Theoretical Physics T2 Quantum Mechanics Course of Lectures by Reinhold A. Bertlmann Script written by Reinhold A. Bertlmann and Nicolai Friis T2{Script of Sommersemester 2008 ii E-mail: [email protected] Homepage: http://homepage.univie.ac.at/Reinhold.Bertlmann/ Contents 1 Wave{Particle Duality 1 1.1 Planck's Law of Black Body Radiation . .1 1.1.1 Quantization of Energy . .1 1.1.2 Black Body Radiation . .2 1.1.3 Derivation of Planck's Law . .5 1.2 The Photoelectric Effect . .7 1.2.1 Facts about the Photoelectric Effect . .7 1.2.2 Einstein's Explanation for the Photoelectric Effect . .8 1.2.3 The Millikan Experiment . .9 1.3 The Compton Effect . 11 1.3.1 The Compton Shift Formula . 12 1.3.2 The Experiment of Compton . 13 1.4 Bohr's Theses . 13 1.5 Wave Properties of Matter . 16 1.5.1 Louis de Broglie's Hypothesis . 16 1.5.2 Electron Diffraction from a Crystal . 17 1.6 Heisenberg's Uncertainty Principle . 19 1.6.1 Heisenberg's Microscope . 19 1.6.2 Energy{Time Uncertainty Principle . 20 1.7 Double{Slit Experiment . 20 1.7.1 Comparison of Classical and Quantum Mechanical Results . 20 1.7.2 Interpretation of Quantum Mechanical Results . 22 1.7.3 Interferometry with C60−Molecules . 24 2 Time{Dependent Schr¨odingerEquation 27 2.1 Wave Function and Time{Dependent Schr¨odingerEquation . 27 2.2 Continuity Equation . 30 2.3 Observables . 33 2.3.1 Operators . 33 2.3.2 Scalar Product . 34 2.4 Expectation Values . 36 2.4.1 Expectation Values of Operators . 36 2.4.2 Uncertainty of Observables . 39 2.5 Commutator of Operators . 41 2.6 Uncertainty Principle . 43 iii iv CONTENTS 2.6.1 Uncertainty Relation for Observables . 43 2.6.2 Energy{Time Uncertainty . 45 2.7 Ehrenfest Theorem and Virial Theorem . 47 2.7.1 Ehrenfest Theorem . 47 2.7.2 Virial Theorem . 48 2.8 Time Evolution of the Wave Packet . 49 2.8.1 Motion of Plane Waves and Wave Packets . 49 2.8.2 Spreading of the Free Wave Packet . 52 3 Mathematical Formalism of Quantum Mechanics 55 3.1 Hilbert Space . 55 3.1.1 Finite{Dimensional Hilbert Spaces . 55 3.1.2 Infinite{Dimensional Hilbert Spaces . 56 3.2 Dirac Notation . 58 3.3 Projection Operators . 59 3.3.1 Projectors for Discrete Spectra . 59 3.3.2 Projectors for Continuous Spectra . 61 3.4 Eigenvectors and Spectral Theorem . 63 3.4.1 Eigenvalue Equation . 63 3.4.2 Spectral Theorem . 64 3.5 R´esum´e:Axioms and Physical Interpretation . 66 4 Time{Independent Schr¨odingerEquation 69 4.1 Stationary States . 69 4.2 Schr¨odingerEquation as Eigenvalue Equation . 72 4.3 Expansion into Stationary States . 73 4.4 Infinite Potential Well . 74 4.5 Finite Potential Well . 76 4.5.1 Bound States . 77 4.5.2 Scattered States . 84 4.6 Tunnel Effect . 91 4.6.1 Finite Potential Barrier . 91 4.6.2 Tunneling { Transmission Coefficient . 92 4.6.3 Physical Examples of Tunneling . 93 5 Harmonic Oscillator and Coherent States 95 5.1 Harmonic Oscillator . 95 5.1.1 Algebraic Method . 95 5.1.2 Zero Point Energy . 100 5.1.3 Comparison with the Classical Oscillator . 102 5.1.4 Matrix Representation of the Harmonic Oscillator . 104 5.1.5 Analytic Method . 106 5.2 Coherent States . 107 5.2.1 Definition and Properties of Coherent States . 108 5.2.2 Coordinate Representation of Coherent States . 115 CONTENTS v 6 The 3{dimensional Schr¨odingerEquation 121 6.1 Angular Momentum . 121 6.2 Angular Momentum Eigenvalues . 124 6.3 Angular Momentum Eigenfunctions . 127 6.4 The 3{dimensional Schr¨odingerEquation . 131 6.5 Coulomb Potential and H{Atom . 135 7 Spin and Spin{Addition 139 7.1 Stern-Gerlach Experiment { Electron Spin . 139 7.1.1 Electron in a Magnetic Field . 139 7.1.2 Stern-Gerlach Experiment . 140 7.2 Mathematical Formulation of Spin . 143 7.2.1 The Pauli{Matrices . 143 7.2.2 Spin Algebra . 144 7.2.3 Spin Measurements . 146 1 7.3 Two Particles with Spin 2 ........................... 147 8 Perturbation Theory, Zeeman Effect, Stark Effect 151 8.1 Time{Independent Perturbation Theory . 