Advanced Mathematical Methods in Theoretical Physics Wintersemester 2017/2018 Technische Universit¨atBerlin PD Dr. Gernot Schaller March 15, 2021 i This lecture aims at providing physics students and neighboring disciplines with a heuristic toolbox that can be used to tackle all kinds of problems. As time and resources are limited, we will explore only small fractions of different much wider fields, and students are invited to take the lecture only as a starting point. The lecture will try to be as self-contained as possible and aims at providing rather recipes than strict mathematical proofs. It is not an original contribution but an excerpt of many papers, books, other lectures, and own experiences. Only a few of these have been included as references (footnotes describing famous scientists originate from Wikipedia and solely serve for better identification). As successful learning requires practice, the lecture (Mondays) will be accompanied by exer- cises. Students should turn in individual solutions to the exercises at the beginning of Wednesday seminars (computer algebra may be used if applicable). Students may earn points for the exercise sheets, to be admitted to the final exam, they should have earned 50% of the points. After passing the final exam, the students earn in total six ECTS credit points. To get these credit points, students therefore need to get 50 % of the exercise points and 50 % to pass the final exam. The lecture script will be made available online at http://www1.itp.tu-berlin.de/schaller/. Corrections and suggestions for improvements should be addressed to [email protected]. The tentative content of the lecture is as follows: Integration Integral Transforms Ordinary Differential Equations Partial Differential Equations Statistics Operators ii Contents 1 Integration of Functions1 1.1 Heuristic Introduction into Complex Analysis . .1 1.1.1 Differentiation over C ...............................1 1.1.2 Integration in the complex plane C .......................4 1.1.3 Cauchy integral theorem . .5 1.1.4 Cauchy's integral formula . .6 1.1.5 Examples . .7 1.1.6 The Laurent Expansion . .9 1.1.7 The Residue Theorem . 11 1.1.8 Principal Value integrals . 14 1.2 Useful Integration Tricks . 16 1.2.1 Standard Integrals . 16 1.2.2 Integration by Differentiation . 17 1.2.3 Saddle point approximation . 18 1.3 The Euler MacLaurin summation formula . 19 1.3.1 Bernoulli functions . 19 1.3.2 The Euler-MacLaurin formula . 21 1.3.3 Application Examples . 22 1.4 Numerical Integration . 24 1.4.1 The Trapezoidal Rule . 25 1.4.2 The Simpson Rule . 26 1.4.3 Monte-Carlo integration . 29 2 Integral Transforms 31 2.1 Fourier Transform . 31 2.1.1 Continuous Fourier Transform . 31 2.1.2 Important Fourier Transforms . 33 2.1.3 Applications of the convolution theorem . 34 2.1.4 Discrete Fourier Transform . 35 2.2 Laplace Transform . 36 2.2.1 Definition . 36 2.2.2 Properties . 40 2.2.3 Applications . 43 3 Ordinary Differential Equations 51 3.1 Linear ODEs with constant coefficients . 52 3.1.1 Properties of the Matrix Exponential . 52 iii iv CONTENTS 3.1.2 Numerical Computation of the matrix exponential . 55 3.2 The adiabatically driven Schr¨odingerequation . 57 3.3 Periodic Linear ODEs . 60 3.4 Nonlinear ODEs . 62 3.4.1 Separable nonlinear ODEs . 62 3.4.2 Fixed-Point Analysis . 63 3.5 Numerical Solution . 69 3.5.1 Runge-Kutta algorithm . 71 3.5.2 Leapfrog integration . 71 3.5.3 Adaptive stepsize control . 72 3.6 A note on Large Systems . 73 4 Special Partial Differential Equations 75 4.1 Separation Ansatz . 76 4.1.1 Diffusion Equation . 76 4.1.2 Damped Wave Equation . 78 4.2 Fourier Transform . 81 4.2.1 Example: Reaction-Diffusion Equation . 81 4.2.2 Example: Unbounded Wave Equation . 82 4.2.3 Example: Fokker-Planck equation . 82 4.3 Green's functions . 84 4.3.1 Example: Poisson Equation . 84 4.3.2 Example: Wave Equation . 85 4.4 Nonlinear Equations . 88 4.4.1 Example: Fisher-Kolmogorov-Equation . 88 4.4.2 Example: Korteweg-de-Vries equation . 