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 20
... hence M is a symmetric matrix. 17. Show that a nonsquare matrix cannot have an inverse. 18. Verify that the elements of the inverse of a matrix M are given by ME W I where M j, is the cofactor of the element mi, of M. 19. Show that ...
... hence M is a symmetric matrix. 17. Show that a nonsquare matrix cannot have an inverse. 18. Verify that the elements of the inverse of a matrix M are given by ME W I where M j, is the cofactor of the element mi, of M. 19. Show that ...
Página 27
... Hence, before a solution to (2.23) can exist, it is necessary that the eigenvalue be so chosen as to satisfy (2.24). This determinantal equation is called the characteristic equation for the matrix A. Sometimes it is known as the ...
... Hence, before a solution to (2.23) can exist, it is necessary that the eigenvalue be so chosen as to satisfy (2.24). This determinantal equation is called the characteristic equation for the matrix A. Sometimes it is known as the ...
Página 42
... that of a nondiagonalizable A. ( ) _ 3 1 Z A _(z_1 o)_(1o) (“hp -3 Z-—1_01 1 Q 3 Example 3 Then from there results 1. 1 Hence 94' = e4“ __ __ § __. 411=1122=ZII (112:0 3 1121: (Z — 1)” 42 SYSTEMS WITH A FINITE NUMBER OF DEGREE 0F FREEDOM.
... that of a nondiagonalizable A. ( ) _ 3 1 Z A _(z_1 o)_(1o) (“hp -3 Z-—1_01 1 Q 3 Example 3 Then from there results 1. 1 Hence 94' = e4“ __ __ § __. 411=1122=ZII (112:0 3 1121: (Z — 1)” 42 SYSTEMS WITH A FINITE NUMBER OF DEGREE 0F FREEDOM.
Página 43
Gerald Goertzel, Nunzio Tralli. 1. 1 Hence 94' = e4“ __ __ § __ Zr dz __ i ( )11 ( )22 27". Z l e e (eA'ha = 0 = 3te' z=1 r __1_ 3 r _ i t (8A)fl_21ri§(Z—1)2ez dZ_3dZez e' 0 3te' e' The point to be noted is the appearance of a term ...
Gerald Goertzel, Nunzio Tralli. 1. 1 Hence 94' = e4“ __ __ § __ Zr dz __ i ( )11 ( )22 27". Z l e e (eA'ha = 0 = 3te' z=1 r __1_ 3 r _ i t (8A)fl_21ri§(Z—1)2ez dZ_3dZez e' 0 3te' e' The point to be noted is the appearance of a term ...
Página 45
... Hence 20(2) _ u(O) = AU(Z) (3.15) as in (3.11a). c. Solve Eq. (3.15) for U(Z). d. Calculate u(t) = j; eZ'U(Z) dZ (3.16) 2111 . This procedure can also be applied to equations which are not in the matrix notation and which contain higher ...
... Hence 20(2) _ u(O) = AU(Z) (3.15) as in (3.11a). c. Solve Eq. (3.15) for U(Z). d. Calculate u(t) = j; eZ'U(Z) dZ (3.16) 2111 . This procedure can also be applied to equations which are not in the matrix notation and which contain higher ...
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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