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Quantum.Mechanics.Classical.Results.Modern.Systems.and.Visualized.Examples
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Contents Part I The Quantum Paradigm 1 1 A First Look at Quantum Physics 3 1.1 How this Book Approaches Quantum Mechanics 3 1.2 Essential Relativity 8 1.3 Quantum Physics: as a Fundamental Constant 10 1.4 Semiclassical Model of the Hydrogen Atom 17 1.5 Dimensional Analysis 21 1.6 Questions and Problems 23 2 Classical Waves 34 2.1 The Classical Wave Equation 34 2.2 Wave Packets and Periodic Solutions 36 2.2.1 General Wave Packet Solutions 36 2.2.2 Fourier Series 38 2.3 Fourier Transforms 43 2.4 Inverting the Fourier transform: the Dirac δ-function 46 2.5 Dispersion and Tunneling 51 2.5.1 Velocities for Wave Packets 51 2.5.2 Dispersion 53 2.5.3 Tunneling 56 2.6 Questions and Problems 57 3 The Schrödinger Wave Equation 65 3.1 The Schrödinger Equation 65 3.2 Plane Waves and Wave Packet Solutions 67 3.2.1 Plane Waves and Wave Packets 67 3.2.2 The Gaussian Wave Packet 70 3.3 “Bouncing” Wave Packets 75 3.4 Numerical Calculation of Wave Packets 77 3.5 Questions and Problems 79 x CONTENTS 4 Interpreting the Schrödinger Equation 84 4.1 Introduction to Probability 84 4.1.1 Discrete Probability Distributions 84 4.1.2 Continuous Probability Distributions 87 4.2 Probability Interpretation of the Schrödinger Wavefunction 91 4.3 Average Values 96 4.3.1 Average Values of Position 96 4.3.2 Average Values of Momentum 98 4.3.3 Average Values of Other Operators 100 4.4 Real Average Values and Hermitian Operators 102 4.5 The Physical Interpretation of φ(p) 104 4.6 Energy Eigenstates, Stationary States, and the Hamiltonian Operator 107 4.7 The Schrödinger Equation in Momentum Space 111 4.7.1 Transforming the Schrödinger Equation Into Momentum Space 111 4.7.2 Uniformly Accelerating Particle 114 4.8 Commutators 116 4.9 The Wigner Quasi-Probability Distribution 118 4.10 Questions and Problems 121 5 The Infinite Well: Physical Aspects 134 5.1 The Infinite Well in Classical Mechanics: Classical Probability Distributions 134 5.2 Stationary States for the Infinite Well 137 5.2.1 Position-Space Wavefunctions for the Standard Infinite Well 137 5.2.2 Expectation Values and Momentum-Space Wavefunctions for the Standard Infinite Well 140 5.2.3 The Symmetric Infinite Well 144 5.3 The Asymmetric Infinite Well 146 5.4 Time-Dependence of General Solutions 151 5.4.1 Two-State Systems 151 5.4.2 Wave Packets in the Infinite Well 154 5.4.3 Wave Packets Versus Stationary States 157 5.5 Questions and Problems 157 6 The Infinite Well: Formal Aspects 166 6.1 Dirac Bracket Notation 166 6.2 Eigenvalues of Hermitian Operators 167 6.3 Orthogonality of Energy Eigenfunctions 168 6.4 Expansions in Eigenstates 171 CONTENTS xi 6.5 Expansion Postulate and Time-Dependence 175 6.6 Parity 181 6.7 Simultaneous Eigenfunctions 183 6.8 Questions and Problems 185 7 Many Particles in the Infinite Well: The Role of Spin and Indistinguishability 192 7.1 The Exclusion Principle 192 7.2 One-Dimensional Systems 193 7.3 Three-Dimensional Infinite Well 195 7.4 Applications 198 7.4.1 Conduction Electrons in a Metal 198 7.4.2 Neutrons and Protons in Atomic Nuclei 200 7.4.3 White Dwarf and Neutron Stars 200 7.5 Questions and Problems 206 8 Other One-Dimensional