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 24
... write 1 S_1=(l) 11,:(1'1'3) Similarly, a second eigencolumn and eigenvalue are found to be 1 s.2=(_1) 42=(1_g) The initial conditions, as given by u(0), may now be written in the form indicated in (2.11) as a linear superposition of ...
... write 1 S_1=(l) 11,:(1'1'3) Similarly, a second eigencolumn and eigenvalue are found to be 1 s.2=(_1) 42=(1_g) The initial conditions, as given by u(0), may now be written in the form indicated in (2.11) as a linear superposition of ...
Página 25
... write an arbitrary column u as a linear combination of the eigenvectors of A sufiices to enable the evaluation off(A) and the solution of the initial value problems earlier considered. It will now be shown that an arbitrary column u may ...
... write an arbitrary column u as a linear combination of the eigenvectors of A sufiices to enable the evaluation off(A) and the solution of the initial value problems earlier considered. It will now be shown that an arbitrary column u may ...
Página 26
... writes f(A) = f(A)$$_1 = $f(1\)~'_1 (211) a result equivalent to (2.16). If (2.21) is applied to the function f(A) = A, there results the remarkable relation A = sAs'l (2.22) in which A has been written in terms of the diagonal matrix ...
... writes f(A) = f(A)$$_1 = $f(1\)~'_1 (211) a result equivalent to (2.16). If (2.21) is applied to the function f(A) = A, there results the remarkable relation A = sAs'l (2.22) in which A has been written in terms of the diagonal matrix ...
Página 29
... Write down the answer: u(t) = e“”u(0) = eA'sx = seA'x = 2n: s_,~e""x,~ i=1 A slightly different form for steps 3 and 4 is often useful: Step 3a Find r, the inverse of s, so that sr = 1. That is, solve the n sets of equations Es.irif=6 ...
... Write down the answer: u(t) = e“”u(0) = eA'sx = seA'x = 2n: s_,~e""x,~ i=1 A slightly different form for steps 3 and 4 is often useful: Step 3a Find r, the inverse of s, so that sr = 1. That is, solve the n sets of equations Es.irif=6 ...
Página 30
Gerald Goertzel, Nunzio Tralli. Step 40 Write down the answer: 2"“ = eA'sr = seA'r = Z s_,-e“'r,_ i=1 It may be noted that step 2 does not determine the matrix s uniquely, since each of the eigencolumns remains an eigencolumn when ...
Gerald Goertzel, Nunzio Tralli. Step 40 Write down the answer: 2"“ = eA'sr = seA'r = Z s_,-e“'r,_ i=1 It may be noted that step 2 does not determine the matrix s uniquely, since each of the eigencolumns remains an eigencolumn when ...
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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