Classical and Quantum Mechanics

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<p>Notes on Classical and Quantum MechanicsJos Thijssen</p> <p>February 10, 2005(560 pages)</p> <p>Available beginning of 1999</p> <p>PrefaceThese notes have been developed over several years for use with the courses Classical and Quantum Mechanics A and B, which are part of the third year applied physics degree program at Delft University of Technology. Part of these notes stem from courses which I taught at Cardiff University of Wales, UK. These notes are intended to be used alongside standard textbooks. For the classical part, several texts can be used, such as the books by Hand and Finch (Analytical Mechanics, Cambridge University Press, 1999) and Goldstein (Classical Mechanics, third edition, Addison Wesley, 2004), the older book by Corben and Stehle (Classical Mechanics, second edition, Dover, 1994, reprint of 1960 edition), and the textbook by Kibble and Berkshire, (Classical Mechanics, 5th edition, World Scientic, 2004). The part on classical mechanics is more self-contained than the quantum part, although consultation of one or more of the texts mentioned is essential for a thorough understanding of this eld. For the quantum mechanics part, we use the book by D. J. Grifths (Introduction to Quantum Mechanics, Second Edition, Pearson Education International/Prentice Hall, 2005). This is a very nice, student-friendly text which, however, has two drawbacks. Firstly, the informal way in which the material is covered, has led to a non-consistent use of Dirac notation; very often, the wavefunction formalism is used instead of the linear algebra notation. Secondly, the book does not go into modern applications of quantum mechanics, such as quantum cryptography and quantum computing. Hopefully these notes remedy that situation. Other books which are useful for learning this material from are Introductory Quantum Mechanics by Liboff (fourth edition, Addison Wesley, 2004) and Quantum Mechanics by Bransden and Joachain (second edition, Prentice Hall, 2000). Many more standard texts are availbale we nally mention here Quantum Mechanics by Basdevant and Dalibard (Springer, 2002) and, by the same authors, The Quantum Mechanics Solver (Springer, 2000). Finally, the older text by Messiah (North Holland, 1961) the books by Cohen-Tannoudji, Diu and Lalo (2 vols., John Wiley, 1996), by Gasiorowicz (John Wiley, 3rd edition, 2003) and by Merzbacher e (John Wiley, 1997) can all be recommended. Not all the material in these notes can be found in undergraduate standard texts. In particular, the chapter on the relation between classical and quantum mechanics, and those on quantum cryptography and on quantum information theory are not found in all books listed here, although Liboffs book contains a chapter on the last two subjects. If you want to know more about these new developments, consult Quantum Computing and Quantum Information by Nielsen and Chuang (Cambridge, 2000). Along with these notes, there is a large problem set, which is more essential than the notes themselves. There are many things in life which you can only learn by doing it yourself. Nobody would seriously believe you can master any sport or playing a musical instrument by reading books. For physics, the situation is exactly the same. You have to learn the subject by doing it yourself even by failing to solve a difcult problem you learn a lot, since in that situation you start thinking about the structure of the subject. In writing these notes I had numerous discussions with and advice from Herre van der Zant and Miriam Blaauboer. I hope the resulting set of notes and problems will help students learn and appreciate the beautiful theory of classical and quantum mechanics. i</p> <p>ContentsPreface 1 Introduction: Newtonian mechanics and conservation laws 1.1 Newtons laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Systems of point particles symmetries and conservation laws . . . . . . . . . . . . Lagrange and Hamilton formulations of classical mechanics 2.1 Generalised coordinates and virtual displacements . . . . . . . . . . . 