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Quantummechanical path integrals with Wiener measure for all polynomial Hamiltonians. II

Ingrid C. Daubechies, John R. Klauder

Published Jan 1, 2012• 0 citations
OpenAlex

Abstract

Related Articles Quantum time of arrival Goursat problem J. Math. Phys. 53, 043702 (2012) Effective-mass Klein-Gordon-Yukawa problem for bound and scattering states J. Math. Phys. 52, 092101 (2011) Efficient quantum trajectory representation of wavefunctions evolving in imaginary time J. Chem. Phys. 135, 034104 (2011) Green's function for the time-dependent scattering problem in the fractional quantum mechanics J. Math. Phys. 52, 042103 (2011) AKNS hierarchy, Darboux transformation and conservation laws of the 1D nonautonomous nonlinear Schrodinger equations J. Math. Phys. 52, 043502 (2011)

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Related Articles Quantum time of arrival Goursat problem J. Math. Phys. 53, 043702 (2012) Effective-mass Klein-Gordon-Yukawa problem for bound and scattering states J. Math. Phys. 52, 092101 (2011) Efficient quantum trajectory representation of wavefunctions evolving in imaginary time J. Chem. Phys. 135, 034104 (2011) Green's function for the time-dependent scattering problem in the fractional quantum mechanics J. Math. Phys. 52, 042103 (2011) AKNS hierarchy, Darboux transformation and conservation laws of the 1D nonautonomous nonlinear Schrodinger equations J. Math. Phys. 52, 043502 (2011)
Abstract We solved the Klein–Gordon equation for a generalized inverse quadratic Yukawa (GIQY) potential via path integrals approach. We applied approximations to deal with the terms 1/r 2 and 1/r. A path integral representation of Green’s function relating to a particle moving in a mixture of equal vector and scalar potentials was established. To integrate Green’s function, a space-time transformation was successfully used, and the present problem was reduced to a previously known modified Pöschl-Teller potential problem. The integrated Green’s function contains important information about the considered system, from which we obtained energy eigenvalues and the corresponding normalized eigenfunctions for various values of n and ℓquantum numbers. Numerical results, Schrödinger solutions, and special cases such as the modified screened Coulomb plus inversely quadratic Yukawa potential, Kratzer potential, Yukawa potential, and Coulomb potential were also considered. These results are largely consistent with previous studies.
The Boltzmann evolution of a wavefunction can be recast as imaginary-time dynamics of the quantum trajectory ensemble. The quantum effects arise from the momentum-dependent quantum potential--computed approximately to be practical in high-dimensional systems--influencing the trajectories in addition to the external classical potential [S. Garashchuk, J. Chem. Phys. 132, 014112 (2010)]. For a nodeless wavefunction represented as ψ(x, t) = exp(-S(x, t)/ħ) with the trajectory momenta defined by ∇S(x, t), analysis of the Lagrangian and Eulerian evolution shows that for bound potentials the former is more accurate while the latter is more practical because the Lagrangian quantum trajectories diverge with time. Introduction of stationary and time-dependent components into the wavefunction representation generates new Lagrangian-type dynamics where the trajectory spreading is controlled improving efficiency of the trajectory description. As an illustration, different types of dynamics are used to compute zero-point energy of a strongly anharmonic well and low-lying eigenstates of a high-dimensional coupled harmonic system.
Recently, a self-contained trajectory-based formulation of non-relativistic quantum mechanics was developed [Ann. Phys. 315, 505 (2005); Chem. Phys. 370, 4 (2010); J. Chem. Phys. 136, 031102 (2012)], that makes no use of wavefunctions or complex amplitudes of any kind. Quantum states are represented as ensembles of real-valued quantum trajectories that extremize a suitable action. Here, the trajectory-based approach is developed into a viable, generally covariant, relativistic quantum theory for single (spin-zero, massive) particles. Central to this development is the introduction of a new notion of global simultaneity for accelerated particles--together with basic postulates concerning probability conservation and causality. The latter postulate is found to be violated by the Klein-Gordon equation, leading to its well-known problems as a single-particle theory. Various examples are considered, including the time evolution of a relativistic Gaussian wavepacket.