Path integration in non-relativistic quantum mechanics

Abstract A new method for computing path integrals explicitly is developed and applied to problems in non-relativistic quantum mechanics, such as: wave functions, propagators on configuration spaces and on phase space, caustic problems, bound states. Path integrals for paths on curved spaces and for paths on multiply-connected spaces are computed.

[1]  P. Combe,et al.  On the cylindrical approximation of the feynman path integral , 1978 .

[2]  The Semiclassical Expansion of the Anharmonic-Oscillator Propagator. , 1979 .

[3]  W. T. Martin,et al.  Transformations of Wiener integrals under a general class of linear transformations , 1945 .

[4]  S. Albeverio,et al.  Mathematical theory of Feynman path integrals , 1976 .

[5]  J. Gervais,et al.  Real-time approach to instanton phenomena: (I). Multidimensional potentials with degenerate absolute minima , 1978 .

[6]  C. Garrod,et al.  Hamiltonian Path-Integral Methods , 1966 .

[7]  Jean Zinn-Justin,et al.  Perturbation theory at large order. I. The phi/sup 2//sup N/ interaction , 1977 .

[8]  R. Feynman,et al.  Quantum Mechanics and Path Integrals , 1965 .

[9]  W. Miller Semiclassical quantization of nonseparable systems: A new look at periodic orbit theory , 1975 .

[10]  V. Maslov Stationary-phase method for Feynman's continual integral , 1970 .

[11]  S. Albeverio,et al.  Oscillatory integrals and the method of stationary phase in infinitely many dimensions, with applications to the classical limit of quantum mechanics I , 1977 .

[12]  A. Maheshwari The Generalized Wiener-Feynman Path Integrals , 1976 .

[13]  A. Maheshwari Comment on 'exact evaluation of a path integral relating to an electron gas in a random potential' , 1975 .

[14]  G. Parisi,et al.  Discrepancies from Asymptotic Series and Their Relation to Complex Classical Trajectories. , 1978 .

[15]  B. Simon On positive eigenvalues of one‐body schrödinger operators , 1969 .

[16]  Joseph B. Keller,et al.  The Feynman integral , 1975 .

[17]  M. Mizrahi Correspondence rules and path integrals , 1979 .

[18]  C. Dewitt,et al.  Feynman Functional Integrals for Systems of Indistinguishable Particles , 1971 .

[19]  Maurice M. Mizrahi,et al.  Phase space path integrals, without limiting procedure , 1978 .

[20]  Kiyosi Itô Wiener Integral and Feynman Integral , 1961 .

[21]  Michael Martin Nieto,et al.  Phase and Angle Variables in Quantum Mechanics , 1968 .

[22]  E. J. McShane,et al.  Integrals devised for special purposes , 1963 .

[23]  J. Keller,et al.  Geometrical theory of diffraction. , 1962, Journal of the Optical Society of America.

[24]  L. Faddeev The Feynman integral for singular Lagrangians , 1969 .

[25]  Stephen W. Hawking,et al.  Path Integral Derivation of Black Hole Radiance , 1976 .

[26]  Jürgen Moser,et al.  Lectures on Celestial Mechanics , 1971 .

[27]  M. Clutton-Brock Feynman's kernel and the classical path , 1965, Mathematical Proceedings of the Cambridge Philosophical Society.

[28]  D. Thouless Vibrational states of nuclei in the random phase approximation , 1961 .

[29]  Tosio Kato Perturbation theory for linear operators , 1966 .

[30]  R. Feynman An Operator calculus having applications in quantum electrodynamics , 1951 .

[31]  Michael V Berry,et al.  Semiclassical approximations in wave mechanics , 1972 .

[32]  Bruce Lee Nelson Relativity, Topology, and Path Integration. , 1978 .

[33]  M. Hirsch,et al.  Differential Equations, Dynamical Systems, and Linear Algebra , 1974 .

[34]  D. McLaughlin Path Integrals, Asymptotics, and Singular Perturbations , 1972 .

[35]  R. Feynman The principle of least action in quantum mechanics , 1942 .

[36]  R. L. Stratonovich,et al.  Topics in the theory of random noise , 1967 .

[37]  V. N. Popov,et al.  Feynman Diagrams for the Yang-Mills Field , 1967 .

[38]  A. Friedman Stochastic Differential Equations and Applications , 1975 .

[39]  M. Kac On Some Connections between Probability Theory and Differential and Integral Equations , 1951 .

[40]  J. Connor Semiclassical theory of molecular collisions : three nearly coincident classical trajectories , 1973 .

[41]  Edmund Taylor Whittaker,et al.  A treatise on the analytical dynamics of particles and rigid bodies , 1927 .

[42]  J. Thorpe,et al.  Lecture Notes on Elementary Topology and Geometry. , 1967 .

[43]  S. Brush Functional Integrals and Statistical Physics , 1961 .

[44]  R. Cameron A Family of Integrals Serving to Connect the Wiener and Feynman Integrals , 1960 .

[45]  Yu. L. Daletskiǐ FUNCTIONAL INTEGRALS CONNECTED WITH OPERATOR EVOLUTION EQUATIONS , 1962 .

[46]  J. Gervais,et al.  WKB wave function for systems with many degrees of freedom: A unified view of solitons and pseudoparticles , 1977 .

