Real Time Path Integral Simulation Methods for Quantum Particles
|Tila||Julkaistu - 29 marraskuuta 2019|
|Nimi||Tampere University Dissertations|
Quantum mechanics represents our current best knowledge of how Nature works. It is especially important for the electronic structure calculations, where classical mechanics breaks down. Even though the theory is well known, only handful of problems can be solved exactly, so approximations and numerical methods are required. Feynman path integral approach offers an intuitively welcome description of nonrelativistic quantum mechanics, rooted in space and time, where quantum many body effects are included transparently. The formalism based on multidimensional integrals naturally calls for the powerful Monte Carlo techniques to be applied when numerical calculations are performed. In this thesis we present a new approach how real time path integral formalism can be applied to simulate dynamics and states of quantum particles, even electrons. We ﬁrst give a brief introduction to the theory of path integrals and Monte Carlo simulations. Much of this theory can be found in textbooks of quantum mechanics but it is included here for completeness, so that this work could serve as a self-contained introduction to anyone interested in path integral simulations. Second, we discuss the imaginary time methods which have proven to be successful in simulations of statistical physics description of the quantum many-particle systems. Finally, we delve into the challenges of the path integrals in real time domain and present the novel methods from the four original papers with demonstrations. The challenges associated with the real time path integral methods are discussed and we present approaches for solving some of these, such as the wave function guided sampling and "widening" of walkers to improve the propagator. We also introduce a novel method for ﬁnding stationary eigenstates of quantum systems, called "incoherent propagation". This approach can be used to ﬁnd the excited states, unlike the conventional Quantum Monte Carlo methods. Presented techniques are then applied to the Hooke’s atom, a system of strong correlation, that is challenging for conventional approaches. Simulations of the ground state and lowest excited
states of Hooke’s atom give excellent results for energetics. We also demonstrate simulation of coherent quantum dynamics at the presence of an external transient electric ﬁeld. We also introduce how conventional diffusion Monte Carlo (DMC) method can be combined with incoherent propagation.
Not having a positive sampling distribution is a problem that plagues real time
path integral calculations. To alleviate this problem, we introduce a novel proba-
bilistic interpretation of the real time propagator and an approach called "real time diffusion Monte Carlo method". This method is demonstrated in simulation of the time evolution of one dimensional harmonic oscillator and in ﬁnding eigenstates of the system using it in conjunction with incoherent propagation.