Comparison of WKB (Wentzel–Kramers–Brillouin) and SWKB solutions of Fokker–Planck equations with exact results; application to electron thermalization

1991 ◽  
Vol 69 (6) ◽  
pp. 712-719 ◽  
Author(s):  
Bernie Shizgal ◽  
Lucio Demeio
Keyword(s):  
Schrödinger Equation ◽  
Probability Distribution Function ◽  
Schrodinger Equation ◽  
Inert Gases ◽  
Time Dependent ◽  
Exact Results ◽  
The One ◽  
Semiclassical Approximations ◽  
Fokker Planck Equations ◽  
Fokker Planck

A comparison of WKB (Wentzel–Kramers–Brillouin) and SWKB eigenfunctions of the Schrödinger equation for potentials in the class encountered in supersymmetric quantum mechanics is presented. The potentials that are studied are those that result from the transformation of a Fokker–Planck eigenvalue problem to a Schrödinger equation. Linear Fokker–Planck equations of the type considered in this paper give the probability distribution function for a large number of physical situations. The time-dependent solutions can be expressed as a sum of exponential terms with each term characterized by an eigenvalue of the Fokker–Planck operator. The specific Fokker–Planck operator considered is the one that describes the thermalization of electrons in the inert gases. The WKB and SWKB semiclassical approximations are compared with exact numerical results. Although the eigenvalues can be very close to the exact values, we find significant departures for the eigenfunctions.

A Refined Speed Limit for the Imaginary-Time Schrödinger Equation

2020 ◽  
Vol 27 (02) ◽  
pp. 2050010
Author(s):  
Jie Sun ◽  
Songfeng Lu
Keyword(s):  
Schrödinger Equation ◽  
Quantum Dynamics ◽  
Schrodinger Equation ◽  
Time Dependent ◽  
Uncertainty Relations ◽  
Speed Limit ◽  
Full Time ◽  
Imaginary Time ◽  
Full Solution ◽  
The One

Recently, Kieu proposed a new class of time-energy uncertainty relations for time-dependent Hamiltonians, which is not only formal but also useful for actually evaluating the speed limit of quantum dynamics. Inspired by this work, Okuyama and Ohzeki obtained a similar speed limit for the imaginary-time Schrödinger equation. In this paper, we refine the latter one to make it be further like that of Kieu formally. As in the work of Kieu, only the initial states and the Hamiltonians, but neither the instantaneous eigenstates nor the full time-dependent wave like functions, which would demand a full solution for a time-dependent system, are required for our optimized speed limit. It turns out to be more helpful for estimating the speed limit of an actual quantum annealing driven by the imaginary-time Schrödinger equation. For certain case, the refined speed limit given here becomes the only useful tool to do this estimation, because the one given by Okuyama and Ohzeki cannot do the same job.

Analytic time-dependent solutions of the one-dimensional Schrödinger equation

2014 ◽  
Vol 82 (10) ◽  
pp. 955-961 ◽  
Author(s):  
Wytse van Dijk ◽  
F. Masafumi Toyama ◽  
Sjirk Jan Prins ◽  
Kyle Spyksma
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Time Dependent ◽  
One Dimensional ◽  
The One

Introduction

2018 ◽  
Author(s):  
Niels Engholm Henriksen ◽  
Flemming Yssing Hansen
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Degrees Of Freedom ◽  
State Level ◽  
Boltzmann Distribution ◽  
Theoretical Description ◽  
Time Dependent ◽  
Nuclear Dynamics ◽  
Molecular Reaction ◽  
Oppenheimer Approximation

This introductory chapter considers first the relation between molecular reaction dynamics and the major branches of physical chemistry. The concept of elementary chemical reactions at the quantized state-to-state level is discussed. The theoretical description of these reactions based on the time-dependent Schrödinger equation and the Born–Oppenheimer approximation is introduced and the resulting time-dependent Schrödinger equation describing the nuclear dynamics is discussed. The chapter concludes with a brief discussion of matter at thermal equilibrium, focusing at the Boltzmann distribution. Thus, the Boltzmann distribution for vibrational, rotational, and translational degrees of freedom is discussed and illustrated.

scholarly journals The Nonlinear Schrödinger Equation for Orthonormal Functions II: Application to Lieb–Thirring Inequalities

2021 ◽  
Author(s):  
Rupert L. Frank ◽  
David Gontier ◽  
Mathieu Lewin
Keyword(s):  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Bound State ◽  
Nonlinear Schrodinger Equation ◽  
Best Constant ◽  
Nonlinear Schrödinger ◽  
The One ◽  
Bound State Case ◽  
Cubic Nonlinear Schrödinger Equation

AbstractIn this paper we disprove part of a conjecture of Lieb and Thirring concerning the best constant in their eponymous inequality. We prove that the best Lieb–Thirring constant when the eigenvalues of a Schrödinger operator $$-\Delta +V(x)$$ - Δ + V ( x ) are raised to the power $$\kappa $$ κ is never given by the one-bound state case when $$\kappa >\max (0,2-d/2)$$ κ > max ( 0 , 2 - d / 2 ) in space dimension $$d\ge 1$$ d ≥ 1 . When in addition $$\kappa \ge 1$$ κ ≥ 1 we prove that this best constant is never attained for a potential having finitely many eigenvalues. The method to obtain the first result is to carefully compute the exponentially small interaction between two Gagliardo–Nirenberg optimisers placed far away. For the second result, we study the dual version of the Lieb–Thirring inequality, in the same spirit as in Part I of this work Gontier et al. (The nonlinear Schrödinger equation for orthonormal functions I. Existence of ground states. Arch. Rat. Mech. Anal, 2021. https://doi.org/10.1007/s00205-021-01634-7). In a different but related direction, we also show that the cubic nonlinear Schrödinger equation admits no orthonormal ground state in 1D, for more than one function.

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