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 32
... procedure of Sec. 2.6 is in terms of a change of variable in such a manner as to greatly simplify the solution of the problem at hand. The problem, written in terms of the new variables, is solved. The old variables are then calculated ...
... procedure of Sec. 2.6 is in terms of a change of variable in such a manner as to greatly simplify the solution of the problem at hand. The problem, written in terms of the new variables, is solved. The old variables are then calculated ...
Página 34
... procedure may be employed. The new variables 0 are introduced by u = S0 11 = S'lu Then Au = f = ASv = SAv or Av = S'lf Ifl,' = 0, then 0 = A1'11 = (S_1)1.f so that a solution may exist only for such f. 1 HA is singular, there exists a ...
... procedure may be employed. The new variables 0 are introduced by u = S0 11 = S'lu Then Au = f = ASv = SAv or Av = S'lf Ifl,' = 0, then 0 = A1'11 = (S_1)1.f so that a solution may exist only for such f. 1 HA is singular, there exists a ...
Página 40
... procedure is illustrated by the following three examples: (1 ') A = e 1 According to step a in the procedure Z — 1 —~ 1 ' 1““) :() —e Z —— 1 qzl 0 Z — 1 — 0 e><qm>=<> —e z — 1 q,, 1 Example 1 A brief computation yields _ _ Z — 1 422. 40 ...
... procedure is illustrated by the following three examples: (1 ') A = e 1 According to step a in the procedure Z — 1 —~ 1 ' 1““) :() —e Z —— 1 qzl 0 Z — 1 — 0 e><qm>=<> —e z — 1 q,, 1 Example 1 A brief computation yields _ _ Z — 1 422. 40 ...
Página 41
... procedure, 1 ii; 2 ~ 1 At = 6.41 ___ __ 2: dz (e )u ( )22 2111 (z-1_€)(z_1+¢)e 1 1 .. = _e(1+e)1+ _2_e(1 e): 2 and (e111)12 = (eA')21 = § m em dZ 1 1 _ ____ _ e(1+e)t_ _ e(1 e)t 2 2 These results have been obtained previously (cf ...
... procedure, 1 ii; 2 ~ 1 At = 6.41 ___ __ 2: dz (e )u ( )22 2111 (z-1_€)(z_1+¢)e 1 1 .. = _e(1+e)1+ _2_e(1 e): 2 and (e111)12 = (eA')21 = § m em dZ 1 1 _ ____ _ e(1+e)t_ _ e(1 e)t 2 2 These results have been obtained previously (cf ...
Página 44
... procedures: a. Define U(Z) by the relation U(Z) =Jwe'z'u(t) dt (3.14) 0 b. Multiply Eq. (3.13) by e—Z' dt and integrate from 0 to 00. Then, the left-hand member becomes face-Z1 d1 zfow + Z) [u(r)e'z'] dt = —u(0) + 20(2) provided that ...
... procedures: a. Define U(Z) by the relation U(Z) =Jwe'z'u(t) dt (3.14) 0 b. Multiply Eq. (3.13) by e—Z' dt and integrate from 0 to 00. Then, the left-hand member becomes face-Z1 d1 zfow + Z) [u(r)e'z'] dt = —u(0) + 20(2) provided that ...
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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