2.2 dAlemberts principle . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 The pendulum . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 The block on the inclined plane . . . . . . . . . . . . . . . . 2.3.3 Heavy bead on a rotating wire . . . . . . . . . . . . . . . . . 2.4 dAlemberts principle in generalised coordinates . . . . . . . . . . . 2.5 Conservative systems the mechanical path . . . . . . . . . . . . . . 2.6 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.1 A system of pulleys . . . . . . . . . . . . . . . . . . . . . . . 2.6.2 Example: the spinning top . . . . . . . . . . . . . . . . . . . 2.7 Non-conservative forces charged particle in an electromagnetic eld 2.7.1 Charged particle in an electromagnetic eld . . . . . . . . . . 2.8 Hamilton mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . 2.9 Applications of the Hamiltonian formalism . . . . . . . . . . . . . . 2.9.1 The three-pulley system . . . . . . . . . . . . . . . . . . . . 2.9.2 The spinning top . . . . . . . . . . . . . . . . . . . . . . . . 2.9.3 Charged particle in an electromagnetic eld . . . . . . . . . . i 1 1 3 8 8 10 11 11 12 14 15 16 20 20 21 23 23 24 27 28 28 29 30 30 33 35 35 36 37 38 39 39 41</p> <p>2</p> <p>. . . . . . . . . . . . . . . . . .</p> <p>. . . . . . . . . . . . . . . . . .</p> <p>. . . . . . . . . . . . . . . . . .</p> <p>. . . . . . . . . . . . . . . . . .</p> <p>. . . . . . . . . . . . . . . . . .</p> <p>. . . . . . . . . . . . . . . . . .</p> <p>. . . . . . . . . . . . . . . . . .</p> <p>. . . . . . . . . . . . . . . . . .</p> <p>3</p> <p>The two-body problem 3.1 Formulation and analysis of the two-body problem . . . . . . . . . . . . . . . . . . 3.2 Solution of the Kepler problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Examples of variational calculus, constraints 4.1 Variational problems . . . . . . . . . . . 4.2 The brachistochrone . . . . . . . . . . . . 4.3 Fermats principle . . . . . . . . . . . . . 4.4 The minimal area problem . . . . . . . . 4.5 Constraints . . . . . . . . . . . . . . . . 4.5.1 Constraint forces . . . . . . . . . 4.5.2 Global constraints . . . . . . . . ii</p> <p>4</p> <p>. . . . . . .</p> <p>. . . . . . .</p> <p>. . . . . . .</p> <p>. . . . . . .</p> <p>. . . . . . .</p> <p>. . . . . . .</p> <p>. . . . . . .</p> <p>. . . . . . .</p> <p>. . . . . . .</p> <p>. . . . . . .</p> <p>. . . . . . .</p> <p>. . . . . . .</p> <p>. . . . . . .</p> <p>. . . . . . .</p> <p>. . . . . . .</p> <p>. . . . . . .</p> <p>. . . . . . .</p> <p>. . . . . . .</p> <p>. . . . . . .</p> <p>. . . . . . .</p> <p>. . . . . . .</p> <p>. . . . . . .</p> <p>. . . . . . .</p> <p>Contents 5 From classical to quantum mechanics 5.1 The postulates of quantum mechanics . . . . . . . . . . . . . . . 5.2 Relation with classical mechanics . . . . . . . . . . . . . . . . . 5.3 The path integral: from classical to quantum mechanics . . . . . . 5.4 The path integral: from quantum mechanics to classical mechanics</p> <p>iii 45 45 47 50 53 55 55 55 60 60 62 63 64 67 69 69 69 71 75 75 78 82 84 87 87 88 92 92 93 95 96 97 100 103 103 103 110 112 114 115 116</p> <p>. . . .</p> <p>. . . .</p> <p>. . . .</p> <p>. . . .</p> <p>. . . .</p> <p>. . . .</p> <p>. . . .</p> <p>. . . .</p> <p>. . . .</p> <p>. . . .</p> <p>6</p> <p>Operator methods for the harmonic oscillator 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 The harmonic oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Angular momentum 7.1 Spectrum of the angular momentum operators 7.2 Orbital angular momentum . . . . . . . . . . 