Introduction to Theoretical Physics: Quantum Mechanics partly extracted from the lecture notes of von H.G. Evertz and W. von der Linden revised and updated by Enrico Arrigoni Version of 29. Mai 2017 1 Inhaltsverzeichnis Table of contents 2 1 Introduction 3 2 Literature 4 3 Failures of classical physics 5 3.1 Blackbodyradiation .................................. ............ 5 3.2 Photoelectriceffect ................................. .............. 11 4 Waves and particles 17 4.1 The double slit experiment with classical particles . ................. 17 4.1.1 Mathematical description . ........... 19 4.2 Light............................................. .......... 19 4.3 Electrons......................................... ............ 20 4.3.1 de Broglie wavelength . ......... 21 5 The wave function and Schr¨odinger equation 23 5.1 Probability density and the wave function . ................ 23 5.2 Waveequationforlight................................ ............. 25 5.3 Euristic derivation of the wave function for massive particles . ..................... 37 5.4 Waveequations ..................................... ............ 41 5.5 Potential......................................... ............ 49 5.6 Time-independent Schr¨odingerequation ................................... 55 5.7 Normalisation....................................... ........... 65 5.8 Summaryofimportantconcepts . ................ 69 5.8.1 (1) Wave-particle dualism . .......... 69 5.8.2 (2) New description of physical quantities . .............. 69 5.8.3 (3) Wave equation for Ψ: Schr¨odingerequation............................ 71 5.8.4 (4) Time independent Schr¨odingerequation ............................. 73 6 Basic potential problems 74 6.1 Boundaryconditionsforthewavefunction . .................. 74 6.2 Constantpotential ................................. .............. 78 6.3 Boundstatesinapotentialwell . ............... 80 6.3.1 Infinite potential well . ........... 80 6.3.2 Finite potential well . ......... 84 6.3.3 Summary:boundstates ............................. ........... 106 6.4 Scatteringatapotentialbarrier . ................. 109 6.4.1 Quantum tunnelling . ........ 110 6.4.2 Resonance....................................... ......... 126 6.5 Classical limit . .......... 128 7 Functions as Vectors 129 7.1 Thescalarproduct .................................. ............. 130 7.2 Operators ........................................ ............ 143 7.3 EigenvalueProblems .................................. ............ 145 7.4 HermitianOperators ................................. ............. 149 7.5 Additional independent variables . ............... 159 8 Dirac notation 159 8.1 Vectors.......................................... ............ 159 8.2 Rulesforoperations................................. .............. 165 8.3 Operators ........................................ ............ 167 8.3.1 HermitianOperators ................................ .......... 173 8.4 Continuousvectorspaces. ................ 181 8.5 Realspacebasis.................................... ............. 195 8.6 Change of basis and momentum representation . ................... 201 8.7 Identityoperator .................................. .............. 207 9 Principles and Postulates of Quantum Mechanics 212 9.1 PostulateI:Wavefunctionorstatevector . .................... 212 9.2 PostulateII:Observables. ................ 212 9.3 PostulateIII:Measureofobservables . ................... 212 9.3.1 Measure of observables, more concretely . ................ 213 9.3.2 Continuousobservables.. .. .. ............ 219 9.4 Expectationvalues .................................. ............. 221 9.4.1 Contiunuousobservables . ............ 225 9.5 Postulate IV: Time evolution . ............. 227 9.5.1 Genericstate ..................................... ......... 235 9.5.2 Furtherexamples .................................. .......... 237 10 Examples and exercises 238 10.1 Wavelengthofanelectron . ............... 238 10.2Photoelectriceffect ................................ ............... 238 10.3 Somepropertiesofawavefunction. .................. 238 10.4 Particle in a box: expectation values . ................ 251 10.5Delta-potential ................................... .............. 259 10.6 Expansion in a discrete (orthogonal) basis . .................. 259 10.7Hermitianoperators ................................ .............. 265 10.8Standarddeviation ................................. .............. 267 10.9 Heisenberg’suncertainty . ................ 269 10.10Qubitsandmeasure................................ ............... 269 10.11Qubits and time evolution . .............. 283 10.12Free-particle evolution . ................ 293 10.13Momentum representation ofx ˆ ........................................ 