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Mathematical Physics Mathematical Physics Bergfinnur Durhuus and Jan Philip Solovej 1 Preface These are notes for the course Mathematical Physics at the university of Copenhagen for students in their second or third year of study. The course consists of two parts the first of which is on Classical Mechanics corresponding to the first three chapters, while the second part is on Quantum Mechanics corresponding to the remaining three chapters. The two parts can be read independently. The main focus is on theoretical developments rather than elaborating on concrete physical systems, which the students are supposed to encounter in regular physics courses. Nevertheless, some examples are included for the purpose of illustration. The prerequisites for the course are standard courses on calculus and linear algebra. We have tried to make the notes essentially self-contained, but presumably some parts, such as the more technical parts of Chapter 2 and the more abstract parts of Chapter 4, may be conceived as difficult without some prior acquaintance with concepts like series expansions, function spaces and normed spaces. The reader may prefer to skip those parts in a first reading and then return for a more thorough study later. October 2014 Bergfinnur Durhuus Jan Philip Solovej 2 Contents 1 Newtonian Mechanics 5 1.1 Classical space-time . 5 1.2 Galilean principle of relativity and inertial systems . 8 1.3 Newton's laws of motion . 10 1.4 Conservative force fields . 15 1.5 Configuration space, phase space and state space . 21 1.6 Digression: Exact and closed differentials . 24 2 Lagrangian Mechanics 35 2.1 Calculus of variations . 35 2.2 Euler-Lagrange equations . 38 2.3 Change of variables . 41 2.4 Hamilton's principle . 44 2.5 Constrained motion . 47 2.6 Appendix . 51 3 Hamiltonian Mechanics 57 3.1 Convex functions . 57 3.2 Legendre transform . 67 3.3 Hamilton's equations . 70 3.4 Noether's Theorem . 76 3.5 Legendre transform in thermodynamics . 78 4 Hilbert Spaces 86 4.1 Inner product spaces . 86 4.2 Orthogonality . 90 4.3 Continuity of the inner product . 93 4.4 Hilbert spaces . 94 4.5 Orthonormal expansions . 99 4.6 Fourier series . 101 5 Operators on Hilbert spaces 107 5.1 Operators on finite dimensional real Hilbert spaces . 107 5.2 Operators on finite dimensional complex Hilbert spaces . 110 5.3 Operators on infinite dimensional Hilbert spaces. 114 5.4 Diagonalizable operators and self-adjoint operators. 123 3 5.5 The Fourier transformation as a unitary operator. 131 6 Quantum mechanics 139 6.1 The quantum mechanical formalism . 139 6.2 Transformations and symmetries in quantum mechanics . 145 6.3 The momentum representation . 147 6.4 Galilei invariance in quantum mechanics . 150 6.5 The free particle . 155 6.6 The free particle on a circle . 156 6.7 The harmonic oscillator . 158 6.8 Stability and the hydrogen atom . 163 Index 174 4 Chapter 1 Newtonian Mechanics In these notes, classical mechanics will be studied as a mathematical model for the descrip- tion of physical systems consisting of a certain (generally finite) number of particles with fixed masses subject to mutual interactions and possibly to external forces as well. From the 17th century onwards, beginning with Newton's foundational work [10], such "Newtonian" models have provided an accurate description of a vast variety of mechanical systems, e. g. the solar system, and continue to be of fundamental importance in physics, even though the advent of quantum mechanics and special relativity in the early 20th century has revealed that Newtonian mechanics should be viewed as an approximation valid only in a certain regime, nowadays called the classical regime. Moreover, classical mechanics continues to be a significant source of motivation and inspiration for mathematical research in subjects such as ordinary differential equations, calculus of variations, and differential geometry. 