7.3 Spin . . . . . . . . . . . . . . . . . . . . . . 7.4 Addition of angular momenta . . . . . . . . . 7.5 Angular momentum and rotations . . . . . .</p> <p>7</p> <p>. . . . .</p> <p>. . . . .</p> <p>. . . . .</p> <p>. . . . .</p> <p>. . . . .</p> <p>. . . . .</p> <p>. . . . .</p> <p>. . . . .</p> <p>. . . . .</p> <p>. . . . .</p> <p>. . . . .</p> <p>. . . . .</p> <p>. . . . .</p> <p>. . . . .</p> <p>. . . . .</p> <p>. . . . .</p> <p>. . . . .</p> <p>. . . . .</p> <p>. . . . .</p> <p>. . . . .</p> <p>. . . . .</p> <p>8</p> <p>Introduction to Quantum Cryptography 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 The idea of classical encryption . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 Quantum Encryption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Scattering in classical and in quantum mechanics 9.1 Classical analysis of scattering . . . . . . . . . 9.2 Quantum scattering with a spherical potential . 9.2.1 Calculation of scattering cross sections 9.2.2 The Born approximation . . . . . . . .</p> <p>9</p> <p>. . . .</p> <p>. . . .</p> <p>. . . .</p> <p>. . . .</p> <p>. . . .</p> <p>. . . .</p> <p>. . . .</p> <p>. . . .</p> <p>. . . .</p> <p>. . . .</p> <p>. . . .</p> <p>. . . .</p> <p>. . . .</p> <p>. . . .</p> <p>. . . .</p> <p>. . . .</p> <p>. . . .</p> <p>. . . .</p> <p>. . . .</p> <p>. . . .</p> <p>10 Symmetry and conservation laws 10.1 Noethers theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 Liouvilles theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 Systems close to equilibrium 11.1 Introduction . . . . . . . . . . . . . . . . 11.2 Analysis of a system close to equilibrium 11.2.1 Example: Double pendulum . . . 11.3 Normal modes . . . . . . . . . . . . . . . 11.4 Vibrational analysis . . . . . . . . . . . . 11.5 The chain of particles . . . . . . . . . . .</p> <p>. . . . . .</p> <p>. . . . . .</p> <p>. . . . . .</p> <p>. . . . . .</p> <p>. . . . . .</p> <p>. . . . . .</p> <p>. . . . . .</p> <p>. . . . . .</p> <p>. . . . . .</p> <p>. . . . . .</p> <p>. . . . . .</p> <p>. . . . . .</p> <p>. . . . . .</p> <p>. . . . . .</p> <p>. . . . . .</p> <p>. . . . . .</p> <p>. . . . . .</p> <p>. . . . . .</p> <p>. . . . . .</p> <p>. . . . . .</p> <p>. . . . . .</p> <p>. . . . . .</p> <p>. . . . . .</p> <p>12 Density operators Quantum information theory 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . 2 The density operator . . . . . . . . . . . . . . . . . . 3 Entanglement . . . . . . . . . . . . . . . . . . . . . . 4 The EPR paradox and Bells theorem . . . . . . . . . . 5 No cloning theorem . . . . . . . . . . . . . . . . . . . 6 Dense coding . . . . . . . . . . . . . . . . . . . . . . 7 Quantum computing and Shors factorisation algorithm</p> <p>. . . . . . .</p> <p>. . . . . . .</p> <p>. . . . . . .</p> <p>. . . . . . .</p> <p>. . . . . . .</p> <p>. . . . . . .</p> <p>. . . . . . .</p> <p>. . . . . . .</p> <p>. . . . . . .</p> <p>. . . . . . .</p> <p>. . . . . . .</p> <p>. . . . . . .</p> <p>. . . . . . .</p> <p>. . . . . . .</p> <p>. . . . . . .</p> <p>. . . . . . .</p> <p>iv</p> <p>Contents</p> <p>Appendix A Review of Linear Algebra 119 1 Hilbert spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 2 Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 Appendix B The time-dependent Schr dinger equation o Appendix C Review of the Schr dinger equation in one dimension o 123 125</p> <p>1</p> <p>Introduction: Newtonian mechanics and conservation lawsIn this lecture course, we shall introduce some mathematical techniques for studying problems in classical mechanics and apply them to several systems. In a previous course, you have already met Newtons laws and some of its applications. In this chapter, we briey review the basic theory, and consider the interpretation of Newtons laws in some detail. Furthermore, we consider conservation laws of classical mechanics which are connected to symmetries of the forces, and derive these conservation laws starting from Newtons laws.