Contents 3 Quantum Mechanics in One Dimension--Bound States I 3.1 Simple Solutions of the Schrodinger Equation The Infinite Square Well-the "Particle-in-a-Box. 47 3.1.2 The Harmonic Oscillator 56 3.2 Penetration of the Classically Forbidden Region 3.2.1 The Infinite Square Well with a Rectangular Barrier Inside 3.3 Retrospective 3.4 References Problems Time-Dependent States in One Dimension 4.1 The Ehrenfest Equation 4.2 The Free Particle Quantum Representation of Par ve 4.3.1 Momentum Representation of the Operatorx 580 4.3.2 The dirac 8-function 4.3.3 Parseval's Theorem The harmonic oscillator revisited-Momentum 4.5 Motion of a Wave Packet 4.5.1 Case l. the free Packet/Particle 4.5.2 Case Il. The Packet/Particle Subjected to a Constant Field 4.5.3 Case Ill. The Packet/Particle Subjected to a Harmonic oscillator Potential 6 Retrospective Problems 5 Stationary States in One Dimension II 3 5.1 The Potential barrier The Potential Step 5.3 The Finite Square Well-Bound States 123 5.4 The morse potential 5.5 The Linear potential 139 5.6 The WKB Approximation 5.6.1 The Nature of the Approximation 5.6 The Connection Formulas for Bound States 148 5.6.3 A Bound State Example--the Linear Potential .. 155 5.6.4 Tunneling 158 omparison with a Rectangular Barrier A Tunneling Example--Predissociation References 65 Problemsxii Contents 3 Quantum Mechanics in One Dimension—Bound States I . . . . . . . . . . . 47 3.1 Simple Solutions of the Schr¨odinger Equation . . . . . . . . . . . . . . . 47 3.1.1 The Infinite Square Well—the “Particle-in-a-Box” . 47 3.1.2 The Harmonic Oscillator . . . . . . . . . . . . . . . . . . . . . . 56 3.2 Penetration of the Classically Forbidden Region . . . . . . . . . . . . . . 69 3.2.1 The Infinite Square Well with a Rectangular Barrier Inside . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 3.3 Retrospective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 3.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 4 Time-Dependent States in One Dimension . . . . . . . . . . . . . . . . . . . . . . . . 83 4.1 The Ehrenfest Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 4.2 The Free Particle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 4.3 Quantum Representation of Particles—Wave Packets . . . . . . . . . 86 4.3.1 Momentum Representation of the Operator x . . . . . 90 4.3.2 The Dirac δ-function . . . . . . . . . . . . . . . . . . . . . . . . . 91 4.3.3 Parseval’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 4.4 The Harmonic Oscillator Revisited—Momentum Eigenfunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 4.5 Motion of a Wave Packet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 4.5.1 Case I. The Free Packet/Particle . . . . . . . . . . . . . . . . 98 4.5.2 Case II. The Packet/Particle Subjected to a Constant Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 4.5.3 Case III. The Packet/Particle Subjected to a Harmonic Oscillator Potential . . . . . . . . . . . . . . . . . . 104 4.6 Retrospective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 5 Stationary States in One Dimension II . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 5.1 The Potential Barrier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 5.2 The Potential Step . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 5.3 The Finite Square Well—Bound States . . . . . . . . . . . . . . . . . . . . . 123 5.4 The Morse Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 5.5 The Linear Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 5.6 The WKB Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 5.6.1 The Nature of the Approximation . . . . . . . . . . . . . . . 145 5.6.2 The Connection Formulas for Bound States . . . . . . 148 5.6.3 A Bound State Example—the Linear Potential . . . . 155 5.6.4 Tunneling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 5.6.5 Comparison with a Rectangular Barrier . . . . . . . . . . 162 5.6.6 A Tunneling Example—Predissociation . . . . . . . . . 163 5.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165