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Quantum Mechanics 1 1 Quantum Mechanics Joachim Burgdorfer¨ and Stefan Rotter 1.1 Introduction 3 1.2 Particle-Wave Duality and the Uncertainty Principle 4 1.3 Schrodinger¨ Equation 6 1.4 Boundary Conditions and Quantization 8 1.5 Angular Momentum in Quantum Mechanics 9 1.6 Formalism of Quantum Mechanics 12 1.7 Solution of the Schrodinger¨ Equation 16 1.7.1 Methods for Solving the Time-Dependent Schrodinger¨ Equation 16 1.7.1.1 Time-Independent Hamiltonian 16 1.7.1.2 Time-Dependent Hamiltonian 17 1.7.2 Methods for Solving the Time-Independent Schrodinger¨ Equation 19 1.7.2.1 Separation of Variables 19 1.7.2.2 Variational Methods 23 1.7.3 Perturbation Theory 24 1.7.3.1 Stationary Perturbation Theory 25 1.7.3.2 Time-Dependent Perturbation Theory 25 1.8 Quantum Scattering Theory 27 1.8.1 Born Approximation 28 1.8.2 Partial-Wave Method 29 1.8.3 Resonances 30 1.9 Semiclassical Mechanics 31 1.9.1 The WKB Approximation 31 1.9.2 The EBK Quantization 33 1.9.3 Gutzwiller Trace Formula 34 1.10 Conceptual Aspects of Quantum Mechanics 35 1.10.1 Quantum Mechanics and Physical Reality 36 1.10.2 Quantum Information 38 1.10.3 Decoherence and Measurement Process 39 1.11 Relativistic Wave Equations 41 1.11.1 The Klein–Gordon Equation 41 1.11.2 The Dirac Equation 42 Encyclopedia of Applied High Energy and Particle Physics.EditedbyReinhardStock Copyright 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN: 978-3-527-40691-3 2 1 Quantum Mechanics Glossary 43 References 44 Further Reading 45 3 1.1 characteristic speed v of the constituents Introduction of the system under study. The domain of nonrelativistic QM is characterized by Quantum mechanics (QM), also called speeds that are small compared with the wave mechanics, is the modern theory of speed of light: matter, of atoms, molecules, solids, and of the interaction of electromagnetic fields v c (1) with matter. It supersedes the classical mechanics embodied in Newton’s laws where c = 3 × 108 ms−1 is the speed of and contains them as a limiting case (the light. For v approaching c, relativistic exten- so-called classical limit of QM). Its develop- sions are necessary. They are referred to ment was stimulated in the early part of the as relativistic wave equations.Becauseofthe twentieth century by the failure of classical possibility of production and destruction of mechanics and classical electrodynamics to particles by conversion of relativistic energy explain important discoveries, such as the E into mass M (or vice versa) according to photoelectric effect, the spectral density of the relation (see chapter 2) blackbody radiation, and the electromag- netic absorption and emission spectra of E = Mc2 (2) atoms. QM has proven to be an extremely successful theory, tested and confirmed a satisfactory formulation of relativistic to an outstanding degree of accuracy for quantum mechanics can only be devel- physical systems ranging in size from sub- oped within the framework of quantum atomic particles to macroscopic samples field theory. This chapter focuses primar- of matter, such as superconductors and ily on nonrelativistic quantum mechanics, superfluids. Despite its undisputed success its formalism and techniques as well as in the description of physical phenomena, applications to atomic, molecular, optical, QM has continued to raise many concep- and condensed-matter physics. A brief dis- tual and philosophical questions, in part cussion of extensions to relativistic wave due to its counterintuitive character, defy- equations is given at the end of the chapter. ing common-sense interpretation. The new ‘‘quantum world’’ will be One distinguishes between nonrelativis- contrasted with the old world of classical tic quantum mechanics and relativistic mechanics. The bridge between the latter quantum mechanics depending on the and the former is provided by the Encyclopedia of Applied High Energy and Particle Physics.EditedbyReinhardStock Copyright 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN: 978-3-527-40691-3 4 1 Quantum Mechanics semiclassical mechanics, whose rigorous unifies the description of distinct entities formulation and application have only of classical physics: mechanics and elec- recently been developed. tromagnetism. Particles, the fundamental constituents of matter, possess, in addition to mechanical properties such as momen- 1.2 tum and angular momentum, wave charac- Particle-Wave Duality and the Uncertainty ter. Conversely, electromagnetic radiation, Principle successfully described by Maxwell’s the- ory of electromagnetic waves, takes on, The starting point of modern quantum under appropriate conditions, the prop- mechanics is the hypothesis of the particle- erty of quasiparticles, the photons, carrying wave duality put forward by de Broglie mechanical energy and momentum. The (1924): the motion of each particle of mass particle and wave