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
... 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 matrices have the properties mentioned earlier in this section for ...
... 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 matrices have the properties mentioned earlier in this section for ...
Página 15
... zero. 3. A set of n homogeneous equations in n unknowns has a nontrivial solution if and only if the determinant of the matrix of coefficients is zero. A nontrivial solution is one in which not all the unknowns have the value zero. 4 ...
... zero. 3. A set of n homogeneous equations in n unknowns has a nontrivial solution if and only if the determinant of the matrix of coefficients is zero. A nontrivial solution is one in which not all the unknowns have the value zero. 4 ...
Página 18
... zero such that k Emm- = 0 (1.48) i=1 The rank of a matrix may now be defined as the maximum number of linearly independent columns which the matrix has. A nonsingular matrix has rank n, since the nonvanishing of the determinant of m ...
... zero such that k Emm- = 0 (1.48) i=1 The rank of a matrix may now be defined as the maximum number of linearly independent columns which the matrix has. A nonsingular matrix has rank n, since the nonvanishing of the determinant of m ...
Página 28
... zero. As such, the characteristic equation is satisfied by n values of 11, some of which may be multiple roots. To ... zero, such that 2 3,11,. = 0 (2.25) The demonstration that this leads to an inconsistency is relatively ...
... zero. As such, the characteristic equation is satisfied by n values of 11, some of which may be multiple roots. To ... zero, such that 2 3,11,. = 0 (2.25) The demonstration that this leads to an inconsistency is relatively ...
Página 33
... zero is called a normal mode. Thus, the jth normal mode is given by “(1) = 8.191(3) = S_,v,(0)e*" For the problem of Sec. 2.3, as just considered, the normal modes are u(t) = (1)6(1+ 0)t 1 1 and u(t) : ( )eu—m —1 For the problem defined ...
... zero is called a normal mode. Thus, the jth normal mode is given by “(1) = 8.191(3) = S_,v,(0)e*" For the problem of Sec. 2.3, as just considered, the normal modes are u(t) = (1)6(1+ 0)t 1 1 and u(t) : ( )eu—m —1 For the problem 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