Methods of Theoretical Physics: I

1 Methods of Theoretical physics : IABSTRACTF irst-order and second-order differential equations; Wronskian; series solutions; ordi-nary and singular points. Orthogonal eigenfunctions and Sturm-Liouville theory. Complexanalysis, contour integration. Integral representationsfor solutions of ODE s. Asymptoticexpansions. Methods of stationary phase and steepest descent. Generalised Whittaker and Watson,A Course of Modern Arfken and H. Weber, mathematical Methods for Morse and H. Feshbach, Methods of Theoretical First and Second-order Differential The Differential Equations of physics .

2 First-order Equations ..42 Separation of Variables in Second-order Linear PDE Separation of variables in Cartesian coordinates .. Separation of variables in spherical polar coordinates.. Separation of variables in cylindrical polar coordinates .. 123 Solutions of the Associated Legendre Series solution of the Legendre equation .. Properties of the Legendre polynomials .. Azimuthally-symmetric solutions of Laplace s equation .. The generating function for the Legendre polynomials .. The associated Legendre functions.

3 The spherical harmonics and Laplace s equation .. Another look at the generating function .. 364 General Properties of Second-order ODE Singular points of the equation .. The Wronskian .. Solution of the inhomogeneous equation .. Series solutions of the homogeneous equation .. Sturm-Liouville Theory .. 655 Functions of a Complex Complex Numbers, Quaternions and Octonions .. Analytic or Holomorphic Functions .. Contour Integration .. Classification of Singularities .. The Oppenheim Formula.

4 Calculus of Residues .. Evaluation of real integrals .. Summation of Series .. Analytic Continuation .. The Gamma Function .. The Riemann Zeta Function .. Asymptotic Expansions .. Method of Steepest Descent .. 1836 Non-linear Differential Method of Isoclinals .. Phase-plane Diagrams .. 1947 Cartesian Vectors and Rotations and reflections of Cartesian coordinate .. The orthogonal groupO(n), and vectors inndimensions .. Cartesian vectors and tensors .. Invariant tensors, and the cross product .. Cartesian Tensor Calculus.

5 21321 First and Second-order Differential The Differential Equations of PhysicsIt is a phenomenological fact that most of the fundamental equations that arise in physicsare of second order in derivatives. These may be spatial derivatives, or time derivatives invarious circumstances. We call the spatial coordinates andtime, theindependentvariablesof the differential equation, while the fields whose behaviour is governed by the equationare called thedependentvariables. Examples of dependent variables are the electromag-netic potentials in Maxwell s equations, or the wave function in quantum mechanics.

6 It isfrequently the case that the equations arelinearin the dependent variables. Consider, forexample, the scalar potential in electrostatics, which satisfies 2 = 4 ( )where is the charge density. The potential appears only linearly in this equation, whichis known as Poisson s equation. In the case where there are nocharges present, so that theright-hand side vanishes, we have the special case of Laplace s linear equations are the Helmholtz equation 2 +k2 = 0, the diffusion equation 2 / t= 0, the wave equation 2 c 2 2 / t2= 0, and the Schr odinger equation h2/(2m)

7 2 +V i h / t= reason for the linearity of most of the fundamental equations in physics can be tracedback to the fact that the fields in the equations do not usuallyact as sources , for example, in electromagnetism the electric and magnetic fields respond to thesources that create them, but they do not themselves act as sources; the electromagneticfields themselves areuncharged; it is the electrons and other particles that carry chargesthat act as the sources, while the photon itself is neutral. There are in fact generalisationsof Maxwell s theory, known as Yang-Mills theories, which play a fundamental r ole in thedescription of the strong and weak nuclear forces, whicharenon-linear.

8 This is preciselybecause the Yang-Mills fields themselves carry the generalised type of electric fundamental theory that has non-linear equations of motion is gravity, describedby Einstein s general theory of relativity. The reason hereis very similar;allforms of energy(mass) act as sources for the gravitational field. In particular, the energy in the gravitationalfield itself acts as a source for gravity, hence the non-linearity. Of course in the Newtonianlimit the gravitational field is assumed to be very weak, and all the non-linearities fact there is every reason to believe that if one looks in sufficient detail then eventhe linear Maxwell equations will receive higher-order non-linear modifications.

9 Our best3candidate for a unified theory of all the fundamental interactions is string theory, and theway in which Maxwell s equations emerge there is as a sort of low-energy effective theory,which will receive higher-order non-linear corrections. However, at low energy scales, theseterms will be insignificantly small, and so we won t usually go wrong by assuming thatMaxwell s equations are good story with theorderof the fundamental differential equations of physics is rathersimilar too. Maxwell s equations, the Schr odinger equation, and Einstein s equations are allof second order in derivatives with respect to (at least someof) the independent variables.

10 Ifyou probe more closely in string theory, you find that Maxwell s equations and the Einsteinequations will also receive higher-order corrections thatinvolve larger numbers of time andspace derivatives, but again, these are insignificant at lowenergies. So in some sense oneshould probably ultimately take the view that the fundamental equations of physics tend tobe of second order in derivatives because those are the only important terms at the energyscales that we normally should certainly expect that at least second derivativeswill be observable, sincethese are needed in order to describe wave-like motion.