Introduction to Quantum Mechanics

The following sections are included:

• Contents

• Preface

The following sections are included:

• Origin and Discovery of Quantum Mechanics

• Contradicting Discretization: Uncertainties

• Particle-Wave Dualism

• Particle-Wave Dualism and Uncertainties
  • Further thought experiments

• Bohr's Complementarity Principle

• Further Examples

The following sections are included:

• Introductory Remarks

• The Hamilton Formalism

• Liouville Equation, Probabilities
  • Single particle consideration
  • Ensemble consideration

• Expectation Values of Observables

• Extension beyond Classical Mechanics

The following sections are included:

• Introductory Remarks

• Hilbert Spaces

• Operators in Hilbert Space

• Linear Functionals and Distributions
  • Interpretation of distributions in physics
  • Properties of functionals and the delta distribution

The following sections are included:

• Introductory Remarks

• Ket and Bra States

• Linear Operators, Hermitian Operators

• Observables

• Representation Spaces and Basis Vectors

The following sections are included:

• Introductory Remarks

• The Density Matrix

• The Probability Density ρ(x, t)

• Schrödinger Equation and Liouville Equation
  • Evaluation of the density matrix

The following sections are included:

• Introductory Remarks

• The One-Dimensional Linear Oscillator

• The Energy Representation of the Oscillator

• The Configuration Space Representation

• The Harmonic Oscillator Equation
  • Derivation of the generating function

The following sections are included:

• Introductory Remarks

• Time-dependent and Time-independent Cases

• The Green's Function of a Free Particle

• Green's Function of the Harmonic Oscillator

• The Inverted Harmonic Oscillator
  • Wave packets
  • A particle's sojourn time T at the maximum

The following sections are included:

• Introductory Remarks

• Asymptotic Series versus Convergent Series
  • The error function and Stokes discontinuities
  • Stokes discontinuities of oscillator functions

• Asymptotic Series from Differential Equations

• Formal Definition of Asymptotic Expansions

• Rayleigh–Schrödinger Perturbation Theory

• Degenerate Perturbation Theory

• Dingle–Müller Perturbation Method

The following sections are included:

• Introductory Remarks

• Reconsideration of Electrodynamics

• Schrödinger and Heisenberg Pictures

• The Liouville Equation

The following sections are included:

• Introductory Remarks

• States and Observables
  • Uncertainty relation for observables A, B

• One-Dimensional Systems
  • The translation operator U(a)

• Equations of Motion

• States of Finite Lifetime

• The Interaction Picture

• Time-Dependent Perturbation Theory

• Transitions into the Continuum

• General Time-Dependent Method

The following sections are included:

• Introductory Remarks

• Separation of Variables, Angular Momentum
  • Separation of variables

• Representation of Rotation Group

• Angular Momentum:Angular Representation

• Radial Equation for Hydrogen-like Atoms

• Discrete Spectrum of the Coulomb Potential
  • The eigenvalues
  • Laguerre polynomials: Various definitions in use!
  • The eigenfunctions
  • Hydrogen-like atoms in parabolic coordinates

• Continuous Spectrum of Coulomb Potential
  • The Rutherford formula

• Scattering of a Wave Packet

• Scattering Phase and Partial Waves

The following sections are included:

• Introductory Remarks

• Continuity Equation and Conditions

• The Short-Range Delta Potential

• Scattering from a Potential Well

• Degenerate Potentials and Tunneling

The following sections are included:

• Introductory Remarks

• The Freely Falling Particle: Quantization
  • Superposition of de Broglie waves
  • Probability distribution at large times

• Stationary States

• The Saddle Point or Stationary Phase Method

The following sections are included:

• Introductory Remarks

• Classical Limit and Hydrodynamics Analogy

• The WKB Method
  • The approximate WKB solutions
  • Turning points and matching of WKB solutions
  • Linear approximation and matching

• Bohr–Sommerfeld–Wilson Quantization

• Further Examples

The following sections are included:

• Introductory Remarks

• The Power Potential

• The Three-Dimensional Wave Function

The following sections are included:

• Introductory Remarks

• Regge Trajectories

• The S-Matrix

• The Energy Expansion

• The Sommerfeld–Watson Transform

• Concluding Remarks

The following sections are included:

• Introductory Remarks

• Cosine Potential: Weak Coupling Solutions
  • The Floquet exponent
  • Four types of periodic solutions

• Cosine Potential: Strong Coupling Solutions
  • Preliminary remarks
  • The solutions
  • The eigenvalues
  • The level splitting

• Elliptic and Ellipsoidal Potentials
  • Introduction
  • Solutions and eigenvalues
  • The level splitting
  • Reduction to Mathieu functions

• Concluding Remarks

The following sections are included:

• Introductory Remarks

• The Inverted Double Well Potential
  • Defining the problem
  • Three pairs of solutions
  • Matching of solutions
  • Boundary conditions at the origin
  • Boundary conditions at infinity
  • The complex eigenvalues

• The Double Well Potential
  • Defining the problem
  • Three pairs of solutions
  • Matching of solutions
  • Boundary conditions at the minima
  • Boundary conditions at the origin
  • Eigenvalues and level splitting
  • General Remarks

The following sections are included:

