# Feynman Path Integrals in Quantum Mechanics and Statistical Physics

Path Integral Formalism Intuitive ApproachProbability AmplitudeDouble Slit ExperimentPhysical StateProbability AmplitudeRevisit Double Slit ExperimentDistinguishabilitySuperposition PrincipleRevisit the Double Slit Experiment/Superposition PrincipleOrthogonalityOrthonormalityChange of BasisGeometrical Interpretation of State VectorCoordinate TransformationProjection OperatorContinuous SpectrumMatrix Representation of Linear OperatorsMatrix ElementLinear Self-Adjoint (Hermitian Conjugate) OperatorsProduct of Hermitian OperatorsContinuous SpectrumSchturm-Liouville Problem: Eigenstates and EigenvaluesRevisit Linear Self-Adjoint (Hermitian) OperatorsUnitary TransformationMean (Expectation) Value and Matrix DensityDegeneracyDensity OperatorCommutativity of OperatorsOperators in Phase SpaceConfiguration SpacePosition and Wave FunctionMomentum SpaceClassical ActionTransition AmplitudePath Integration in Phase SpaceFrom the Schrödinger Equation to Path IntegrationTrotter Product FormulaTransition AmplitudeHamiltonian Formulation of Path IntegrationPath Integral SubtletiesMid-point RuleLagrangian Formulation of Path IntegrationComplex Gaussian IntegralTransition AmplitudeLaw for Consecutive EventsSemigroup Property of the Transition AmplitudeStationary and Quasi-Classical ApproximationsStationary Phase Method/Fourier IntegralContribution from Non-Degenerate Stationary PointsUnique Stationary PointQuasi-Classical Approximation/Fluctuating PathFree Particle Classical Action and Transition AmplitudeFree Particle Classical ActionFree Particle Transition AmplitudeFrom Path Integrals to Quantum MechanicsFree and Driven Harmonic Oscillator Classical Action and Transition AmplitudeFree Oscillator Classical ActionDriven or Forced Harmonic Oscillator Classical ActionFree and Driven Harmonic Oscillator Transition AmplitudeFluctuation Contribution to Transition AmplitudeMaslov CorrectionGeneralized Feynman Path IntegrationCoordinate RepresentationFree Particle Transition AmplitudeGaussian Functional Feynman Path IntegralsCharged Particle in a Magnetic FieldFrom Path Integration to the Schrödinger EquationWave FunctionSchrödinger EquationThe Schrödinger Equation’s Green’s FunctionTransition Amplitude for a Time-Independent HamiltonianRetarded Green FunctionQuasi-Classical ApproximationWentzel-Kramer-Brillouin (WKB) MethodCondition of Applicability of the Quasi-Classical ApproximationBounded Quasi-Classical MotionQuasi-Classical QuantizationPath Integral LinkPotential WellPotential BarrierQuasi-Classical Derivation of the PropagatorReflection and Tunneling via a BarrierTransparency of the Quasi-Classical BarrierHomogenous FieldMotion in a Central Symmetric FieldPolar EquationRadial Equation for a Spherically Symmetric Potential in Three DimensionsMotion in a Coulombic FieldHydrogen AtomFree Particle and Harmonic OscillatorEigenfunction and EigenvalueFree ParticleTransition Amplitude for a Particle in a Homogenous FieldHarmonic OscillatorTransition Amplitude HermiticityMatrix Element of a Physical Operator via Functional IntegralMatrix Representation of the Transition Amplitude of a Forced Harmonic OscillatorCharged Particle Interaction with PhononsPath Integral Perturbation TheoryTime-Dependent PerturbationTransition ProbabilityTime-Energy Uncertainty RelationDensity of Final StateTransition RateContinuous Spectrum due to a Constant PerturbationHarmonic PerturbationTransition Matrix ElementFunctional DerivativeFunctional Derivative of the Action FunctionalFunctional