Potentials 210 8.1 Singular Potentials 210 8.1.1 Continuity of ψ(x) 210 8.1.2 Single δ-function Potential 212 8.1.3 Twin δ-function Potential 213 8.1.4 Infinite Array of δ-functions: Periodic Potentials and the Dirac Comb 216 8.2 The Finite Well 221 8.2.1 Formal Solutions 221 8.2.2 Physical Implications and the Large x Behavior of Wavefunctions 225 8.3 Applications to Three-Dimensional Problems 230 8.3.1 The Schrödinger Equation in Three Dimensions 230 8.3.2 Model of the Deuteron 231 8.4 Questions and Problems 234 9 The Harmonic Oscillator 239 9.1 The Importance of the Simple Harmonic Oscillator 239 9.2 Solutions for the SHO 243 9.2.1 Differential Equation Approach 243 9.2.2 Properties of the Solutions 247 9.3 Experimental Realizations of the SHO 249 9.4 Classical Limits and Probability Distributions 251 xii CONTENTS 9.5 Unstable Equilibrium: Classical and Quantum Distributions 254 9.6 Questions and Problems 255 10 Alternative Methods of Solution and Approximation Methods 260 10.1 Numerical Integration 261 10.2 The Variational or Rayleigh–Ritz Method 266 10.3 The WKB method 273 10.3.1 WKB Wavefunctions 274 10.3.2 WKB Quantized Energy Levels 277 10.4 Matrix Methods 278 10.5 Perturbation Theory 286 10.5.1 Nondegenerate States 286 10.5.2 Degenerate Perturbation Theory 293 10.5.3 Time-Dependent Perturbation Theory 295 10.6 Questions and Problems 299 11 Scattering 307 11.1 Scattering in One-Dimensional Systems 307 11.1.1 Bound and Unbound States 307 11.1.2 Plane Wave Solutions 310 11.2 Scattering from a Step Potential 310 11.3 Scattering from the Finite Square Well 315 11.3.1 Attractive Well 315 11.3.2 Repulsive Barrier 319 11.4 Applications of Quantum Tunneling 321 11.4.1 Field Emission 321 11.4.2 Scanning Tunneling Microscopy 324 11.4.3 α-Particle Decay of Nuclei 325 11.4.4 Nuclear Fusion Reactions 328 11.5 Questions and Problems 330 12 More Formal Topics 333 12.1 Hermitian Operators 333 12.2 Quantum Mechanics, Linear Algebra, and Vector Spaces 337 12.3 Commutators 341 12.4 Uncertainty Principles 343 12.5 Time-Dependence and Conservation Laws in Quantum Mechanics 346 12.6 Propagators 352 12.6.1 General Case and Free Particles 352 12.6.2 Propagator and Wave Packets for the Harmonic Oscillator 353 CONTENTS xiii 12.7 Timescales in Bound State Systems: Classical Period and Quantum Revival Times 357 12.8 Questions and Problems 360 13 Operator and Factorization Methods for the Schrödinger Equation 370 13.1 Factorization Methods 370 13.2 Factorization of the Harmonic Oscillator 371 13.3 Creation and Annihilation Operators 377 13.4 Questions and Problems 380 14 Multiparticle Systems 384 14.1 Generalities 384 14.2 Separable Systems 387 14.3 Two-Body Systems 389 14.3.1 Classical Systems 390 14.3.2 Quantum Case 391 14.4 Spin Wavefunctions 394 14.5 Indistinguishable Particles 396 14.6 Questions and Problems 407 Part II The Quantum World 413 15 Two-Dimensional Quantum Mechanics 415 15.1 2D Cartesian Systems 417 15.1.1 2D Infinite Well 418 15.1.2 2D Harmonic Oscillator 422 15.2 Central Forces and Angular Momentum 423 15.2.1 Classical Case 423 15.2.2 Quantum Angular Momentum in 2D 425 15.3 Quantum Systems with Circular Symmetry 429 15.3.1 Free Particle 429 15.3.2 Circular Infinite Well 432 15.3.3 Isotropic Harmonic Oscillator 435 15.4 Questions and Problems 437 16 The Schrödinger Equation