297 10.14Groundstateofthehydrogenatom . ................... 305 10.15Excited isotropic states of the hydrogen atom . ..................... 317 10.16Tight-bindingmodel ................................ .............. 327 11 Some details 345 11.1 Probability density . ............. 345 11.2 Fourier representation of the Dirac delta . ................... 347 11.3 Transition from discrete to continuum . .................. 349 1 Introduction .... Quantum mechanics is of central importance for our understanding of nature. As we will see, even simple experiments show that the classical deterministic approach with its well-defined properties of matter is incorrect. This is most obvious at the microscopic scale, in the regime of atoms and elementary particles, which can only be described with the help of quantum mechanics. But of course also the macroscopic world is defined by quantum mechanics, which is important in phenomena like e.g. a laser, an LED, superconductivity, ferro-magnetism, nuclear magnetic resonance (MRI in medicine), or even large objects like neutron stars. One of the central propositions of quantum mechanics is, that only statements about probabilities can be made, unlike in classical physics, where one can predict the behaviour of a system by solving the equations of motion. The corre- sponding equation of motion in quantum mechanics is the Schr¨odinger’s equation, which describes so-called probability amplitudes instead of deterministic locations. Just like every other theory quantum mechanics cannot be derived, not any less than Newton’s laws can be. The development of the theory follows experimental observations, often times in a long process of trial and error. In such a development it is common that new terms are formed to describe the physical behaviour. If a theory does not only describe previous observations but can make own predictions, further experiments can be performed to verify their validity. If these predictions were indeed correct, the theory is furthermore confirmed, however not “proven“, since there could be further experiments that wouldn’t be predicted correctly. If a prediction of a theory is, however, not correct, then the theory is falsified. The in many aspects at first very peculiar quantum mechanics has so far splendidly withstood all experimental examinations, unlike some previously proposed alternatives (with e.g. ,,hidden variables”). In latest years there has been a rapid development in the application of experimentally increasingly well controllable, fundamental quantum mechanics, e.g. for quantum information science, with some spectacular experiments (,,quantum teleportation”), which specifically uses the non-local properties of quantum mechanics. Fundamental quantum mecha- nical phenomena are also increasingly interesting for specifically designed applications like quantum cryptography or quantum computers. 2 Literature .... R. Shankar, Principles of Quantum Mechanics, 1994. • (Pages 107-112 and 157-164 for parts in german of lecture notes) C. Claude Cohen-Tannoudji ; Bernard Diu ; Franck Laloe.¨ , Quantum Mechanics, 1977. • (Pages 67-78 for parts in german of lecture notes) J.L. Basdevant, J. Dalibard, Quantum Mechanics, 2002. • J.J. Sakurai, Modern Quantum Mechanics, 1994. • J.S. Townsend, A Modern Approach to Quantum Mechanics, 1992. • L.E. Ballentine, Quantum Mechanics: A Modern Development, 1998. • 3 Failures of classical physics .... 3.1 Blackbody radiation At high temperatures matter (for example metals) emit a continuum radiation spectrum. The color they emit is pretty much the same at a given temperature independent of the particular substance. An idealized description is the so-called blackbody model, which describes a perfect absorber and emitter of radiation. In a blackbody, electromagnetic waves of all wavevectors k are present. One can consider a wave with wavevector k as an independent oscillator (mode). 4000K u(ω) 3000K 2000K ω Energy density u(ω) of blackbody radiation at different temperatures: The energy distribution u(ω) vanishes at small and large ω, • there is a maximum in between. The maximum frequency ω (“color”) of the distribution obeys the law (Wien’s law) ω = const. T • max max Classical understanding For a given frequency ω (= 2πν), there are many oscillators (modes) k having that frequency. Since ω = c k the | | number (density) n(ω) of oscillators with frequency ω is proportional to the surface of a sphere with radius ω/c, i. e. n(ω) ω2 ∝ The energy equipartition law of statistical physics tells us that at temperature T each mode is excited to the same energy KB T . Therefore, at temperature T the energy density u(ω, T ) at a certain frequency ω would be given by u(ω, T ) K T ω2 ∝ B (Rayleigh hypothesis). 2 KBT ω 4000K u(ω) 3000K