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 vii
... Diagonalizable Matrices . . . 21 2.1 Solution by Taylor Series . . . . . . . . 21 2.2 Eigenvalues and Eigencolumns . . . . . . . 22 2.3 Superposition . . . . . . . . . . 23 2.4 Completeness . . . . . . . . . . 24 2.5 Diagonalization of ...
... Diagonalizable Matrices . . . 21 2.1 Solution by Taylor Series . . . . . . . . 21 2.2 Eigenvalues and Eigencolumns . . . . . . . 22 2.3 Superposition . . . . . . . . . . 23 2.4 Completeness . . . . . . . . . . 24 2.5 Diagonalization of ...
Página 20
... “Determinants and Matrices,” 5th ed., Interscience Publishers, Inc., New York, 1948. 15. Verify that (M_l)i1 = CHAPTER 2 Solution for Diagonalizable Matrices 2.1 Solution by Taylor 20 SYSTEMS WITH A FINITE NUMBER or DEGREES or FREEDOM.
... “Determinants and Matrices,” 5th ed., Interscience Publishers, Inc., New York, 1948. 15. Verify that (M_l)i1 = CHAPTER 2 Solution for Diagonalizable Matrices 2.1 Solution by Taylor 20 SYSTEMS WITH A FINITE NUMBER or DEGREES or FREEDOM.
Página 21
Gerald Goertzel, Nunzio Tralli. CHAPTER 2 Solution for Diagonalizable Matrices 2.1 Solution by Taylor Series A standard method for the solution of the differential equation 12(1) = Au(t) (2.1) yields the solution in the form of a power ...
Gerald Goertzel, Nunzio Tralli. CHAPTER 2 Solution for Diagonalizable Matrices 2.1 Solution by Taylor Series A standard method for the solution of the differential equation 12(1) = Au(t) (2.1) yields the solution in the form of a power ...
Página 23
... to a simple illustration. As an example, consider A=<; i) use) Then, as the reader will wish to verify, A<1>=~+g><1> so that, following (2.10), one may write 1 S_1=(l) 11,:(1'1'3) SOLUTION FOR DIAGONALIZABLE MATRICES.
... to a simple illustration. As an example, consider A=<; i) use) Then, as the reader will wish to verify, A<1>=~+g><1> so that, following (2.10), one may write 1 S_1=(l) 11,:(1'1'3) SOLUTION FOR DIAGONALIZABLE MATRICES.
Página 25
... = 2616,; whencef(A) =f(A)I = E[f(A)6_,]6,_. Since by hypothesis the quantities in the brackets are known, f(A) has been explicitly determined, and is thus known. (0 0) 1 0 1 This is clearly not a SOLUTION FOR DIAGONALIZABLE MATRICES 25.
... = 2616,; whencef(A) =f(A)I = E[f(A)6_,]6,_. Since by hypothesis the quantities in the brackets are known, f(A) has been explicitly determined, and is thus known. (0 0) 1 0 1 This is clearly not a SOLUTION FOR DIAGONALIZABLE MATRICES 25.
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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