Thermal Schrödinger Equation: Efficient Tool for Simulation of Many-Body Quantum Dynamics at Finite Temperature

2017 ◽  
Vol 529 (12) ◽  
pp. 1700200 ◽  
Author(s):  
Maxim F. Gelin ◽  
Raffaele Borrelli
Keyword(s):  
Schrödinger Equation ◽  
Finite Temperature ◽  
Quantum Dynamics ◽  
Schrodinger Equation ◽  
Many Body ◽  
Efficient Tool ◽  
Many Body Quantum Dynamics

QUANTUM DYNAMICS OF WAVE PACKET IN HARMONIC POTENTIAL

2005 ◽  
Vol 19 (24) ◽  
pp. 3745-3754
Author(s):  
ZHAN-NING HU ◽  
CHANG SUB KIM
Keyword(s):  
Schrödinger Equation ◽  
Wave Packet ◽  
Quantum Dynamics ◽  
Schrodinger Equation ◽  
Analytic Solution ◽  
Nonlinear Differential Equations ◽  
Time Dependent ◽  
Dimensional System ◽  
Two Dimensional ◽  
Vibrational States

In this paper, the analytic solution of the time-dependent Schrödinger equation is obtained for the wave packet in two-dimensional oscillator potential. The quantum dynamics of the wave packet is investigated based on this analytic solution. To our knowledge, this is the first time we solve, analytically and exactly this kind of time-dependent Schrödinger equation in a two-dimensional system, in which the Gaussian parameters satisfy the coupled nonlinear differential equations. The coherent states and their rotations of the system are discussed in detail. We find also that this analytic solution includes four kinds of modes of the evolutions for the wave packets: rigid, rotational, vibrational states and a combination of the rotation and vibration without spreading.

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.

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