[47]  E. M. Lifshitz,et al.  Classical theory of fields , 1952 .

[48]  Green's Function for an Electron in a Lattice , 1969 .

[49]  M. Mizrahi Generalized Hermite polynomials , 1975 .

[50]  J. S. Dowker Quantum mechanics and field theory on multiply connected and on homogeneous spaces , 1972 .

[51]  J. Gervais,et al.  Point canonical transformations in the path integral , 1976 .

[52]  M. Mizrahi On Path Integral Solutions of the Schrodinger Equation, Without Limiting Procedure , 1976 .

[53]  J. Connor Evaluation of multidimensional canonical integrals in semiclassical collision theory , 1973 .

[54]  L. Schulman A Path integral for spin , 1968 .

[55]  Path integration in phase space , 1977 .

[56]  M. Kac On distributions of certain Wiener functionals , 1949 .

[57]  I. Segal Distributions in Hilbert space and canonical systems of operators , 1958 .

[58]  J. Dollard,et al.  Product integrals and the Schrödinger equation , 1977 .

[59]  Stanislav Molchanov,et al.  DIFFUSION PROCESSES AND RIEMANNIAN GEOMETRY , 1975 .

[60]  C. Dewitt Feynman's path integral , 1972 .

[61]  D. Saari,et al.  Stable and Random Motions in Dynamical Systems , 1975 .

[62]  Mark S. C. Reed,et al.  Method of Modern Mathematical Physics , 1972 .

[63]  Y. Kifer On the Asymptotics of the Transition Density of Processes with Small Diffusion , 1977 .

[64]  B. Gaveau Principe de moindre action, propagation de la chaleur et estimees sous elliptiques sur certains groupes nilpotents , 1977 .

[65]  Kiyosi Itô Generalized uniform complex measures in the Hilbertian metric space with their application to the Feynman integral , 1967 .

[66]  D. Thouless Stability conditions and nuclear rotations in the Hartree-Fock theory , 1960 .

[67]  R. Balian,et al.  Solution of the Schrodinger Equation in Terms of Classical Paths , 1974 .

[68]  Cécile DeWitt-Morette,et al.  The semiclassical expansion , 1976 .

[69]  Joseph B. Keller,et al.  Corrected bohr-sommerfeld quantum conditions for nonseparable systems , 1958 .

[70]  H. Matsumoto,et al.  THE GOLDSTONE THEOREM AND DYNAMICAL REARRANGEMENT OF SYMMETRY IN THE PATH - INTEGRAL FORMALISM , 1974 .

[71]  K. Yosida,et al.  Brownian Motion on the Surface of the 3-Sphere , 1949 .

[72]  R. Ikeda,et al.  Proton magnetic resonance studies of quantum mechanical tunnelling of ammonium ions in several ammonium salts , 1973 .

[73]  N. Wiener The Average value of a Functional , 1924 .

[74]  M. Berry,et al.  Calculating the bound spectrum by path summation in action-angle variables , 1977 .

[75]  M. Misheloff,et al.  Path-integral formulation of scattering theory , 1975 .

[76]  E. M. Lifshitz,et al.  Quantum mechanics: Non-relativistic theory, , 1959 .

[77]  M. Gutzwiller,et al.  Periodic Orbits and Classical Quantization Conditions , 1971 .

[78]  Brosl Hasslacher,et al.  Particle spectrum in model field theories from semiclassical functional integral techniques , 1975 .

[79]  J. Connor Asymptotic evaluation of multidimensional integrals for the S matrix in the semiclassical theory of inelastic and reactive molecular collisions , 1973 .

[80]  The classical action in nonrelativistic quantum mechanics , 1977 .

[81]  Bryce S. DeWitt,et al.  Dynamical theory of groups and fields , 1964 .

[82]  I. Gel'fand,et al.  Integration in Functional Spaces and its Applications in Quantum Physics , 1960 .

[83]  R. Feynman,et al.  Space-Time Approach to Non-Relativistic Quantum Mechanics , 1948 .

[84]  George F. R. Ellis,et al.  The Large Scale Structure of Space-Time , 2023 .

[85]  J. Gillis,et al.  Probability and Related Topics in Physical Sciences , 1960 .

[86]  Choquet Bruhat,et al.  Analysis, Manifolds and Physics , 1977 .

[87]  A. Truman Feynman path integrals and quantum mechanics as h/→0 , 1976 .

[88]  Edward Nelson Feynman Integrals and the Schrödinger Equation , 1964 .

[89]  A. Salam,et al.  Equivalent Formulations of Massive Vector Field Theories , 1970 .

[90]  F. Bayen,et al.  Quantum mechanics as a deformation of classical mechanics , 1977 .

[91]  D. McLaughlin Complex Time, Contour Independent Path Integrals, and Barrier Penetration , 1972 .

[92]  N. G. van Kampen,et al.  Stochastic differential equations , 1976 .

[93]  Feynman path integrals , 1974 .

[94]  Cecile Morette,et al.  On the definition and approximation of Feynman's path integrals , 1951 .

[95]  W. Thirring Classical Dynamical Systems , 1978 .

[96]  Leonard Meirovitch,et al.  Methods of analytical dynamics , 1970 .

[97]  R. Balian,et al.  Asymptotic evaluation of the Green's function for large quantum numbers , 1971 .