dimensional Laplacian in polar coordinates can be found in MO, p. 145. The transformed equation is
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (10)
where u is now u(p, φ, t).
Again separating variables by writing u = R(ρ)Φ(φ)z(t), we find that each of the functions R,Φ,z satisfy ordinary differential equations,
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (11)
In these equations, k2 and v2 are as yet undetermined separation constants. If k2 > 0, the time dependence will be oscillatory, as desired.
To specify an eigenvalue problem we must prescribe boundary, and/or other, conditions on the solution u(ρ, φ, t). The spatial boundary condition is u = 0 for ρ = a, independent of φ and t. Hence
R(a) = 0 (12)
But the other spatial variable φ is limited in its physical range (0 = ≤ φ ≤ 2π), and, for a circular membrane, has no role in the boundary conditions. Instead, the physical requirement of single-valuedness of u(ρ, φ, t) inside ρ = a makes the condition
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (13)
where ρ is any integer. The solution for Φ(φ) in (11) is seen to be
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (14)
The single-valuedness requirement (13) means therefore that
V = m (15)
where m = 0, ± 1, ± 2,.... We thus see that the separation constant v2 is forced to take on discrete values that are the squares of integers by the physical conditions of the problem.
Consideration of the radial equation will complete the specifications of the eigenvalue problem. If we define the dimensionless variable x = kITLρITL the radial equation in (11) becomes
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (16)
The condition (15) has already been imposed. Equation (16) is Bessel's equation, discussed in Appendix A. But before writing down the explicit solution, we make some qualitative observations. Regardless of the detailed shape of the boundary of the membrane, the physical aspects of the problem imply that the displacement cannot increase indefinitely in any given direction. There will be, in fact, local maxima and minima in the distorted surface of the membrane as it oscillates. In mathematical terms, this means that along a radial line, for example, the curvature of the curve of displacement versus distance cannot always be convex with respect to the line of zero displacement. In particular, if the displacement starts from zero at some point on the membrane and has positive curvature [(1/u)(d2u/ds2) > 0] nearby, then the curvature must change sign at least once between that point and the boundary in order to vanish at the boundary. Figure 2-1 shows qualitatively the necessary behavior. The oscillatory behavior in space is reflected mathematically in the fact that the wave equation (9) is a hyperbolic differential equation.
For the vibrating string the oscillatory character of the sines and cosines is taken for granted. But the properties of the radial equation (16) may not be so obvious. It is useful to make a slight change of dependent variable by introducing
y(x) = [square root of x] R(x)
Then y satisfies the differential equation
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (17)
For physical reasons we look for solutions that vanish at x = 0. The qualitative properties of y(x) are readily seen from (17). For small x the (1/x2) term dominates and, provided m [not equal] 0, gives a positive curvature. (In fact, for x < 1, y ~ xm+1/2). But for large x (x > m), the equation is approximately simple harmonic with solution y ~ sin (x + a). Thus the solution y(x) must appear as shown in Figure 2-2, with positive curvature near the origin and negative curvature beyond some point x ~ m. The boundary condition (12) can evidently be satisfied by choosing k in x = kρ so that ka is equal to one of the values of x where y = 0.
The actual solution of (16) in terms of Bessel functions is
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (18)
where Jm and Nm are the Bessel and Neumann functions of order m (see Appendix A). Since Nm diverges at ρ = 0, the coefficient B must vanish. The boundary condition (12) implies that
Jm (ka) = 0
Thus the separation constant k can take on only discrete values given by
kmn = xmn/a (19)
where xmn are the roots of Jm(x) = 0, tabulated in Table A-1 of Appendix A. For each m value determined by the azimuthal variation, there is a countably infinite set of values of k corresponding to n = 1,2,3,.... Thus, the totality of discrete eigenvalues of k form a doubly infinite, countable set.
The eigenfrequencies of oscillation are, from (11), given by
ωmn = xmn(v/a) (20)
The normal modes of oscillation associated with these frequencies are
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (21)
The most general solution for a single-valued displacement satisfying the boundary conditions is
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (22)
The coefficients Amn,Bmn and the phase angle am will be determined by the initial conditions, u(ρ, φ, 0) = F(ρ, φ) and [partial derivative]u/[partial derivative]t(ρ, φ, 0) = G(ρ, φ), as will be indicated in Sec. 3-4.
Lest the reader give undue weight to the difference between boundary conditions and single-valuedness in determining eigenvalues, we remark that a simple change in the physical problem can convert one into the other. For example, instead of a circular membrane of radius a, consider a pie-shaped membrane bounded by the sides ρ = a, φ = 0, and φ = φ0, where φ0< 2π. Then the requirement of zero displacement at all edges requires that the solution (14) for the azimuthal variation be
Φ(φ) = sin(vφ)
where v = mπ/φ0, with m = 1,2,3,.... The radial functions, eigenfrequencies, etc., are also suitably changed. Thus, both the separation constants ? and k are, in this case, determined by the boundary conditions. The requirement of single-valuedness does not enter explicitly.
2-3 SMALL OSCILLATIONS OF A MECHANICAL SYSTEM
As an example of an eigenvalue problem involving somewhat different mathematics, we consider the small oscillations of a mechanical system with N degrees of freedom about its equilibrium configuration. For simplicity we shall assume that the forces are derivable from a potential and that the kinetic energy is a simple diagonal quadratic form. Thus, if the generalized coordinates are qi(i = 1,2, ..., N), the kinetic energy is
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (23)
where the effective masses associated with all degrees of freedom have been assumed to be unity. (The more general problem of different masses or a nondiagonal form for T can be found in H. Goldstein, "Classical Mechanics," Addison-Wesley, Reading, Mass., 1960, Chap. 10, or K. R. Symon, "Mechanics," 2nd ed., Addison-Wesley, Reading, Mass., 1960, pp. 477ff.)
For small oscillations around equilibrium (qj = 0 for all j) the potential energy can be expanded in a Taylor's series,
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (24)
In (24) the linear term in the q's is missing because of the definition of equilibrium. The constant coefficients kij are effective spring constants given by ([MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]) and can be viewed as the elements of a real symmetric matrix. If we keep only the quadratic terms in (24), the Lagrangian (or Newtonian) equations of motion are
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (25)
The set (25) of N-linear, coupled, second-order, ordinary differential equations describes the small-amplitude [because we neglected higher terms in (24)] oscillations of the system around equilibrium. In case the reader is confused by the compact notation in (25), we display the array of equations explicitly:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (26)
In writing (26) care has been taken to group terms involving a single coordinate together in a systematic manner.
To determine the eigenfrequencies and the associated normal modes of oscillation, we assume that oscillations can exist with a frequency ω where ω is a constant to be determined. Then the equations (25) become a set of algebraic equations,
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (27)
where the q's are now amplitudes of oscillation. From the theory of linear algebraic equations we know that the set (27) of homogeneous equations has a solution only if the determinant of the coefficients vanishes:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (28)
Equation (28) is called the secular or the characteristic equation. The left-hand side is a polynomial of degree N in ω2 with real coefficients. Thus (28) has N roots for ω2. These are the eigenvalues of the problem. It can be shown that if kij = kji, the roots are all real.
Problem: Three equal masses are connected together by four springs with the same force constant as shown in the figure:
Find the eigenfrequencies of longitudinal oscillation and the normal mode of motion associated with each frequency.
Excerpted from Mathematics for Quantum Mechanics by John David Jackson. Copyright © 1990 John David Jackson. Excerpted by permission of Dover Publications, Inc..
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