89 4.5 Numerical Solution: Finite Differences . 90 4.5.1 Explicit Forward-Time Discretization . 91 4.5.2 Implicit Centered-Time discretization . 92 4.5.3 Indexing in higher dimensions . 94 4.5.4 Nonlinear PDEs . 94 5 Master Equations 99 5.1 Rate Equations . 99 5.1.1 Example 1: Fluctuating two-level system . 100 5.1.2 Example 2: Interacting quantum dots . 100 5.2 Density Matrix Formalism . 101 5.2.1 Density Matrix . 101 5.2.2 Dynamical Evolution in a closed system . 102 5.2.3 Most general evolution in an open system . 104 5.3 Lindblad Master Equation . 104 5.3.1 Example: Master Equation for a driven cavity . 106 5.3.2 Superoperator notation . 107 5.4 Full Counting Statistics in master equations . 108 5.4.1 Phenomenologic Identification of Jump Terms . 108 5.4.2 Example: Single-Electron-Transistor . 111 5.5 Entropy and Thermodynamics . 112 CONTENTS v 5.5.1 Spohn's inequality . 112 5.5.2 Phenomenologic definition of currents . 113 5.5.3 Thermodynamics of Lindblad equations . 114 5.5.4 Nonequilibrium thermodynamics . 116 6 Canonical Operator Transformations 119 6.1 Bogoliubov transformations . 119 6.1.1 Example: Diagonalization of a homogeneous chain . 121 6.2 Jordan-Wigner Transform . 121 6.3 Collective Spin Models . 128 6.3.1 Example: The Lipkin-Meshkov-Glick model . 131 6.3.2 Holstein-Primakoff transform . 133 vi CONTENTS Chapter 1 Integration of Functions In this chapter, we first briefly review the properties of complex numbers and functions of complex numbers. We will find that these properties can be used to solve e.g. challenging integrals in an elegant fashion. 1.1 Heuristic Introduction into Complex Analysis Complex analysis treats complex-valued functions of a complex variable, i.e., maps from C to C. As will become obvious in the following, there are fundamental differences compared to real functions, which however can be used to simplify many relationships extremely. In the following, we will therefore for a complex-valued function f(z) always use the partition z = x + iy : fx; y 2 Rg 2 f(z) = u(x; y) + iv(x; y): fu; v : R ! Rg ; (1.1) where x and y denote real and imaginary part of the complex variable z, and u and v the real and imaginary part of the function f(z), respectively. 1.1.1 Differentiation over C The differentiability of complex-valued functions is treated in complete analogy to the real case. One fundamental difference however is that the limit f(z0) − f(z) f 0(z) := lim (1.2) z0!z z0 − z has in the complex plane an infinite number of ways to approach z0 ! z, whereas in the real case there are just two such paths (left and right limit). In analogy to the real case we define the derivative of a complex function f(z) via the difference quotient when the limit in Eq. (1.2) a.) exists and b.) is independent of the chosen path. In this case, the function f(z) is then called in z complex differentiable. If f(z) is complex- differentiable in a U"-environment of z0, it is also called holomorphic in z0. Demanding that a function is holomorphic has strong consequences, see Fig. 1.1. It implies 1 2 CHAPTER 1. INTEGRATION OF FUNCTIONS iy z’=z+i ∆ y Figure 1.1: For the complex-differentiable func- tion f(x + iy) = u(x; y) + iv(x; y) we consider the 0 path z ! z once parallel to the real and once z z’=z+ ∆ x parallel to the imaginary axis. For a complex- differentiable function, the result must be inde- pendent of the chosen path. x that the derivative can be expressed by u(x + ∆x; y) + iv(x + ∆x; y) − u(x; y) − iv(x; y) f 0(z) = lim ∆x!0 ∆x u(x + ∆x; y) − u(x; y) v(x + ∆x; y) − v(x; y) = lim + i lim ∆x!0 ∆x ∆x!0 ∆x @u @v = + i : (1.3) @x @x On the other hand one must also have u(x; y + ∆y) + iv(x; y + ∆y) − u(x; y) − iv(x; y) f 0(z) = lim ∆y!0 i∆y u(x; y + ∆y) − u(x; y) v(x; y + ∆y) − v(x; y) = lim + i lim ∆y!0 i∆y ∆y!0 i∆y @u @v = −i + ; (1.4) @y @y since f(z) has been assumed complex differentiable. Now, comparing the real and imaginary parts of Eqns. (1.3) and (1.4) one obtains the Cauchy 1-Riemann 2 differential equations @u @v @u @v = ; = − ; (1.5) @x @y @y @x.