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 14
... follows: A matrix is singular if and only if its determinantis zero. This definition yields also a test to tell when a matrix is singular—1e, one need merely evaluate the determinant. It will be seen in Sec. 1.8 that nonsingular ...
... follows: A matrix is singular if and only if its determinantis zero. This definition yields also a test to tell when a matrix is singular—1e, one need merely evaluate the determinant. It will be seen in Sec. 1.8 that nonsingular ...
Página 16
... follows directly from property 2, these equations have a unique solution for each column of q. Thus there exists a unique q. To find the unique p one proceeds in a similar 16 SYSTEMS WITH A FINITE NUMBER OF DEGREES OF FREEDOM.
... follows directly from property 2, these equations have a unique solution for each column of q. Thus there exists a unique q. To find the unique p one proceeds in a similar 16 SYSTEMS WITH A FINITE NUMBER OF DEGREES OF FREEDOM.
Página 17
... follows from properties 1 and 2 taken in conjunction with the nonvanishing of the determinant of m. It has just been shown that p and q as required exist. To demonstrate their equality is trivial PM = (pm)q = 14 =4 = p(mq) = M = P Thus ...
... follows from properties 1 and 2 taken in conjunction with the nonvanishing of the determinant of m. It has just been shown that p and q as required exist. To demonstrate their equality is trivial PM = (pm)q = 14 =4 = p(mq) = M = P Thus ...
Página 18
... follows again from property 3, noting that m having rank n means that mx = O has no solution other than x = 0. Any n + l n-columns are linearly dependent Thus, call these n + l n-columns m_,, i = 1,2, . . . , n, and y. If m is singular ...
... follows again from property 3, noting that m having rank n means that mx = O has no solution other than x = 0. Any n + l n-columns are linearly dependent Thus, call these n + l n-columns m_,, i = 1,2, . . . , n, and y. If m is singular ...
Página 22
... follows that f(A)“ =f (1):: (2.8) The result given in (2.8), which holds for any u which is an eigencolumn of A [i.e. , any u satisfying (2.7)], and for any f with convergent-power-series expansion, will be the basis of the work in the ...
... follows that f(A)“ =f (1):: (2.8) The result given in (2.8), which holds for any u which is an eigencolumn of A [i.e. , any u satisfying (2.7)], and for any f with convergent-power-series expansion, will be the basis of the work in the ...
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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