Path integrals, stochastic differential equations and operator ordering in supersymmetric quantum mechanics

1990 ◽  
Vol 23 (13) ◽  
pp. 2927-2937
Author(s):  
M O'Connor
Keyword(s):  
Quantum Mechanics ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Path Integrals ◽  
Supersymmetric Quantum Mechanics ◽  
Operator Ordering

scholarly journals Deformed Shape Invariant Superpotentials in Quantum Mechanics and Expansions in Powers of ℏ

Symmetry ◽  
2020 ◽  
Vol 12 (11) ◽  
pp. 1853
Author(s):  
Christiane Quesne
Keyword(s):  
Differential Equation ◽  
Quantum Mechanics ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Supersymmetric Quantum Mechanics ◽  
Invariance Condition ◽  
Partial Differential ◽  
Shape Invariance ◽  
Shape Invariant ◽  
Infinite Set

We show that the method developed by Gangopadhyaya, Mallow, and their coworkers to deal with (translational) shape invariant potentials in supersymmetric quantum mechanics and consisting in replacing the shape invariance condition, which is a difference-differential equation, which, by an infinite set of partial differential equations, can be generalized to deformed shape invariant potentials in deformed supersymmetric quantum mechanics. The extended method is illustrated by several examples, corresponding both to ℏ-independent superpotentials and to a superpotential explicitly depending on ℏ.

Stochastic differential equations: Langevin, Fokker–Planck (FP) equations

2021 ◽  
pp. 831-856
Author(s):  
Jean Zinn-Justin
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Langevin Equation ◽  
Diffusion Processes ◽  
Random Noise ◽  
Path Integrals ◽  
Brst Symmetry ◽  
Random Motion ◽  
Definition Of ◽  
Fokker Planck

This chapter is devoted to the study of Langevin equations, first order in time differential equations, which depend on a random noise, and which belong to a class of stochastic differential equations that describe diffusion processes, or random motion. From a Langevin equation, a Fokker–Planck (FP) equation for the probability distribution of the solutions, at given time, of the Langevin equation can be derived. It is shown that observables averaged over the noise can also be calculated from path integrals, whose integrands define automatically positive measures. The path integrals involve dynamic actions that have automatically a Becchi–Rouet–Stora–Tyutin (BRST) symmetry and, when the driving force derives from a potential, exhibit the simplest form of supersymmetry. In some cases, like Brownian motion on Riemannian manifolds, difficulties appear in the precise definition of stochastic equations, quite similar to the quantization problem encountered in quantum mechanics (QM). Time discretization provides one possible solution to the problem.

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