in Three Dimensions 448 16.1 Spherical Coordinates and Angular Momentum 449 16.2 Eigenfunctions of Angular Momentum 454 16.2.1 Methods of Derivation 454 xiv CONTENTS 16.2.2 Visualization and Applications 463 16.2.3 Classical Limit of Rotational Motion 465 16.3 Diatomic Molecules 467 16.3.1 Rigid Rotators 467 16.3.2 Molecular Energy Levels 469 16.3.3 Selection Rules 472 16.4 Spin and Angular Momentum 475 16.5 Addition of Angular Momentum 482 16.6 Free Particle in Spherical Coordinates 491 16.7 Questions and Problems 492 17 The Hydrogen Atom 501 17.1 Hydrogen Atom Wavefunctions and Energies 501 17.2 The Classical Limit of the Quantum Kepler Problem 507 17.3 Other “Hydrogenic” Atoms 513 17.3.1 Rydberg Atoms 513 17.3.2 Muonic Atoms 515 17.4 Multielectron Atoms 517 17.4.1 Helium-Like Atoms 519 17.4.2 Lithium-Like Atoms 524 17.4.3 The Periodic Table 527 17.5 Questions and Problems 529 18 Gravity and Electromagnetism in Quantum Mechanics 540 18.1 Classical Gravity and Quantum Mechanics 540 18.2 Electromagnetic Fields 543 18.2.1 Classical Electric and Magnetic Fields 543 18.2.2 E and B Fields in Quantum Mechanics 548 18.3 Constant Electric Fields 550 18.4 Atoms in Electric Fields: The Stark Effect 552 18.4.1 Classical Case 552 18.4.2 Quantum Stark Effect 555 18.5 Constant Magnetic Fields 561 18.6 Atoms in Magnetic Fields 564 18.6.1 The Zeeman Effect: External B Fields 564 18.6.2 Spin-Orbit Splittings: Internal B Fields 569 18.6.3 Hyperfine Splittings: Magnetic Dipole–Dipole Interactions 574 18.7 Spins in Magnetic Fields 576 18.7.1 Measuring the Spinor Nature of the Neutron Wavefunction 576 18.7.2 Spin Resonance 578 CONTENTS xv 18.8 The Aharonov–Bohm Effect 583 18.9 Questions and Problems 586 19 Scattering in Three Dimensions 596 19.1 Classical Trajectories and Cross-Sections 597 19.2 Quantum Scattering 603 19.2.1 Cross-Section and Flux 603 19.2.2 Wave Equation for Scattering and the Born Approximation 606 19.3 Electromagnetic Scattering 612 19.4 Partial Wave Expansions 619 19.5 Scattering of Particles 624 19.5.1 Frames of Reference 625 19.5.2 Identical Particle Effects 631 19.6 Questions and Problems 635 A Dimensions and MKS-type Units for Mechanics, Electricity and Magnetism, and Thermal Physics 641 A.1 Problems 642 B Physical Constants, Gaussian Integrals, and the Greek Alphabet 644 B.1 Physical Constants 644 B.2 The Greek Alphabet 646 B.3 Gaussian Probability Distribution 646 B.4 Problems 648 C Complex Numbers and Functions 649 C.1 Problems 651 D Integrals, Summations, and Calculus Results 653 D.1 Integrals 653 D.2 Summations and Series Expansions 658 D.3 Assorted Calculus Results 661 D.4 Real Integrals by Contour Integration 661 D.5 Plotting 664 D.6 Problems 665 E Special Functions 666 E.1 Trigonometric and Exponential Functions 666 E.2 Airy Functions 667 xvi CONTENTS E.3 Hermite Polynomials 668 E.4 Cylindrical Bessel Functions 669 E.5 Spherical Bessel Functions 669 E.6 Legendre Polynomials 669 E.7 Generalized Laguerre Polynomials 670 E.8 The Dirac δ-Function 671 E.9 The Euler Gamma Function 672 E.10 Problems 672 F Vectors, Matrices, and Group Theory 674 F.1 Vectors and Matrices 674 F.2 Group Theory 679 F.3 Problems 679 G Hamiltonian Formulation of Classical Mechanics 680 G.1 Problems 685 REFERENCES 687 INDEX 695 |
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