The Matrix Perturbation Method in Quantum Mechanics

This book provides an alternative approach to time-independent perturbation theory in non-relativistic quantum mechanics. It allows easy application to any initial condition because it is based on an approximation to the evolution operator and may also be used on unitary evolution operators for the unperturbed Hamiltonian in the case where the eigenvalues cannot be found. This flexibility sets it apart from conventional perturbation theory. The matrix perturbation method also gives new theoretical insights; for example, it provides corrections to the energy and wave function in one operation. Another notable highlight is the facility to readily derive a general expression for the normalization constant at m -th order, a significant difference between the approach within and those already in the literature. Another unique aspect of the matrix perturbation method is that it can be extended directly to the Lindblad master equation. The first and second-order corrections are obtained for this equation and the method is generalized for higher orders. An alternative form of the Dyson series, in matrix form instead of integral form, is also obtained. Throughout the book, several benchmark examples and practical applications underscore the potential, accuracy and good performance of this novel approach. Moreover, the method's applicability extends to some specific time-dependent Hamiltonians. This book represents a valuable addition to the literature on perturbation theory in quantum mechanics and is accessible to students and researchers alike. Preface Introduction: Matrix Perturbation Method—A New Approach to Time-Independent Quantum Perturbation Theory References Contents 1 Standard Time-Independent Perturbation Theory 1.1 Introduction 1.2 Discrete Non-degenerated Spectrum 1.2.1 Example: The One Dimensional Harmonic Oscillator with a Cubic Perturbation 1.3 Discrete Degenerated Spectrum 1.3.1 Example: The Three-Dimensional Isotropic Harmonic Oscillator with an xy Perturbation 1.3.2 Example: The Stark Effect in the Hydrogen Atom References 2 Standard Time-Dependent Perturbation Theory 2.1 Introduction 2.2 Method of Variation of Constants 2.2.1 Time-Dependent Perturbation Theory When the Spectrum Is Discrete 2.2.1.1 Finite Time Perturbation 2.2.1.2 Constant Perturbation 2.2.2 Time-Dependent Perturbation Theory When the Spectrum Is Discrete and Continuous 2.2.2.1 Monochromatic Perturbation 2.3 Method of Dyson Series 2.3.1 Constant Perturbation 2.4 Transition Probabilities 2.4.1 Constant Perturbation References 3 The Matrix Perturbation Method 3.1 Introduction 3.2 Matrix Approach to the Perturbation Theory 3.2.1 First-Order Correction 3.2.2 Second-Order Correction 3.2.3 Higher Order Corrections 3.3 Normalization Constant 3.4 Connection with the Standard Time-Independent Perturbation Theory 3.5 The Dyson Series in the Matrix Method References 4 Examples of the Matrix Perturbation Method 4.1 Introduction 4.2 The Harmonic Oscillator Perturbed by a Linear Anharmonic Term 4.2.1 The Exact Solution 4.2.1.1 A Coherent State as Initial State 4.2.2 The Approximated Perturbative Solution 4.2.2.1 Zero-Order Term 4.2.2.2 First-Order Term 4.2.2.3 Second-Order Term 4.2.2.4 The Perturbative Solution Up to Second Order 4.2.3 Comparison Between the Exact and the Approximated Solutions 4.2.3.1 The Exact and the Approximated Solutions for ω=1, λ=0.01, and α=4 at Different Times 4.2.3.2 The Exact and the Approximated Solutions for ω=1, λ=0.01, and α=10 at Different Times 4.2.3.3 The Exact and the Approximated Solutions for ω=1, λ=0.1, and α=10 at Different Times 4.2.3.4 Does the Coincidence Regime Depend on the Initial Condition? 4.3 The Harmonic Oscillator Plus a Cubic Potential 4.3.1 First Order 4.3.1.1 A Number State m as Initial State 4.3.1.2 A Coherent State α as Initial State 4.3.2 Second Order 4.3.3 Initial Condition Equal to a Number State 4.4 The Repulsive Quadratic Potential Plus a Linear Term 4.4.1 Exact Solution of the Repulsive Quadratic Potential 4.4.2 Exact Solution to the Quadratic Repulsive Potential Plus a Linear Term 4.4.3 Perturbative Solution 4.4.3.1 Zero-Order Correction 4.4.3.2 First-Order Correction 4.4.3.3 Second-Order Correction 4.4.3.4 The Solution to Second Order 4.4.3.5 A Coherent State as Initial State 4.4.3.6 The Normalized Solution 4.4.3.7 A Cat State as Initial State 4.4.4 Comparison of the Exact and the Perturbative Solutions References 5 Applications of the Matrix Perturbation Method 5.1 Introduction 5.2 Trapped Ion Hamiltonian 5.2.1 High Intensity Regime 5.2.1.1 First-Order Correction 5.2.1.2 Second-Order Correction 5.2.1.3 Comparison of the Perturbative Solution with the Small Rotation Approximation Solution 5.3 Perturbative Solution for the Rabi Model 5.4 The Binary Waveguide Array 5.4.1 Exact Solution 5.4.2 Small Rotation Approximated Solution 5.4.3 Matrix Perturbative Solution 5.4.3.1 Zero-Order Perturbative Solution 5.4.3.2 First-Order Perturbative Solution 5.4.3.3 Second-Order Perturbative Solution 5.4.3.4 Third-Order Perturbative Solution 5.4.4 Comparison of the Perturbative Solution with the Exact Solution and with the Small Rotation Solution References 6 The Matrix Perturbation Method for the Lindblad Master Equation 6.1 Introduction 6.2 Lindblad Master Equation 6.2.1 First-Order Correction 6.2.2 Second-Order Correction 6.2.3 Higher Orders 6.3 Lossy Cavity Filled with a Kerr Medium 6.3.1 Exact Solution 6.3.2 Perturbative Solution 6.3.2.1 First-Order Correction 6.3.2.2 Second-Order Correction 6.4 Comparison Between the Exact and the Approximated Solution References 7 Eliminating the Time Dependence for a Class of Time-Dependent Hamiltonians 7.1 Introduction 7.2 First Case 7.2.1 A Particle with Strongly Pulsating Mass Moving in a Linear Potential 7.2.1.1 Exact Solution 7.2.1.2 Perturbative Solution 7.2.1.3 Comparison Between the Exact and Perturbative Solutions 7.2.2 A Particle with Exponentially Increasing Mass in the Presence of a Linear Potential 7.2.2.1 Exact Solution 7.2.2.2 Perturbative Solution 7.2.2.3 Comparison Between the Exact and Perturbative Solutions 7.3 Second Case 7.3.1 Example 7.3.1.1 Exact Solution 7.3.1.2 Perturbative Solution 7.3.1.3 Comparison Between the Exact and Perturbative Solutions 7.3.2 Other Examples 7.3.2.1 Harmonic Oscillator with a Quadratically Growing Mass 7.3.2.2 Forced Caldirola–Kanai Oscillator 7.3.2.3 Particle with a Hyperbolic Growing Mass References Index