Mathematical Methods for Physics and Engineering : A Comprehensive Guide

19.3 EXERCISES that would involve a number of non-trivial integrals if tackled using explicit wavefunctions.
Given that the fir ...

QUANTUM OPERATORS For a particle of massmmoving in a one-dimensional potentialV(x), prove Ehrenfest’s theorem: d〈px〉 dt =− 〈 dV ...

19.3 EXERCISES Now evaluate the expectation value using the eigenvalue properties ofH,namely H|r〉=Er|r〉, and deduce thesum rule ...

QUANTUM OPERATORS 19.4 Hints and answers 19.1 Show that the commutator is anti-Hermitian. 19.3 Use the Hermitian conjugate of th ...

20 Partial differential equations: general and particular solutions In this chapter and the next the solution of differential eq ...

PDES: GENERAL AND PARTICULAR SOLUTIONS 20.1 Important partial differential equations Most of the important PDEs of physics are s ...

20.1 IMPORTANT PARTIAL DIFFERENTIAL EQUATIONS u x x T T ∆s x+∆x θ 1 θ 2 Figure 20.1 The forces acting on an element of a string ...

PDES: GENERAL AND PARTICULAR SOLUTIONS 20.1.2 The diffusion equation The diffusion equation κ∇^2 u= ∂u ∂t (20.2) describes the t ...

20.1 IMPORTANT PARTIAL DIFFERENTIAL EQUATIONS The second term,f(r,t), represents a varying density of heat sources throughout th ...

PDES: GENERAL AND PARTICULAR SOLUTIONS describes the quantum mechanical wavefunctionu(r,t) of a non-relativistic particle of mas ...

20.3 GENERAL AND PARTICULAR SOLUTIONS equations by cross-multiplication, obtaining ∂p ∂y ∂ui ∂x = ∂p ∂x ∂ui ∂y , or, for our spe ...

PDES: GENERAL AND PARTICULAR SOLUTIONS is of the form A(x, y) ∂u ∂x +B(x, y) ∂u ∂y +C(x, y)u=R(x, y), (20.9) whereA(x, y),B(x, y ...

20.3 GENERAL AND PARTICULAR SOLUTIONS which, when substituted into the PDE (20.9), give [ A(x, y) ∂p ∂x +B(x, y) ∂p ∂y ] df(p) d ...

PDES: GENERAL AND PARTICULAR SOLUTIONS Each is a valid solution (the freedom of choice of form arises from the fact thatu is spe ...

20.3 GENERAL AND PARTICULAR SOLUTIONS If we take, for example,h(x, y)=expy, which clearly satisfies (20.15), then the general so ...

PDES: GENERAL AND PARTICULAR SOLUTIONS for ODEs, we require that the particular solution is not already contained in the general ...

20.3 GENERAL AND PARTICULAR SOLUTIONS homogeneous equation isu(x, y)=f(x^2 +y^2 ) for arbitrary functionf. Now by inspection a p ...

PDES: GENERAL AND PARTICULAR SOLUTIONS We now tackle the problem of solving some types of second-order PDE with constant coeffic ...

20.3 GENERAL AND PARTICULAR SOLUTIONS the terms ready for substitution into (20.20), we obtain ∂u ∂x =a df(p) dp , ∂u ∂y =b df(p ...

PDES: GENERAL AND PARTICULAR SOLUTIONS will be satisfactory solutions of the equation and that the general solution will be u(x, ...