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 23
... obtained. Thus, let the eigencolumns of A be denoted s_,-, so that As, = Ls, (2.10) Further, suppose that there are enough such eigencolumns that one may write, for the specified initial conditions 14(0), “(0) = 5.1111 + 3.2% + ...
... obtained. Thus, let the eigencolumns of A be denoted s_,-, so that As, = Ls, (2.10) Further, suppose that there are enough such eigencolumns that one may write, for the specified initial conditions 14(0), “(0) = 5.1111 + 3.2% + ...
Página 25
... obtain a = s'lu (2.14) Thus, if A has n linearly independent eigencolumns, an arbitrary column may be written as a superposition of these eigencolumns. For this reason, the eigencolumns are said to form a complete set. As shall be seen ...
... obtain a = s'lu (2.14) Thus, if A has n linearly independent eigencolumns, an arbitrary column may be written as a superposition of these eigencolumns. For this reason, the eigencolumns are said to form a complete set. As shall be seen ...
Página 33
... obtained by inspection. Thus, consider the new variables y = x1 _ x2 2 = x1 + x2 which correspond to the masses of Fig. 1.2 moving together (y = 0 implies that x1 = x2) or oppositely (z = 0). Each of these motions may be expected to ...
... obtained by inspection. Thus, consider the new variables y = x1 _ x2 2 = x1 + x2 which correspond to the masses of Fig. 1.2 moving together (y = 0 implies that x1 = x2) or oppositely (z = 0). Each of these motions may be expected to ...
Página 34
... obtain the equations satisfied by y and z, the two Eqs. (1.4) are added and subtracted to yield m§+kz=0 mi+(k—2ko)Y=0 2.8 The Steady-state Solution As was seen in Sec. 1.1, one is sometimes concerned with the solution of Au = f (2.28) ...
... obtain the equations satisfied by y and z, the two Eqs. (1.4) are added and subtracted to yield m§+kz=0 mi+(k—2ko)Y=0 2.8 The Steady-state Solution As was seen in Sec. 1.1, one is sometimes concerned with the solution of Au = f (2.28) ...
Página 39
Gerald Goertzel, Nunzio Tralli. From (3.2) one may obtain d" n! g f(Z) — = — —— dZ n 2 0 3.3 da"f(a) 2m 0 (z - a)"+1 ( ) Further, for n 2 0 1 11 since the integral is analytic within C. 3.3 Application to Matrices The assertion made in ...
Gerald Goertzel, Nunzio Tralli. From (3.2) one may obtain d" n! g f(Z) — = — —— dZ n 2 0 3.3 da"f(a) 2m 0 (z - a)"+1 ( ) Further, for n 2 0 1 11 since the integral is analytic within C. 3.3 Application to Matrices The assertion made in ...
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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