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E-grāmata: Feynman Path Integrals in Quantum Mechanics and Statistical Physics [Taylor & Francis e-book]

(University of Dschang, Cameroon)
  • Formāts: 400 pages, 58 Line drawings, black and white
  • Izdošanas datums: 20-Apr-2021
  • Izdevniecība: CRC Press
  • ISBN-13: 9781003145554
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  • Taylor & Francis e-book
  • Cena: 266,81 €*
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  • Standarta cena: 381,15 €
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Feynman Path Integrals in Quantum Mechanics and Statistical PhysicsPalielināt
  • Formāts: 400 pages, 58 Line drawings, black and white
  • Izdošanas datums: 20-Apr-2021
  • Izdevniecība: CRC Press
  • ISBN-13: 9781003145554
Citas grāmatas par šo tēmu:
Taylor & Francis e-booksMore info about Taylor & Francis e-books
Rokasgrāmatas mājas lapa: https://www.taylorfrancis.com/books/9781003145554
"This book provides an ideal introduction to the use of Feynman Path Integrals in the fields of quantum mechanics and statistical physics. It is written for graduate students and researchers in physics, mathematical physics, applied mathematics as well as chemistry. The material is presented in an accessible manner for readers with little knowledge of quantum mechanics and no prior exposure to path integrals. It begins with elementary concepts and a review of quantum mechanics that gradually builds the framework for the Feynman path integrals and how they are applied to problems in quantum mechanics and statistical physics. Problem sets throughout the book allow readers to test their understanding and reinforce the explanations of the theory in real situations"--

This book provides an ideal introduction to the use of Feynman path integrals in the fields of quantum mechanics and statistical physics. It is written for graduate students and researchers in physics, mathematical physics, applied mathematics as well as chemistry. The material is presented in an accessible manner for readers with little knowledge of quantum mechanics and no prior exposure to path integrals. It begins with elementary concepts and a review of quantum mechanics that gradually builds the framework for the Feynman path integrals and how they are applied to problems in quantum mechanics and statistical physics. Problem sets throughout the book allow readers to test their understanding and reinforce the explanations of the theory in real situations.

Features:

  • Comprehensive and rigorous yet, presents an easy-to-understand approach.

  • Applicable to a wide range of disciplines.

    • Accessible to those with little, or basic, mathematical understanding.