</p> <p>1.1</p> <p>Newtons laws</p> <p>The aim of a mechanical theory is to predict the motion of objects. It is convenient to start with point particles which have no dimensions. The trajectory of such a point particle is described by its position at each time. Denoting the spatial position vector by r, the trajectory of the particle is given as r(t), a three-dimensional function depending on a one-dimensional coordinate: the time. The velocity is dened as the time-derivative of the vector r(t), and by convention it is denoted as r(t): r(t) = d r(t), dt (1.1)</p> <p>and the acceleration a is dened as the second derivative of the position vector with respect to time: a(t) = r(t). The last concept we must introduce is that of momentum p: it is dened as p = m (t), r (1.3) (1.2)</p> <p>where m is the mass. Although we have an intuitive idea about the meaning of mass, this is also a rather subtle physical concept, as is clear from the frequent confusion of mass with the concept of weight (see below). Now let us state Newtons laws: 1. A body not inuenced by any other matter will move at constant velocity 2. The rate of change of momentum of a body is equal to the force, F: dp = F(r,t). dt 1 (1.4)</p> <p>2</p> <p>Introduction: Newtonian mechanics and conservation laws</p> <p>Table 1.1: Forces for various systems. The symbol mi stand for the mass of point particle i, qi stands for electric charge of particle i, B is a magnetic, and E an electric eld. G, and g are known constants. The gravitational and the electrostatic forces are directed along the line connecting the two particles i = 1, 2.</p> <p>Forces in nature System Gravity Gravity near earths surface Electrostatics Particle in an electromagnetic eld Air friction Force FG = GmM r12 Fg = mg z 1 FC = 4 0 q1 q2 r12 FEM = q (E + r B) Ffr = r</p> <p>3. When a particle exerts a force F on another particle, then the other particle exerts a force on the rst particle which is equal in magnitude but opposite in direction to the force F these forces are directed along the line connecting the two particles. Denoting the particle by indices 1 and 2, and the force exerted on 1 by 2 by F1,2 and the force exerted on 2 by 1 by F2,1 , we have: F1,2 = F2,1 = F1,2 r1,2 . (1.5)</p> <p> where r1,2 is a unit vector pointing from r1 to r2 . The denotes whether the force is repulsive () or attractive (+). Some remarks about these laws are in place. It is questionable whether the second law is really a statement, as a new vector quantity, called force, is introduced, which is not yet dened. Only if we know the force, we can predict how a particle will move. In that sense, a real law is only formed by combining Newtons second law together with an explicit expression for the force. In table 1.1, known forces are given for several systems. Note that the force generally depends on the position r, on the velocity r, and also explicitly on time (e.g. when an external, time-varying eld is present). An implicit dependence on time is further provided by the time dependence of the position vector r(t). In most cases, the mass is taken to be constant, although this is not always true: you may think of a rocket burning its fuel, or disposing of its launching system, or bodies moving at a speed of the order of the speed of light, where the mass deviates from the rest mass. With constant mass, the second law reads: m (t) = F(r,t). r (1.6) In fact, the second law disentangles two ingredients of the motion. One is the mass m, which is a property of the moving particle which is acted upon by the force, and the other the...</p>

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