• Introductory Remarks

• The Potential 1/r⁴ — Case of Small h²
  • Preliminary considerations
  • Small h² solutions in terms of Bessel functions
  • Small h² solutions in terms of hyperbolic functions
  • Notation and properties of solutions
  • Derivation of the S-matrix
  • Evaluation of the S-matrix
  • Calculation of the absorptivity

• The Potential 1/r⁴ — Case of Large h²
  • Preliminary remarks
  • The Floquet exponent for large h²
  • Construction of large-h² solutions
  • The connection formulas
  • Derivation of the S-matrix

• Concluding Remarks

The following sections are included:

• Introductory Remarks

• Cosine Potential: Large Order Behaviour

• Cosine Potential: Complex Eigenvalues
  • The decaying ground state
  • Decaying excited states
  • Relating the level splitting to imaginary E
  • Recalculation of large order behaviour

• Cosine Potential: A Different Calculation

• Anharmonic Oscillators
  • The inverted double well
  • The double well

• General Remarks

The following sections are included:

• Introductory Remarks

• Path Integrals and Green's Functions

• The Green's Function for Potential V=0
  • Configuration space representation
  • Momentum space representation

• Including V in First Order Perturbation

• Rederivation of the Rutherford Formula

• Path Integrals in Dirac's Notation

• Canonical Quantization from Path Integrals

The following sections are included:

• Introductory Remarks

• The Constant Classical Field

• Soliton Theories in One Spatial Dimension

• Stability of Classical Configurations

• Bogomol'nyi Equations and Bounds

• The Small Fluctuation Equation

• Existence of Finite-Energy Solutions

• Ginzburg–Landau Vortices

• Introduction to Homotopy Classes

• The Fundamental Homotopy Group

The following sections are included:

• Introductory Remarks

• Instantons and Anti-Instantons

• The Level Difference

• Field Fluctuations
  • The fluctuation equation
  • Evaluation of the functional integral
  • The Faddeev–Popov constraint insertion
  • The single instanton contribution
  • Instanton–anti-instanton contributions

• Concluding Remarks

The following sections are included:

• Introductory Remarks

• The Bounce in a Simple Example

• The Inverted Double Well: The Bounce and Complex Energy
  • The bounce solution
  • The single bounce contribution
  • Evaluation of the single bounce kernel
  • Sum over an infinite number of bounces
  • Comments

• Inverted Double Well: Constant Solutions

• The Cubic Potential and its Complex Energy

The following sections are included:

• Introductory Remarks

• The Double Well Theory on a Circle
  • Periodic configurations
  • The fluctuation equation
  • The limit of infinite period

• The Inverted Double Well on a Circle
  • Periodic configurations
  • The fluctuation equation
  • The limit of infinite period

• The Sine–Gordon Theory on a Circle
  • Periodic configurations
  • The fluctuation equation

• Conclusions

The following sections are included:

• Introductory Remarks

• The Double Well and Periodic Instantons
  • Periodic configurations and the double well
  • Transition amplitude and Feynman kernel
  • Fluctuations about the periodic instanton
  • The single periodic instanton contribution
  • Sum over instanton–anti-instanton pairs

• The Cosine Potential and Periodic Instantons
  • Periodic configurations and the cosine potential
  • Transition amplitude and Feynman kernel
  • The fluctuation equation and its eigenmodes
  • The single periodic instanton contribution
  • Sum over instanton–anti-instanton pairs

• The Inverted Double Well and Periodic Instantons
  • Periodic configurations and the inverted double well
  • Transition amplitude and Feynman kernel
  • The fluctuation equation and its eigenmodes
  • The single periodic bounce contribution
  • Summing over the infinite number of bounces

• Concluding Remarks

The following sections are included:

• Introductory Remarks

• Constraints: How they arise
  • Singular Lagrangians

• The Hamiltonian of Singular Systems

• Persistence of Constraints in Course of Time

• Constraints as Generators of a Gauge Group

• Gauge Fixing and Dirac Quantization

• The Formalism of Dirac Quantization
  • Poisson and Dirac brackets in field theory

• Dirac Quantization of Free Electrodynamics

• Faddeev–Jackiw Canonical Quantization
  • The method of Faddeev and Jackiw

The following sections are included:

• Introductory Remarks

• Relating Period to Temperature

• Crossover in Previous Cases
  • The double well and phase transitions
  • The cosine potential and phase transitions

• Crossover in a Simple Spin Model

• Concluding Remarks

The considerations in the preceding text demonstrate that with perturbation theory one can go significantly beyond the examples which are usually treated in elementary texts on quantum mechanics, like the harmonic oscillator, the Coulomb potential and several other simple potentials, and — unlike any other method — one can handle with it such diverse cases as periodic potentials, anharmonic oscillators, screened Coulomb potentials, and even a typical singular potential. In most cases the expansions of interest in physics are asymptotic, in a few cases supplemented by convergent expansions (as in the case of the cosine potential). A crucial point in the formulation of the perturbation method is the full exploitation of the symmetries provided by the parameters of the respective Schrödinger equation. If these are taken into account, the perturbation method can be extended and employed even for the calculation of the behaviour of its large order terms. This is not widely appreciated, and therefore one encounters repeatedly attempts to find a better way with some kind of convergent expansion…

The following sections are included:

• Appendix A Properties of Jacobian Elliptic Functions

• Bibliography

• Index