character are comple- M is associated with a wave (‘‘matter wave’’) mentary to each other. An object behaves of wavelength either like a particle or like a wave. The hypothesis of particle-wave duality h λ = (3a) has been experimentally verified in remark- p able detail. Interference of matter waves due to the indistinguishability of paths where h = 6.63 × 10−34 J s is Planck’s con- was demonstrated first for electrons (Davis- stant, discovered by M. Planck in 1900 in son and Germer, 1927), later for neutrons, his investigation of blackbody radiation, and, most recently, for neutral atoms. The and p = Mv is the classical momentum of particle-wave duality of electromagnetic the particle. Similarly, the frequency ν of radiation and its complementarity have the wave is associated with the mechan- been explored through modern versions ical energy of the particle through the of Young’s double-slit experiment using expression state-of-the-art laser technology and fast E switches. Hellmuth and colleagues (1987) ν = (3b) h succeeded, for example, in switching from particle to wave character and vice versa These relations, ascribing wave properties even after the photon has passed through to particles, are the logical complements the slit (in the experiment, a beam split- to postulates put forward 20 years earlier ter) by activating a switch (a Pockels cell) by Einstein (1905), assigning the charac- in one of the two optical paths of the teristics of particles (‘‘photons,’’ particles interferometer. By use of the switch, the of zero rest mass) to electromagnetic radi- interference patterns are made to appear ation. In fact, Einstein employed, in his or disappear, as expected for a wavelike or explanation of the ‘‘photoelectric effect,’’ particlelike behavior of the photon, respec- the expression E = hν (Eq. 3b) to find the tively. This ‘‘delayed choice’’ experiment energy E of each photon associated with not only demonstrates the particle-wave electromagnetic radiation of frequency ν. duality for photons but also elucidates Photons can eject electrons from the atoms important conceptual aspects of the nonlo- of a material provided that the energy cality of quantum mechanics. The decision exceeds the binding energy of the elec- as to whether the photon will behave tron. This particle-wave duality of QM like a particle or a wave need not be 1.2 Particle-Wave Duality and the Uncertainty Principle 5 made at the double slit itself but can be propagation. The physical interpretation of spatially separated and temporarily post- the amplitude A was provided by Born poned. (1926) in terms of a probability density The consequences of the wave-particle duality for the description of physical phe- P(x) =|ψ(x)|2 =|A|2 (5) nomena are far reaching and multifaceted. One immediate consequence that repre- sents the most radical departure from for finding the particle at the coordinate x. classical physics is the uncertainty princi- The matter wave (Eq. 4) is further charac- = = ple (Heisenberg, 1927). One deeply rooted terized by the wave number k 2π/λ p/ = = notion of classical physics is that all dynam- and the angular frequency ω 2πv E/ . = ical observables can be measured, at least The ‘‘rationalized’’ Planck’s constant = × −34 in principle, with arbitrary accuracy. Uncer- h/2π 1.05 10 J s is frequently used tainties in the measurement are a matter in QM. of experimental imperfection that, in prin- The uncertainty principle is inherent to ciple, can be overcome. The wave nature all wave phenomena. Piano tuners have of particles in QM imposes, however, fun- exploited it for centuries. They sound a damental limitations on the simultaneous vibrating tuning fork of standard frequency accuracy of measurements of dynamical in unison with a piano note of the same variables that cannot be overcome, no mat- nominal frequency and listen to a beat tone ter how much the measurement process between the struck tune and the tuning = can be improved. This uncertainty princi- fork. For a fork frequency of ν 440 Hz = ple is an immediate consequence of the and a string frequency of ν 441 Hz, wave behavior of matter. Consider a wave one beat tone will be heard per second of the form (δν = ν − ν = 1Hz).Thegoalofthetuner is to reduce the number of beats as much − ψ(x) = Aei(kx ωt) (4) as possible. To achieve an accuracy of δν = 0.010.01 Hz, the tuner has to wait for at traveling along the x coordinate least about δt = 100 s (in fact, only a fraction (Figure 1.1). Equation (4) is called a of this time since the frequency mismatch plane wave because the wave fronts form can be detected prior to completion of planes perpendicular to the direction of one beat period) to be sure no beat had Traveling matter wave Reψ(x, t) 0 0 1 2 t x Fig. 1.1 Matter wave traveling 3 along the x direction; the figure 10 shows the real part of ψ(x), 0 4 Eq.
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