    • Preface xi
    1 Path Integral Formalism Intuitive Approach
    		1(12)
    1.1 Probability Amplitude
    		1(12)
    1.1.1 Double Slit Experiment
    		1(1)
    1.1.2 Physical State
    		2(1)
    1.1.3 Probability Amplitude
    		2(1)
    1.1.4 Revisit Double Slit Experiment
    		2(1)
    1.1.5 Distinguishability
    		3(1)
    1.1.6 Superposition Principle
    		3(1)
    1.1.7 Revisit the Double Slit Experiment/Superposition Principle
    		4(1)
    1.1.8 Orthogonality
    		5(1)
    1.1.9 Orthonormality
    		6(1)
    1.1.10 Change of Basis
    		7(1)
    1.1.11 Geometrical Interpretation of State Vector
    		8(1)
    1.1.12 Coordinate Transformation
    		9(1)
    1.1.13 Projection Operator
    		10(1)
    1.1.14 Continuous Spectrum
    		11(2)
    2 Matrix Representation of Linear Operators
    		13(16)
    2.1 Matrix Element
    		13(1)
    2.2 Linear Self-Adjoint (Hermitian Conjugate) Operators
    		13(2)
    2.3 Product of Hermitian Operators
    		15(1)
    2.4 Continuous Spectrum
    		16(1)
    2.5 Schturm-Liouville Problem: Eigenstates and Eigenvalues
    		17(3)
    2.6 Revisit Linear Self-Adjoint (Hermitian) Operators
    		20(1)
    2.7 Unitary Transformation
    		21(1)
    2.8 Mean (Expectation) Value and Matrix Density
    		22(1)
    2.9 Degeneracy
    		23(1)
    2.10 Density Operator
    		24(1)
    2.11 Commutativity of Operators
    		25(4)
    3 Operators in Phase Space
    		29(6)
    3.1 Introduction
    		29(1)
    3.2 Configuration Space
    		30(1)
    3.3 Position and Wave Function
    		31(1)
    3.4 Momentum Space
    		32(1)
    3.5 Classical Action
    		33(2)
    4 Transition Amplitude
    		35(18)
    4.1 Path Integration in Phase Space
    		36(5)
    4.1.1 From the Schrodinger Equation to Path Integration
    		36(3)
    4.1.2 Trotter Product Formula
    		39(2)
    4.2 Transition Amplitude
    		41(12)
    4.2.1 Hamiltonian Formulation of Path Integration
    		41(2)
    4.2.2 Path Integral Subtleties
    		43(1)
    4.2.2.1 Mid-point Rule
    		43(1)
    4.2.3 Lagrangian Formulation of Path Integration
    		44(1)
    4.2.3.1 Complex Gaussian Integral
    		44(2)
    4.2.4 Transition Amplitude
    		46(5)
    4.2.5 Law for Consecutive Events
    		51(1)
    4.2.6 Semigroup Property of the Transition Amplitude
    		51(2)
    5 Stationary and Quasi-Classical Approximations
    		53(24)
    5.1 Stationary Phase Method / Fourier Integral
    		53(3)
    5.2 Contribution from Non-Degenerate Stationary Points
    		56(4)
    5.2.1 Unique Stationary Point
    		58(2)
    5.3 Quasi-Classical Approximation/Fluctuating Path
    		60(7)
    5.3.1 Free Particle Classical Action and Transition Amplitude
    		60(1)
    5.3.1.1 Free Particle Classical Action
    		61(1)
    5.3.1.2 Free Particle Transition Amplitude
    		62(3)
    5.3.1.3 From Path Integrals to Quantum Mechanics
    		65(2)
    5.4 Free and Driven Harmonic Oscillator Classical Action and Transition Amplitude
    		67(4)
    5.4.1 Free Oscillator Classical Action
    		67(2)
    5.4.2 Driven or Forced Harmonic Oscillator Classical Action
    		69(2)
    5.5 Free and Driven Harmonic Oscillator Transition Amplitude
    		71(1)
    5.6 Fluctuation Contribution to Transition Amplitude
    		72(5)
    5.6.1 Maslov Correction
    		74(3)
    6 Generalized Feynman Path Integration
    		77(14)
    6.1 Coordinate Representation
    		77(2)
    6.2 Free Particle Transition Amplitude
    		79(2)
    6.3 Gaussian Functional Feynman Path Integrals
    		81(6)
    6.4 Charged Particle in a Magnetic Field
    		87(4)
    7 From Path Integration to the Schrodinger Equation
    		91(10)
    7.1 Wave Function
    		91(1)
    7.2 Schrodinger Equation
    		92(2)
    7.3 The Schrodinger Equation's Green's Function
    		94(1)
    7.4 Transition Amplitude for a Time-Independent Hamiltonian
    		95(2)
    7.5 Retarded Green Function
    		97(4)
    8 Quasi-Classical Approximation
    		101(34)
    8.1 Wentzel-Kramer-Brillouin (WKB) Method
    		101(11)
    8.1.1 Condition of Applicability of the Quasi-Classical Approximation
    		104(2)
    8.1.2 Bounded Quasi-Classical Motion
    		106(3)
    8.1.3 Quasi-Classical Quantization
    		109(2)
    8.1.4 Path Integral Link
    		111(1)
    8.2 Potential Well
    		112(2)
