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Journal of Physics A Mathematical and General

Stochastic simulation of quantum mechanics

Simon M. Catterall, Ian T. Drummond, R.R. Horgan

Published Sep 7, 1991• 1 citation
Publisher

Abstract

Langevin methods have proved useful in simulating lattice field theories of many kinds. These methods are essentially a practical realization of the stochastic quantization approach to quantum mechanical systems. The authors wish to show that the Langevin method is effective in calculations of non-relativistic quantum mechanical systems. For a one-dimensional system a more conventional approach through the diagonalization of the Hamiltonian matrix using the method of Sturm sequencing is more efficient. However, in two and higher dimensions where the Hamiltonian is no longer tridiagonal the Langevin scheme is more efficient and accurate. It follows therefore that Langevin methods may actually have a great deal to offer in quantum mechanical problems in high dimensions especially in cases where the potential does not have spherical symmetry.

DOI
10.1088/0305-4470/24/17/025

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Langevin methods have proved useful in simulating lattice field theories of many kinds. These methods are essentially a practical realization of the stochastic quantization approach to quantum mechanical systems. The authors wish to show that the Langevin method is effective in calculations of non-relativistic quantum mechanical systems. For a one-dimensional system a more conventional approach through the diagonalization of the Hamiltonian matrix using the method of Sturm sequencing is more efficient. However, in two and higher dimensions where the Hamiltonian is no longer tridiagonal the Langevin scheme is more efficient and accurate. It follows therefore that Langevin methods may actually have a great deal to offer in quantum mechanical problems in high dimensions especially in cases where the potential does not have spherical symmetry.
We present lattice simulations of nonequilibrium quantum fields in Minkowskian space-time. Starting from a nonthermal initial state, the real-time quantum ensemble in (3 + 1) dimensions is constructed by a stochastic process in an additional (5th) "Langevin-time." For the example of a self-interacting scalar field, we show how to resolve apparent unstable Langevin dynamics and compare our quantum results with those obtained in classical field theory. Such a direct simulation method is crucial for our understanding of collision experiments of heavy nuclei or other nonequilibrium phenomena in strongly coupled quantum many-body systems.
The stochastic quantization method is developed in a possible “Heisenberg” operator theory of stochastic processes so designed as to keep a formal analogy to quantum mechanics. The general theory is first formulated for stochastic processes of the Wiener type and then its application to quantization of boson fields is presented. It seems that the operator formalism is convenient to examine implications of the stochastic quantization method. Within the theoretical framework we develop the “Feynman-Dyson” approach to field propagators in perturbation theory. This approach enables us to simplify and systematize practical calculations in comparison with the Langevin equation method. In the Appendix we formulate the Green function approach to field propagators using the characteristic functional.
Previous years researchers began to simulate open quantum system, taking into account the interaction between system and the environment. One approach to deal with this problem is to use the density matrix within the Liouville-von-Neumann formalism or the Markovian variant the Lindblad equations. Another way is to use a stochastic approach where a random force is added to the system. The benefit of the stochastic approach is to solve the dynamics of the system with less time and memory than the density matrix approaches. In this project we want to develop a stochastic approach that can deal with the stochastic wave functions approach. We did this on a 2-level system and found that it works well when comparing to a density matrix approach. Next, we tested a quantum particle connect to a bath of harmonic oscillators using the stochastic approach. We found that a friction term is necessary and applied it. Like in the classical Langevin equations the friction constant and the random force fluctuations are related by the fluctuation-dissipation constant. We showed that with friction the dynamics decays to an ensemble with energy of Egs+kBTE_{gs}+k_BTEgs​+kB​T. However, we also found here are problems. The system seems to absorb energy indefinitely if the temperature is higher than the zero point energy or if the system is a Morse oscillator. Thus more research is required to make this method work.
Algorithms are described for efficiently simulating quantum mechanical systems on quantum computers. A class of algorithms for simulating the Schrödinger equation for interacting many-body systems are presented in some detail. These algorithms would make it possible to simulate nonrelativistic quantum systems on a quantum computer with an exponential speedup compared to simulations on classical computers. Issues involved in simulating relativistic systems of Dirac or gauge particles are discussed.
One of the useful and practical methods for solving quantum-mechanical many- body systems is to recast the full problem into a form of the effective interaction acting within a model space of tractable size. Many of the effective-interaction theories in nuclear physics have been formulated by use of the so called box introduced by Kuo et. al. It has been one of the central problems how to calculate the box accurately' and efficiently. Introducing new basis states, the Hamiltonian is transformed to a block-tridiagonal form in terms of submatrices with small dimension. With the transformed Hamiltonian we derive a recursion method of calculating the box non-perturbatively and analytically. The box given in this study corresponds to a non-perturbative solution for the energy-dependent. Hamiltonian interaction which is often referred to as the Bloch-Horowitz or the Feshbach form. The present approach has a possibility of resolving many of the theoretical and practical difficulties encountered in the calculation of effective-interaction and/or Hamiltonian.