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Feynman Path Integrals in Quantum Mechanics and Statistical Physics

Path Integral Formalism Intuitive Approach Probability Amplitude Double Slit Experiment Physical State Probability Amplitude Revisit Double Slit Experiment Distinguishability Superposition Principle Revisit the Double Slit Experiment/Superposition Principle Orthogonality Orthonormality Change of Basis Geometrical Interpretation of State Vector Coordinate Transformation Projection Operator Continuous Spectrum Matrix Representation of Linear Operators Matrix Element Linear Self-Adjoint (Hermitian Conjugate) Operators Product of Hermitian Operators Continuous Spectrum Schturm-Liouville Problem: Eigenstates and Eigenvalues Revisit Linear Self-Adjoint (Hermitian) Operators Unitary Transformation Mean (Expectation) Value and Matrix Density Degeneracy Density Operator Commutativity of Operators Operators in Phase Space Configuration Space Position and Wave Function Momentum Space Classical Action Transition Amplitude Path Integration in Phase Space From the Schrödinger Equation to Path Integration Trotter Product Formula Transition Amplitude Hamiltonian Formulation of Path Integration Path Integral Subtleties Mid-point Rule Lagrangian Formulation of Path Integration Complex Gaussian Integral Transition Amplitude Law for Consecutive Events Semigroup Property of the Transition Amplitude Stationary and Quasi-Classical Approximations Stationary Phase Method/Fourier Integral Contribution from Non-Degenerate Stationary Points Unique Stationary Point Quasi-Classical Approximation/Fluctuating Path Free Particle Classical Action and Transition Amplitude Free Particle Classical Action Free Particle Transition Amplitude From Path Integrals to Quantum Mechanics Free and Driven Harmonic Oscillator Classical Action and Transition Amplitude Free Oscillator Classical Action Driven or Forced Harmonic Oscillator Classical Action Free and Driven Harmonic Oscillator Transition Amplitude Fluctuation Contribution to Transition Amplitude Maslov Correction Generalized Feynman Path Integration Coordinate Representation Free Particle Transition Amplitude Gaussian Functional Feynman Path Integrals Charged Particle in a Magnetic Field From Path Integration to the Schrödinger Equation Wave Function Schrödinger Equation The Schrödinger Equation’s Green’s Function Transition Amplitude for a Time-Independent Hamiltonian Retarded Green Function Quasi-Classical Approximation Wentzel-Kramer-Brillouin (WKB) Method Condition of Applicability of the Quasi-Classical Approximation Bounded Quasi-Classical Motion Quasi-Classical Quantization Path Integral Link Potential Well Potential Barrier Quasi-Classical Derivation of the Propagator Reflection and Tunneling via a Barrier Transparency of the Quasi-Classical Barrier Homogenous Field Motion in a Central Symmetric Field Polar Equation Radial Equation for a Spherically Symmetric Potential in Three Dimensions Motion in a Coulombic Field Hydrogen Atom Free Particle and Harmonic Oscillator Eigenfunction and Eigenvalue Free Particle Transition Amplitude for a Particle in a Homogenous Field Harmonic Oscillator Transition Amplitude Hermiticity Matrix Element of a Physical Operator via Functional Integral Matrix Representation of the Transition Amplitude of a Forced Harmonic Oscillator Charged Particle Interaction with Phonons Path Integral Perturbation Theory Time-Dependent Perturbation Transition Probability Time-Energy Uncertainty Relation Density of Final State Transition Rate Continuous Spectrum due to a Constant Perturbation Harmonic Perturbation Transition Matrix Element Functional Derivative Functional Derivative of the Action Functional Functional Derivative and Matrix Element Quantum Statistical Mechanics Functional Integral Approach Introduction Density Matrix Partition Function Expectation Value of a Physical Observable Density Matrix Density Matrix in the Energy Representation Partition Function and Density Matrix Path Integral Representation Density Matrix Path Integral Representation Density Matrix Operator Average Value in Phase Space Generalized Gaussian Functional Path Integral in Phase Space Density Matrix via Transition Amplitude Partition Function in the Path integral Representation Particle Interaction with a Driven or Forced Harmonic Oscillator: Partition Function Free Particle Density Matrix and Partition Function Quantum Harmonic Oscillator Density Matrix and Partition Function Quasi-Classical Approximation in Quantum Statistical Mechanics Centroid Effective Potential Expectation Value Feynman Variational Method Polaron Theory Introduction Polaron Energy and Effective Mass Functional Influence Phase Polaron Model Lagrangian Polaron Partition Function Influence Phase via Feynman Functional Integral in The Density Matrix Representation Expectation Value of a Physical Quantity Density matrix Full System Polaron Partition Function in a 3D Structure Model System Polaron Partition Function in a 3D Structure Feynman Inequality and Generating Functional Polaron Characteristics in a 3D Structure Polaron Asymptotic Characteristics Polaron Characteristics in a Quasi-1D Quantum Wire Hamiltonian of the Electron in a Quasi 1D Quantum Wire Lagrangian of the Electron in a Quasi-1D Quantum Wire Partition function of the Electron in a Quasi-1D Quantum Wire Polaron Generating Function Polaron Asymptotic Characteristics Strong Coupling Regime Polaron Characteristics Bipolaron Characteristics in a Quasi-1D Quantum Wire Introduction Bipolaron Diagrammatic Representation Bipolaron Lagrangian Bipolaron Equation of Motion