    8.3 Potential Barrier
    		114(2)
    8.4 Quasi-Classical Derivation of the Propagator
    		116(1)
    8.5 Reflection and Tunneling via a Barrier
    		117(2)
    8.6 Transparency of the Quasi-Classical Barrier
    		119(2)
    8.7 Homogenous Field
    		121(14)
    8.7.1 Motion in a Central Symmetric Field
    		125(1)
    8.7.1.1 Polar Equation
    		125(4)
    8.7.1.2 Radial Equation for a Spherically Symmetric Potential in Three Dimensions
    		129(1)
    8.7.2 Motion in a Coulombic Field
    		130(1)
    8.7.2.1 Hydrogen Atom
    		130(5)
    9 Free Particle and Harmonic Oscillator
    		135(10)
    9.1 Eigenfunction and Eigenvalue
    		135(3)
    9.1.1 Free Particle
    		135(2)
    9.1.2 Transition Amplitude for a Particle in a Homogenous Field
    		137(1)
    9.2 Harmonic Oscillator
    		138(5)
    9.3 Transition Amplitude Hermiticity
    		143(2)
    10 Matrix Element of a Physical Operator via Functional Integral
    		145(8)
    10.1 Matrix Representation of the Transition Amplitude of a Forced Harmonic Oscillator
    		147(6)
    10.1.1 Charged Particle Interaction with Phonons
    		150(3)
    11 Path Integral Perturbation Theory
    		153(20)
    11.1 Time-Dependent Perturbation
    		160(3)
    11.2 Transition Probability
    		163(1)
    11.3 Time-Energy Uncertainty Relation
    		164(2)
    11.4 Density of Final State
    		166(2)
    11.4.1 Transition Rate
    		166(2)
    11.5 Continuous Spectrum due to a Constant Perturbation
    		168(1)
    11.6 Harmonic Perturbation
    		169(4)
    12 Transition Matrix Element
    		173(6)
    13 Functional Derivative
    		179(12)
    13.1 Functional Derivative of the Action Functional
    		181(2)
    13.2 Functional Derivative and Matrix Element
    		183(8)
    14 Quantum Statistical Mechanics Functional Integral Approach
    		191(8)
    14.1 Introduction
    		191(1)
    14.2 Density Matrix
    		191(1)
    14.2.1 Partition Function
    		191(1)
    14.3 Expectation Value of a Physical Observable
    		192(1)
    14.4 Density Matrix
    		192(2)
    14.5 Density Matrix in the Energy Representation
    		194(5)
    15 Partition Function and Density Matrix Path Integral Representation
    		199(20)
    15.1 Density Matrix Path Integral Representation
    		199(6)
    15.1.1 Density Matrix Operator Average Value in Phase Space
    		199(2)
    15.1.1.1 Generalized Gaussian Functional Path Integral in Phase Space
    		201(1)
    15.1.2 Density Matrix via Transition Amplitude
    		202(3)
    15.2 Partition Function in the Path Integral Representation
    		205(4)
    15.3 Particle Interaction with a Driven or Forced Harmonic Oscillator: Partition Function
    		209
    15.4 Free Particle Density Matrix and Partition Function
    		221
    15.5 Quantum Harmonic Oscillator Density Matrix and Partition Function
    		214(5)
    16 Quasi-Classical Approximation in Quantum Statistical Mechanics
    		219(1)
    16.1 Centroid Effective Potential
    		220(5)
    16.2 Expectation Value
    		225(4)
    17 Feynman Variational Method
    		229(8)
    18 Polaron Theory
    		237(100)
    18.1 Introduction
    		237(2)
    18.2 Polaron Energy and Effective Mass
    		239(2)
    18.3 Functional Influence Phase
    		241(5)
    18.3.1 Polaron Model Lagrangian
    		243(1)
    18.3.2 Polaron Partition Function
    		243(3)
    18.4 Influence Phase via Feynman Functional Integral in the Density Matrix Representation
    		246(9)
    18.4.1 Expectation Value of a Physical Quantity
    		246(1)
    18.4.1.1 Density Matrix
    		246(9)
    18.5 Full System Polaron Partition Function in a 3D Structure
    		255(1)
    18.6 Model System Polaron Partition Function in a 3D Structure
    		256(1)
    18.7 Feynman Inequality and Generating Functional
    		257(2)
    18.8 Polaron Characteristics in a 3D Structure
    		259(6)
    18.8.1 Polaron Asymptotic Characteristics
    		264(1)
    18.9 Polaron Characteristics in a Quasi-1D Quantum Wire
    		265(4)
    18.9.1 Hamiltonian of the Electron in a Quasi 1D Quantum Wire
    		265(1)
    18.9.1.1 Lagrangian of the Electron in a Quasi-1D Quantum Wire
    		266(1)
    18.9.1.2 Partition Function of the Electron in a Quasi-1D Quantum Wire
    		267(2)
    18.10 Polaron Generating Function
    		269(1)
    18.11 Polaron Asymptotic Characteristics