151 8.1.1 First Order Corrections . 152 8.1.2 Second Order Corrections . 154 8.2 Zeeman Effect . 154 8.3 Stark Effect . 155 9 Density Matrices 159 9.1 General Properties of Density Matrices . 159 9.2 Pure and Mixed States . 160 9.2.1 Pure States . 161 9.2.2 Mixed States . 161 9.3 Time Evolution of Density Matrices . 162 10 Entanglement of Quantum Systems 165 10.1 Entanglement and Separability . 165 10.2 EPR Paradox . 166 10.3 Bell Inequalities . 167 10.4 Quantum Teleportation . 171 10.4.1 Qubits . 171 10.4.2 Quantum Teleportation Set-Up . 172 vi CONTENTS Chapter 1 Wave{Particle Duality 1.1 Planck's Law of Black Body Radiation 1.1.1 Quantization of Energy The foundation of quantum mechanics was laid in 1900 with Max Planck's discovery of the quantized nature of energy. When Planck developed his formula for black body radiation he was forced to assume that the energy exchanged between a black body and its thermal (electromagnetic) radiation is not a continuous quantity but needs to be restricted to discrete values depending on the (angular-) frequency of the radiation. Planck's formula can explain { as we shall see { all features of the black body radiation and his finding is phrased in the following way: Proposition 1.1.1 Energy is quantized and given in units of ~!: E = ~! Here ! denotes the angular frequency ! = 2πν. We will drop the prefix "angular" in the following and only refer to it as the frequency. We will also bear in mind the connection to the wavelength λ given by c = λν, where c is the speed of light, and to the 1 period T given by ν = T . The fact that the energy is proportional to the frequency, rather than to the intensity of the wave { what we would expect from classical physics { is quite counterintuitive. The proportionality constant ~ is called Planck's constant: h = = 1; 054 × 10−34 Js = 6; 582 × 10−16 eVs (1.1) ~ 2π h = 6; 626 × 10−34 Js = 4; 136 × 10−15 eVs : (1.2) 1 2 CHAPTER 1. WAVE{PARTICLE DUALITY 1.1.2 Black Body Radiation A black body is by definition an object that completely absorbs all light (radiation) that falls on it. This property makes a black body a perfect source of thermal radiation. A very good realization of a black body is an oven with a small hole, see Fig. 1.1. All radiation that enters through the opening has a very small probability of leaving through it again. Figure 1.1: Scheme for realization of a black body as a cavity Thus the radiation coming from the opening is just the thermal radiation, which will be measured in dependence of its frequency and the oven temperature as shown in Fig. 1.2. Such radiation sources are also referred to as (thermal) cavities. The classical law that describes such thermal radiation is the Rayleigh-Jeans law which expresses the energy density u(!) in terms of frequency ! and temperature T . kT 2 Theorem 1.1 (Rayleigh-Jeans law) u(!) = π2c3 ! where k denotes Boltzmann's constant, k = 1; 38 × 10−23 JK−1. Boltzmann's constant plays a role in classical thermo-statistics where (ideal) gases are analyzed whereas here we describe radiation. The quantity kT has the dimension of en- ergy, e.g., in a classical system in thermal equilibrium, each degree of freedom (of motion) 1 has a mean energy of E = 2 kT { Equipartition Theorem. From the expression of Theorem 1.1 we immediately see that the integral of the energy density over all possible frequencies is divergent, Z 1 d! u(!) ! 1 ; (1.3) 0 1.1. PLANCK'S LAW OF BLACK BODY RADIATION 3 Figure 1.2: Spectrum of a black body for different temperatures, picture from: http://commons.wikimedia .org/wiki/Image:BlackbodySpectrum lin 150dpi en.png which would imply an infinite amount of energy in the black body radiation. This is known as the ultraviolet catastrophe and the law is therefore only valid for small frequen- cies. For high frequencies a law was found empirically by Wilhelm Wien in 1896. 3 −B ! Theorem 1.2 (Wien's law) u(!) ! A! e T for ! ! 1.