Transformation into Normal Coordinates Diagonalization of the Lagrangian Bipolaron Partition Function Bipolaron Generating Function Bipolaron Asymptotic Characteristics Polaron Characteristics in a Quasi-0D Spherical Quantum Dot Introduction Polaron Lagrangian Normal Modes Lagrangian Diagonalization Transformation to Normal Coordinates Polaron Partition Function Generating Function Bipolaron Characteristics in a Quasi-0D Spherical Quantum Dot Introduction Model Lagrangian Model Lagrangian Equation of Motion and Normal Modes Diagonalization of the Lagrangian Partition Function Full System Influence Phase Bipolaron Energy Generating Function Bipolaron Characteristics Polaron Characteristics in a Cylindrical Quantum Dot System Hamiltonian Transformation to Normal Coordinates Lagrangian Diagonalization Polaron Energy/Partition Function Polaron Generating Function Polaron Energy Bipolaron Characteristics in a Cylindrical Quantum Dot System Hamiltonian Model System Action Functional Equation of Motion / Normal Modes Lagrangian Diagonalization Bipolaron Partition Function Bipolaron Generating Function Bipolaron Energy Polaron Characteristics in a Quasi-0D Cylindrical Quantum Dot with Asymmetrical Parabolic Potential Polaron Energy Bipolaron Characteristics in a Quasi-0D Cylindrical Quantum Dot with Asymmetrical Parabolic Potential Polaron in a Magnetic Field Multiphoton Absorption by Polarons in a Spherical Quantum Dot Theory of Multiphoton Absorption by Polarons Basic Approximations Absorption Coefficient Polaronic Kinetics in a Spherical Quantum Dot Kinetic Theory of Gases Distribution Function Principle of Detailed Equilibrium Transport Phenomenon and Boltzmann-Lorentz Kinetic Equation Transport Relaxation Time Boltzmann H-Theorem Thermal Conductivity Diffusion

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From Path Integrals to Quantum Mechanics
From equation 4.99 chapter 4, we can find the link between the coordinates and the wave parameters. The type of the coordinates qifl and q: contributing most to the transition amplitude are those satisfying the relation in 5.70 and tailor the phase of the transition amplitude to have an...
(Feynman Path Integrals in Quantum Mechanics and Statistical Physics)
The Feynman Path Integrals Hypothesis
The second example is the generation of the Feynman Path Integral, a hypothesis about the behaviour of quantum particles—in particular about the paths, or in a sense the 'trajectories', of quantum particles. It aims at solving the problems of QM data by starting from the problem of the infinite self-energies...
(Model-Based Reasoning in Science and Technology)
Generalized Feynman Path Integration
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(Feynman Path Integrals in Quantum Mechanics and Statistical Physics)
Gaussian Functional Feynman Path Integrals
We consider the following generalized Feynman path integral: where, the Lagrangian: is quadratic both in the set of generalized coordinates q as well as generalized velocities q and f (f') is the driving force. Such an integral in 6.30 is therefore Gaussian and so may...
(Feynman Path Integrals in Quantum Mechanics and Statistical Physics)
THE IMPACT OF QUANTUM MECHANICS
As discussed in regard to research by Jahn and Dunne (1997), one of the most powerful ways of connecting the objective and subjective realms is to examine recent work in quantum mechanics. The current resurgence of quantum mechanics is quite clear in its premise of stating that the observer is never...
(Affective Teaching in Nursing)
Quantum Mechanical Analysis of Selected NMR Pulse Sequences
NMR pulse sequences play a central role in the modern NMR technique; they allow the manipulation of the nuclear spin Hamiltonian with great versatility, setting up the conditions necessary for the desired NMR observable to be detected. The pulse sequences to be discussed next include the one pulse sequence,...
(NMR of liquid crystal dendrimers)
Quantum Mechanics
In quantum theory, the state of a physical system is typically represented by a nonzero vector in a complex Hilbert space H, but all non-zero scalar multiples of a vector represent the same state. Thus, the states constitute the projective space associated to H. Because of the freedom to adjust...
(Martin Davis on Computability, Computational Logic, and Mathematical Foundations)
Molecular Structure and Quantum Mechanics
Our other possible locus for emergence in chemistry is molecular structure. Are molecules anything more than systems of charged particles, interacting according to the laws of quantum mechanics? I seem to be in a minority of one in giving that question an affirmative answer, and certainly so when I defend...
(Philosophical and Scientific Perspectives on Downward Causation)
Path Integrals
The path integral approach has been tried out to quantize gravity. The major problem here is the definition of measure on the space of all paths in consideration. We could foliate the spacetime and consider all possible three-geometries on each spacelike surface. The amplitude in question is that of...
(The story of collapsing stars : black holes, naked singularities, and the cosmic play of quantum gravity)
Bell’s six possible worlds of quantum mechanics
Towards the end of his life Bell published two rather general papers on the foundations of quantum theory, and, as they are of considerable interest, we review them briefly in this chapter. The first, published in 1986, is titled ‘Six Possible Worlds of Quantum Mechanics’, and he used this paper...
(The New Quantum Age: From Bell’s Theorem to Quantum Computation and Teleportation)