    		270(3)
    18.12 Strong Coupling Regime Polaron Characteristics
    		273(3)
    18.13 Bipolaron Characteristics in a Quasi-ID Quantum Wire
    		276(13)
    18.13.1 Introduction
    		276(2)
    18.13.2 Bipolaron Diagrammatic Representation
    		278(1)
    18.13.3 Bipolaron Lagrangian
    		278(2)
    18.13.4 Bipolaron Equation of Motion
    		280(2)
    18.13.5 Transformation into Normal Coordinates
    		282(1)
    18.13.5.1 Diagonalization of the Lagrangian
    		282(1)
    18.13.6 Bipolaron Partition Function
    		283(2)
    18.13.7 Bipolaron Generating Function
    		285(1)
    18.13.8 Bipolaron Asymptotic Characteristics
    		286(3)
    18.14 Polaron Characteristics in a Quasi-OD Spherical Quantum Dot
    		289(6)
    18.14.1 Introduction
    		289(1)
    18.14.2 Polaron Lagrangian
    		290(1)
    18.14.3 Normal Modes
    		290(1)
    18.14.4 Lagrangian Diagonalization
    		291(1)
    18.14.4.1 Transformation to Normal Coordinates
    		291(1)
    18.14.5 Polaron Partition Function
    		292(1)
    18.14.6 Generating Function
    		293(2)
    18.15 Bipolaron Characteristics in a Quasi-OD Spherical Quantum Dot
    		295(5)
    18.15.1 Introduction
    		295(1)
    18.15.2 Model Lagrangian
    		296(1)
    18.15.3 Model Lagrangian
    		296(1)
    18.15.3.1 Equation of Motion and Normal Modes
    		296(1)
    18.15.4 Diagonalization of the Lagrangian
    		297(2)
    18.15.5 Partition Function
    		299(1)
    18.15.6 Full System Influence Phase
    		300(1)
    18.16 Bipolaron Energy
    		300(4)
    18.16.1 Generating Function
    		300(1)
    18.16.2 Bipolaron Characteristics
    		301(3)
    18.17 Polaron Characteristics in a Cylindrical Quantum Dot
    		304(6)
    18.17.1 System Hamiltonian
    		304(1)
    18.17.2 Transformation to Normal Coordinates
    		305(1)
    18.17.2.1 Lagrangian Diagonalization
    		305(1)
    18.17.3 Polaron Energy/Partition Function
    		306(1)
    18.17.4 Polaron Generating Function
    		307(1)
    18.17.5 Polaron Energy
    		308(2)
    18.18 Bipolaron Characteristics in a Cylindrical Quantum Dot
    		310(5)
    18.18.1 System Hamiltonian
    		310(1)
    18.18.1.1 Model System Action Functional
    		310(1)
    18.18.1.2 Equation of Motion / Normal Modes
    		311(1)
    18.18.1.3 Lagrangian Diagonalization
    		312(1)
    18.18.1.4 Bipolaron Partition Function
    		312(1)
    18.18.1.5 Bipolaron Generating Function
    		313(1)
    18.18.1.6 Bipolaron Energy
    		313(2)
    18.19 Polaron Characteristics in a Quasi-OD Cylindrical Quantum Dot with Asymmetrical Parabolic Potential
    		315(1)
    18.20 Polaron Energy
    		316(4)
    18.21 Bipolaron Characteristics in a Quasi-OD Cylindrical Quantum Dot with Asymmetrical Parabolic Potential
    		320(4)
    18.22 Polaron in a Magnetic Field
    		324(13)
    19 Multiphoton Absorption by Polarons in a Spherical Quantum Dot
    		337(14)
    19.1 Theory of Multiphoton Absorption by Polarons
    		337(1)
    19.2 Basic Approximations
    		338(1)
    19.3 Absorption Coefficient
    		339(12)
    20 Polaronic Kinetics in a Spherical Quantum Dot
    		351(14)
    21 Kinetic Theory of Gases
    		365(26)
    21.1 Distribution Function
    		365(1)
    21.2 Principle of Detailed Equilibrium
    		365(4)
    21.3 Transport Phenomenon and Boltzmann-Lorentz Kinetic Equation
    		369(4)
    21.4 Transport Relaxation Time
    		373(2)
    21.5 Boltzmann H-Theorem
    		375(3)
    21.6 Thermal Conductivity
    		378(2)
    21.7 Diffusion
    		380(6)
    21.8 Electron-Phonon System Equation of Motion
    		386(5)
    References 391(4)
    Index 395
    Lukong Cornelius Fai is professor of theoretical physics at the Department of Physics, Faculty of Sciences, University of Dschang. He is Head of Condensed Matter and Nanomaterials as well as Mesoscopic and Multilayer Structures Laboratory. He was formerly a senior associate at the Abdus Salam International Centre for Theoretical Physics (ICTP), Italy. He holds a Masters of Science in Physics and Mathematics (June 1991) as well as a Doctor of Science in Physics and Mathematics (February 1997) from Moldova State University. He is an author of over a hundred scientific publications and three textbooks
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