Path integral solution of the Schrödinger equation in curvilinear coordinates: A straightforward procedure

1996 ◽  
Vol 37 (9) ◽  
pp. 4310-4319
Author(s):  
J. LaChapelle
Keyword(s):  
Schrödinger Equation ◽  
Path Integral ◽  
Schrodinger Equation ◽  
Integral Solution ◽  
Curvilinear Coordinates

PATH INTEGRAL CALCULATION OF GREEN’S FUNCTION FOR SCHRÖDINGER EQUATION IN UNITARY GAUGE

1992 ◽  
Vol 07 (31) ◽  
pp. 7775-7786
Author(s):  
L. ROZANSKY
Keyword(s):  
Schrödinger Equation ◽  
Green’S Function ◽  
Green's Function ◽  
Path Integral ◽  
Schrodinger Equation ◽  
Preexponential Factor ◽  
Unitary Gauge ◽  
Gauge Fixing

Green’s function of Schrödinger equation is represented as a time-reparametrization invariant path integral. Unitary gauge fixing enables us to get the WKB preexponential factor without calculating determinants of operators containing derivatives.

scholarly journals Review of Feynman’s Path Integral in Quantum Statistics: from the Molecular Schrödinger Equation to Kleinert’s Variational Perturbation Theory

2014 ◽  
Vol 15 (4) ◽  
pp. 853-894 ◽  
Author(s):  
Kin-Yiu Wong
Keyword(s):  
Schrödinger Equation ◽  
Path Integral ◽  
Schrodinger Equation ◽  
Stochastic Methods ◽  
Phosphoryl Transfer ◽  
Many Body ◽  
Molecular Systems ◽  
Pi Method ◽  
Variational Perturbation ◽  
Kp Theory

AbstractFeynman’s path integral reformulates the quantum Schrödinger differential equation to be an integral equation. It has been being widely used to compute internuclear quantum-statistical effects on many-body molecular systems. In this Review, the molecular Schrödinger equation will first be introduced, together with the Born-Oppenheimer approximation that decouples electronic and internuclear motions. Some effective semiclassical potentials, e.g., centroid potential, which are all formulated in terms of Feynman’s path integral, will be discussed and compared. These semiclassical potentials can be used to directly calculate the quantum canonical partition function without individual Schrödinger’s energy eigenvalues. As a result, path integrations are conventionally performed with Monte Carlo and molecular dynamics sampling techniques. To complement these techniques, we will examine how Kleinert’s variational perturbation (KP) theory can provide a complete theoretical foundation for developing non-sampling/non-stochastic methods to systematically calculate centroid potential. To enable the powerful KP theory to be practical for many-body molecular systems, we have proposed a new path-integral method: automated integration-free path-integral (AIF-PI) method. Due to the integration-free and computationally inexpensive characteristics of our AIF-PI method, we have used it to perform ab initio path-integral calculations of kinetic isotope effects on proton-transfer and RNA-related phosphoryl-transfer chemical reactions. The computational procedure of using our AIF-PI method, along with the features of our new centroid path-integral theory at the minimum of the absolute-zero energy (AMAZE), are also highlighted in this review.

scholarly journals Kajian Integral Lintasan Levy dalam Mekanika Kuantum Fraksional untuk Membentuk Persamaan Schrodinger Fraksional

2020 ◽  
Vol 5 (1) ◽  
pp. 80-86
Author(s):  
Chandra Halim ◽  
M. Farchani Rosyid
Keyword(s):  
Stochastic Process ◽  
Quantum Mechanics ◽  
Fractal Dimension ◽  
Schrödinger Equation ◽  
Path Integral ◽  
Schrodinger Equation ◽  
Fractional Quantum ◽  
Standard Quantum ◽  
Fractional Schrödinger Equation ◽  
Fractional Schrodinger Equation

The implementation of Lévy path integral generated by Lévy stochastic process on fractional Schrödinger equation has been investigated in the framework of fractional quantum mechanics. As the comparison, the implementation of Feynmann path integral generated by Wiener stochastic process on Schrödinger equation also has been investigated in the framework of standard quantum mechanics. There are two stochastic processes. There are Lévy stochastic and Wiener stochastic process. Both of them are able to produce fractal. In fractal’s concept, there is a value known as fractal dimension. The implementation of fractal dimension is the diffusion equation obtained by using Fokker Planck equation. In this paper, Lévy and Wiener fractal dimension have been obtained. There are  for Lévy and 2 for Wiener/Brown fractal dimension. Fractional quantum mechanics is generalization of standard quantum mechanics. A fractional quantum mechanics state is represented by wave function from fractional Schrödinger equation. Fractional Schrödinger equation is obtained by using kernel of Lévy path integral generated by Lévy stochastic process. Otherwise, standard quantum mechanics state is represented by wave function from standard Schrödinger equation. Standard Schrödinger equation is obtained by using kernel of Feynmann path integral generated by Wiener/Brown stochastic process.  Both Lévy and Feynmann Kernel have been investigated and the outputs are the Fourier Integral momentum phase of those kernels. We find that the forms of those kernels have similiraty. Therefore, we obtain Schrödinger equation from Lévy and Feynmann Kernel and also the comparison of Lévy energy in fractional quantum mechanics and particle energy in standard quantum mechanics.

scholarly journals Lévy path integral approach to the fractional Schrödinger equation with delta-perturbed infinite square well

2015 ◽  
Vol 36 ◽  
pp. 1560015 ◽  
Author(s):  
M. M. I. Nayga ◽  
J. P. H. Esguerra
Keyword(s):  
Schrödinger Equation ◽  
Path Integral ◽  
Schrodinger Equation ◽  
Perturbation Series ◽  
Integral Approach ◽  
Fractional Schrödinger Equation ◽  
Path Integral Approach ◽  
Square Well ◽  
Energy Dependent ◽  
Fractional Schrodinger Equation

Using a path integral approach, we consider a fractional Schrödinger equation with delta-perturbed infinite square well. The Lévy path integral, which is generalized from the Feynman path intergal for the propagator, is expanded into a perturbation series. From this, the energy-dependent Green's function is obtained.

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