1.1 Classical space-time We shall begin { rather abstractly { by introducing the notion of space-time in classical mechanics. A central paradigm of the classical framework is that time is absolute, which reflects the universality of simultaneous events in classical space-time (in striking contrast to special relativity where the simultaneity of events is relative). The basic ingredient of the mathematical model of classical space-time we are about to describe is a set, M, called the universe, and whose elements are referred to as events. It is assumed that the events can be labeled by four real numbers, i. e., there exists a bijective 4 map x : M! R , called a coordinate system on M. Composing one coordinate system x 4 4 with a bijective map : R ! R we obtain a second coordinate system y = ◦ x. Thus coordinate systems are far from being unique. We call the coordinate transformation from x-coordinates to y-coordinates. Next, we postulate the existence of two functions T : M × M ! R and d : D ! R, where D = f(p; q) 2 M × M j T (p; q) = 0g: 5 Chapter 1 Newtonian Mechanics 6 The function T is called the time-difference function and T (p; q) has the physical interpre- tation of the lapse of time between the event p and the event q as measured by standard clocks. Two events p and q are said to be simultaneous, if T (p; q) = 0 holds. Thus D is the set of pairs of simultaneous events, and the physical interpretation of d(p; q) is the (spatial) distance between the simultaneous events p and q. Note that the distance between non-simultaneous events has no physical meaning and is not defined. In view of the physical interpretation of the functions T and d, these must be subject to further constraints, which bring in the Euclidean structure of three-dimensional space. This is achieved by postulating the existence of a coordinate system x = (x1; x2; x3; t) such that d(p; q) = kx(p) − x(q)k ; (p; q) 2 D; (1.1) and T (p; q) = t(q) − t(p) ; p; q 2 M ; (1.2) 3 where we use the notation x = (x1; x2; x3) and k · k denotes the Euclidean norm in R defined by q 2 2 2 kxk = x1 + x2 + x3 : A coordinate system x fulfilling (1.1) and (1.2) will be called a Galilean coordinate system. Likewise, we call x1(p); x2(p); x3(p) the space-coordinates and t(p) the time-coordinate of the event p 2 M. The relation (1.1) expresses the Euclidean nature of the distance d. A few remarks on the Euclidean norm and its properties will be useful for the following discussion. We define 3 the scalar product (also called inner product) of two vectors x; y 2 R , written as hx; yi or simply as x · y, by x · y = hx; yi = x1y1 + x2y2 + x3y3 : Note that kxk2 = hx; xi > 0 for x 6= 0 = (0; 0; 0) ; 3 and it is clear from the definition that hx; yi is a linear real-valued function of x 2 R , for fixed y, as well as of y for fixed x. Moreover, we have hx; yi = hy; xi. We say, that h·; ·i is 3 a symmetric and positive definite bilinear form on R . 3 Given two non-zero vectors x; y 2 R , we define the angle θ 2 [0; π] between x and y through the formula x · y cos θ = : (1.3) kxkkyk Note that the definition (1.3) presupposes that the right-hand side belongs to the interval [−1; 1], which is a consequence of the so-called Cauchy-Schwarz inequality; see Exercise 1.1 d). Furthermore, we say that x and y are orthogonal, written as x ? y, if x · y = 0. 3 3 A mapping S : R ! R is called an isometry if it preserves the Euclidean distance, that is if 3 kS(x) − S(y)k = kx − yk ; x; y 2 R : 3 3 It can be shown (see Exercise 1.2) that any isometry S : R ! R is of the form S(x) = a + S0(x) ; Chapter 1 Newtonian Mechanics 7 3 where a 2 R is a fixed vector and S0 is a linear isometry, also called an orthogonal transformation. Moreover, any such orthogonal transformation is either a rotation around an axis through 0 or it is a rotation around an axis through 0 composed with a reflection in a plane through 0 (see Exercise 1.4). A matrix that represents an orthogonal transformation 3 with respect to the standard basis for R is called an orthogonal matrix, and the set of orthogonal matrices is denoted by O(3). They are discussed further in Exercises 1.3 and 1.4. The following result gives a characterization of coordinate transformations between Galilean coordinate systems. Proposition 1.1. Let x and y be two Galilean coordinate systems. Then there exists a 3 constant t0 2 R, a function ' : R ! R and a function A : R ! O(3), such that y = ◦ x, where 3 (x; t) = ('(t) + A(t)x; t − t0) ; x 2 R ; t 2 R : (1.4) Remark. We see from equation (1.4) that the time coordinates with respect to x and y are related by a constant shift. This property is commonly expressed by saying that \time is absolute" in classical physics, since time is uniquely determined once a zero-point has been chosen. Consequently, we shall in the following frequently restrict attention to systems with a common time coordinate. Proof of Proposition 1.1. Choose a fixed event q0 2 M and write x = (x; t) and y = (y; s). Then, according to (1.2), we have t(p) − t(q0) = s(p) − s(q0) ; p 2 M ; and hence s(p) = t(p) − t0 ; p 2 M ; (1.5) where t0 = t(q0) − s(q0).
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