Applied Mathematical Methods in Theoretical Physics by Michio Masujima PDF

By Michio Masujima

ISBN-10: 3527405348

ISBN-13: 9783527405343

All there's to understand approximately sensible research, essential equations and calculus of adaptations in one volume.

This complex textbook is split into elements: the 1st on essential equations and the second one at the calculus of adaptations. It starts with a brief advent to sensible research, together with a brief assessment of complicated research, earlier than carrying on with a scientific dialogue of alternative varieties of equations, resembling Volterra vital equations, singular indispensable equations of Cauchy sort, crucial equations of the Fredholm style, with a distinct emphasis on Wiener-Hopf fundamental equations and Wiener-Hopf sum equations.

After a number of comments at the historic improvement, the second one half starts off with an advent to the calculus of adaptations and the connection among fundamental equations and purposes of the calculus of diversifications. It additional covers purposes of the calculus of adaptations built within the moment 1/2 the 20 th century within the fields of quantum mechanics, quantum statistical mechanics and quantum box conception.

through the ebook, the writer provides over a hundred and fifty difficulties and routines - many from such branches of physics as quantum mechanics, quantum statistical mechanics, and quantum box theory—together with outlines of the recommendations in each one case. precise recommendations are given, supplementing the fabrics mentioned commonly textual content, permitting difficulties to be solved making direct use of the strategy illustrated. the unique references are given for tough difficulties. the result's whole insurance of the mathematical instruments and methods utilized by physicists and utilized mathematicians.

meant for senior undergraduates and first-year graduates in technology and engineering, this can be both invaluable as a reference and self-study advisor.

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5) We also require that G(x, t; x , t ) = 0 for t < t . 6) Note that the initial condition at t = −∞ is satisfied as well as causality. Note also that the set of equations could be obtained by the methods we used in the previous two examples. To solve the above equations, Eqs. 7) +∞  dk +∞ dω +ikx +iωt ˜   G(x, t; x , t ) = G(k, ω; x , t ). 5), we find +∞ G(x, t; x , t ) = −∞ dk 2π +∞ −∞ dω 2π −1 ω + k2 eik(x−x ) eiω(t−t ) . 8) Where do we use the condition that G(x, t; x , t ) = 0 for t < t ?

Then, from Eq. 5), we obtain G(r − r ; E)V (r )ψ(r ) dr . 6) Here G(r − r ; E) is given by G(r − r ; E) = ∞ dt exp −∞ iE(t − t ) Kret (t − t ; r − r ). Setting Kret (t − t ; r − r ) = {ip(r − r ) − iE(t − t )} dEd3 p exp K(E, p), (2π)4 δ(t − t )δ 3 (r − r ) = dEd3 p {ip(r − r ) − iE(t − t )} exp , (2π)4 substituting into Eq. 3), and writing H0 = p2 /2m, we obtain E− p2 2m K(E, p) = 1. 7) 54 2 Integral Equations and Green’s Functions The solution consistent with the retarded boundary condition is K(E, p) = 1 E− (p2 /2m) + iε , with ε positive infinitesimal.

24) Lx G(x, x ) = δ(x − x ). 25) Take the inner product of Eq. 24) with G(x, x ) from the left and Eq. 25) with G(x, x ) from the right. (G(x, x ), Lx G(x, x )) = (G(x, x ), δ(x − x )), (Lx G(x, x ), G(x, x )) = (δ(x − x ), G(x, x )). Since Lx is assumed to be self-adjoint, subtracting the two equations above results in G∗ (x , x ) = G(x , x ). , G(x , x ) is symmetric. 4 Green’s Function for Time-Dependent Scattering Problem The time-dependent Schrödinger equation takes the following form after setting 2m = 1, i ∂2 ∂ + 2 ∂t ∂x ψ(x, t) = V (x, t)ψ(x, t).

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