Derivative and Matrix ElementQuantum Statistical Mechanics Functional Integral ApproachIntroductionDensity MatrixPartition FunctionExpectation Value of a Physical ObservableDensity MatrixDensity Matrix in the Energy RepresentationPartition Function and Density Matrix Path Integral RepresentationDensity Matrix Path Integral RepresentationDensity Matrix Operator Average Value in Phase SpaceGeneralized Gaussian Functional Path Integral in Phase SpaceDensity Matrix via Transition AmplitudePartition Function in the Path integral RepresentationParticle Interaction with a Driven or Forced Harmonic Oscillator: Partition FunctionFree Particle Density Matrix and Partition FunctionQuantum Harmonic Oscillator Density Matrix and Partition FunctionQuasi-Classical Approximation in Quantum Statistical MechanicsCentroid Effective PotentialExpectation ValueFeynman Variational MethodPolaron TheoryIntroductionPolaron Energy and Effective MassFunctional Influence PhasePolaron Model LagrangianPolaron Partition FunctionInfluence Phase via Feynman Functional Integral in The Density Matrix RepresentationExpectation Value of a Physical QuantityDensity matrixFull System Polaron Partition Function in a 3D StructureModel System Polaron Partition Function in a 3D StructureFeynman Inequality and Generating FunctionalPolaron Characteristics in a 3D StructurePolaron Asymptotic CharacteristicsPolaron Characteristics in a Quasi-1D Quantum WireHamiltonian of the Electron in a Quasi 1D Quantum WireLagrangian of the Electron in a Quasi-1D Quantum WirePartition function of the Electron in a Quasi-1D Quantum WirePolaron Generating FunctionPolaron Asymptotic CharacteristicsStrong Coupling Regime Polaron CharacteristicsBipolaron Characteristics in a Quasi-1D Quantum WireIntroductionBipolaron Diagrammatic RepresentationBipolaron LagrangianBipolaron Equation of MotionTransformation into Normal CoordinatesDiagonalization of the LagrangianBipolaron Partition FunctionBipolaron Generating FunctionBipolaron Asymptotic CharacteristicsPolaron Characteristics in a Quasi-0D Spherical Quantum DotIntroductionPolaron LagrangianNormal ModesLagrangian DiagonalizationTransformation to Normal CoordinatesPolaron Partition FunctionGenerating FunctionBipolaron Characteristics in a Quasi-0D Spherical Quantum DotIntroductionModel LagrangianModel LagrangianEquation of Motion and Normal ModesDiagonalization of the LagrangianPartition FunctionFull System Influence PhaseBipolaron EnergyGenerating FunctionBipolaron CharacteristicsPolaron Characteristics in a Cylindrical Quantum DotSystem HamiltonianTransformation to Normal CoordinatesLagrangian DiagonalizationPolaron Energy/Partition FunctionPolaron Generating FunctionPolaron EnergyBipolaron Characteristics in a Cylindrical Quantum DotSystem HamiltonianModel System Action FunctionalEquation of Motion / Normal ModesLagrangian DiagonalizationBipolaron Partition FunctionBipolaron Generating FunctionBipolaron EnergyPolaron Characteristics in a Quasi-0D Cylindrical Quantum Dot with Asymmetrical Parabolic PotentialPolaron EnergyBipolaron Characteristics in a Quasi-0D Cylindrical Quantum Dot with Asymmetrical Parabolic PotentialPolaron in a Magnetic FieldMultiphoton Absorption by Polarons in a Spherical Quantum DotTheory of Multiphoton Absorption by PolaronsBasic ApproximationsAbsorption CoefficientPolaronic Kinetics in a Spherical Quantum DotKinetic Theory of GasesDistribution FunctionPrinciple of Detailed EquilibriumTransport Phenomenon and Boltzmann-Lorentz Kinetic EquationTransport Relaxation TimeBoltzmann H-TheoremThermal ConductivityDiffusion