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 25
... consider only such matrices as do. In the light of (2.14), the value of f(A)u may be written concisely as HA)“ = 12sz (LXs'lhfl (2-15) . Since this equation is true for arbitrary u, it follows from the footnote at the beginning of this ...
... consider only such matrices as do. In the light of (2.14), the value of f(A)u may be written concisely as HA)“ = 12sz (LXs'lhfl (2-15) . Since this equation is true for arbitrary u, it follows from the footnote at the beginning of this ...
Página 30
... considers the problem solved in Sec. 2.3: 1 g c A - (g I) u(0) _ (d) Step1 iA-i|=1_'1 g |=(1—1>2— 2=0 1—1 8 11=1+g h=l—g Step2 (A—11)s.1=(_g g )(s“)=0 g “g 521 whence a solution is Similarly (A _ 1.9,, = (*7 g)(s“) = 0 1 ...
... considers the problem solved in Sec. 2.3: 1 g c A - (g I) u(0) _ (d) Step1 iA-i|=1_'1 g |=(1—1>2— 2=0 1—1 8 11=1+g h=l—g Step2 (A—11)s.1=(_g g )(s“)=0 g “g 521 whence a solution is Similarly (A _ 1.9,, = (*7 g)(s“) = 0 1 ...
Página 32
... in terms of the new variables, is solved. The old variables are then calculated from their relation to the new. To see how this works, consider diagonalizable A A = 32 SYSTEMS WITH A FINITE NUMBER OF DEGREES or FREEDOM.
... in terms of the new variables, is solved. The old variables are then calculated from their relation to the new. To see how this works, consider diagonalizable A A = 32 SYSTEMS WITH A FINITE NUMBER OF DEGREES or FREEDOM.
Página 33
... 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 exist independently. (The first ...
... 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 exist independently. (The first ...
Página 39
... Consider fl)— dZ (3.4) 2111 1 Z — A where C encloses each zero of |Z — Al and no singularity off(Z). It will be shown that (3.4) is f(A) in the two cases where f(A) has been otherwise defined. When f(A) has not been otherwise defined ...
... Consider fl)— dZ (3.4) 2111 1 Z — A where C encloses each zero of |Z — Al and no singularity off(Z). It will be shown that (3.4) is f(A) in the two cases where f(A) has been otherwise defined. When f(A) has not been otherwise defined ...
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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