Stochastic Quantization of Brownian Particle Motion Obeying Kramers Equation

概要

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The stochastic quantization problem of Brownian particle motion that follows Kramers equation is investigated in one-dimension using invariant related unitary transformation method. The explicit Hamiltonians for Langevin equation are constructed separately for the systems that have several different types of the external forces. In the case which does not have external forces, the spectrum of Schrödinger solution is continuous since the particles are unbound in a finite region. However, when a external force is given by the (time-dependent) harmonic potential, the particles are bound inside the potential, leading the spectrum of quantum wave functions to be discrete. The bound–unbound transitions are important since it is connected to metal–insulator transitions that can be achievable for certain compound semiconductors by increasing the doping concentrations at low temperatures. Additionally, when the Hamiltonian involves a higher order term of $x$ as well as harmonic potential term, we have executed the ordinary perturbation expansion in order to obtain the approximate quantum solutions.
2010-06-15