Some Mathematical Methods of PhysicsCourier Corporation, 2014 M03 5 - 320 páginas This well-rounded, thorough treatment for advanced undergraduates and graduate students introduces basic concepts of mathematical physics involved in the study of linear systems. The text emphasizes eigenvalues, eigenfunctions, and Green's functions. Prerequisites include differential equations and a first course in theoretical physics. The three-part presentation begins with an exploration of systems with a finite number of degrees of freedom (described by matrices). In part two, the concepts developed for discrete systems in previous chapters are extended to continuous systems. New concepts useful in the treatment of continuous systems are also introduced. The final part examines approximation methods — including perturbation theory, variational methods, and numerical methods — relevant to addressing most of the problems of nature that confront applied physicists. Two Appendixes include background and supplementary material. 1960 edition. |
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Página 4
... satisfies Eq. (1.3) is called a harmonic oscillator. F Key I R ck %MN{}—MN—{}Mw% k m k, m k a Figure 1.1 Figure 1.2 A somewhat more complicated system is that of two harmonic oscillators coupled to each other, diagramed in Fig. 1.2. The ...
... satisfies Eq. (1.3) is called a harmonic oscillator. F Key I R ck %MN{}—MN—{}Mw% k m k, m k a Figure 1.1 Figure 1.2 A somewhat more complicated system is that of two harmonic oscillators coupled to each other, diagramed in Fig. 1.2. The ...
Página 11
... satisfies the relations Im = ml = m (1.33) It should be noted here that unless m is square, two distinct matrices I are involved in (1.33). The unit matrix I has the following form—all of (1.32) its elements along the main diagonal (the ...
... satisfies the relations Im = ml = m (1.33) It should be noted here that unless m is square, two distinct matrices I are involved in (1.33). The unit matrix I has the following form—all of (1.32) its elements along the main diagonal (the ...
Página 17
... satisfies PM” = 6i. (1'43) or written out 2 mijik = 6w The matrix of coefficients is the matrix mT, so that the existence of a unique solution for each row of p follows from properties 1 and 2 taken in conjunction with the nonvanishing ...
... satisfies PM” = 6i. (1'43) or written out 2 mijik = 6w The matrix of coefficients is the matrix mT, so that the existence of a unique solution for each row of p follows from properties 1 and 2 taken in conjunction with the nonvanishing ...
Página 27
... satisfies the equation Au = lu (223) for some nonzero u and some number 1. But (cf. Sec. 1.7) the necessary and sufficient condition that (2.23) have a nontrivial solution u is that the determinant of coefficients vanish. Thus, it must ...
... satisfies the equation Au = lu (223) for some nonzero u and some number 1. But (cf. Sec. 1.7) the necessary and sufficient condition that (2.23) have a nontrivial solution u is that the determinant of coefficients vanish. Thus, it must ...
Página 44
... (t) = 51; eZ'U(Z) dZ (3.10) where U(Z) satisfies i U(Z) = q(Z)“(0) = (Z — A)'1u(0) (3-11) or (z - A)U(Z) = “(0) (3.11.1) Substitution for q(Z) in (3.11) by means of (3.8) yields U(Z) =f eA'e_z'u(O) dt =J e'Z'u(t) (it (3.12) 0 0 ...
... (t) = 51; eZ'U(Z) dZ (3.10) where U(Z) satisfies i U(Z) = q(Z)“(0) = (Z — A)'1u(0) (3-11) or (z - A)U(Z) = “(0) (3.11.1) Substitution for q(Z) in (3.11) by means of (3.8) yields U(Z) =f eA'e_z'u(O) dt =J e'Z'u(t) (it (3.12) 0 0 ...
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applied approximate arbitrary base vectors basis Bessel function boundary conditions Chap chapter coefficients column commute complete consider constant continuous systems contour corresponding cylindrical functions defined definition denoted determinant diagonal diagonalizable differential equation Dirac notation domain eigen eigencolumns eigenfunctions eigenvalue equation eigenvector elements evaluate expansion find finite number first follows formula Fourier given Green’s function Hence Hermitian matrix Hermitian operator infinite integral Introduction inverse Laplacian linear operator linearly independent lowest eigenvalue matrix McGraw-Hill Book Company membrane method multiplication nonsingular normal normal matrix Note number of degrees obtained orthonormality conditions perturbation plane procedure QUANTUM MECHANICS relations representation result Ritz method satisfies satisfy scattering solve specified spherical spherical harmonics string Substitution theorem theory tion trial functions vanish variable vector space verified wave write written yields York zero