Applications in Quantum Mechanics

Theory of the electronic and transport properties of graphene under a periodic electric or magnetic field

Using the formalism of supersymmetric quantum mechanics, we give an exact solution for a family of one-dimensional periodic potentials, which are the supersymmetric partners of the potential proportional to the trigonometric function cos(2x) such that the Schrodinger equation for this potential is named the Mathieu equation mathematically. We show that the new potentials are distinctly different from their original ones. However, both have the same energy band structure. All the potentials obtained in this paper are free of singularities.

Chapter 1 - Electrons in One-Dimensional Periodic Potentials

Minigap plasmons in a two-dimensional electron gas subjected to a one-dimensional periodic potential

It has been observed that electron in a perfect crystal moves in a spatially periodic field of force due to the ions and the averaged effect of all the electrons. This work shows the investigative work done to determine the energy band structure of an electron in a one-dimensional periodic potential. The application of the Kronig-Penney model was applied to an electron state in a delta-like potential. To fully understand the Kronig-Penney model, the concept of Bloch’s theorem was first introduced to describe the conduction of electrons in solids. It has been found that the periodic potential introduces gaps in the reduced representation with an increasing number of potential well/barrier strengths. It has been observed that the regions of non-propagating states, which give rise to energy band gaps, become larger with decreasing values.

The effective "miniband" mass approximation is shown to break down in strong magnetic fields when the magnetic length is smaller than the superlattice period. Under these conditions a real-space description must be used and features in the inhomogeneously broadened cyclotron resonance can be indexed by the position of the cyclotron orbit with respect to the barriers and wells. The line shape agrees with a calculation of the quantum states in a magnetic field in a one-dimensional periodic potential and these experiments give us a rare view of Landau states in the limit where the periodic potential has a period that is larger than the cyclotron diameter.

We have theoretically calculated a tunable electron wave filter based on graphene superlattices with one-dimensional periodic potentials. It is found that such structure with defect layer possess an unusual tunneling state occurring inside the forbidden gap, and the tunneling state can be modulated by changing the width of the defect layer. We have also achieved independently tunable multichannel electron wave filters by constructing fractal structures. The modulation of the transmission gap by width, period number, and incidence angle may lead to potential applications in graphene-based electronic devices.

Effective diffusion coefficient of a Brownian particle in a periodic potential

We study the diffusion of a quantum Brownian particle in a one-dimensional periodic potential with substitutional disorder. The particle is coupled to a dissipative environment, which induces a frictional force proportional to the velocity. The dynamics for arbitrary temperature is studied by using Feynman's influence-functional theory. We calculate the mobility to lowest order in the disorder and strength of the periodic potential. It is shown that for weak dissipation the linear mobility, which vanishes atT=0 due to localization effects, may exhibit a maximum and a subsequent minimum with increasing temperature. The relation to the diffusion of heavy particles in metals or doped semiconductors is briefly discussed.

The diffusive motion of particles in a one-dimensional periodic potential is treated using the Fokker-Planck equation method. First, a concise form of the Fokker-Planck equation and of the correlation function for this problem is set up. By expanding the distribution function into suitable eigenfunctions, a general method for calculating the correlation functions is then given. Finally, explicit calculations are presented for the velocity correlation functions, some of these are compared with those which were obtained by continued fraction methods.

The Weierstrass elliptic function expansion method and its applications in nonlinear wave equations

In recent years, there has been an increase in the application of quantum mechanics (QM) methods to describe properties related to the ADMET profile of small molecules. The application of these methods allows calculating useful descriptors and physiochemical properties contributing to ADMET prediction. Considering that QM methods are the only one that describe the electronic state of a molecules, such methods are particularly useful for studying the metabolism of drugs; furthermore, the introduction of mixed QM and molecular mechanics (QM/MM) is also increasing the understanding of drug interaction with cytochromes from a mechanistic point of view. Finally, combining the increase number of experimental data with machine learning algorithms and QM-derived descriptors allowed the creation of an end-user software capable of affecting the drug discovery process.

Chaos-like properties such as noise or fluctuations arising from quantum effects have recently investigated on a single barrier potential and on a one-dimensional periodic potential barrier. They used either the planewave of light or a wavepacket (pulse) passing through the potential barrier, demonstrating the chaos created by the bounded one-dimensional multibarrier potential (Bar and Horwitz, 2002; Hondou and Sawada, 1995), and proved it is an ordered and periodic phenomenon. Also another group demonstrated unexpected behavior of a dissipative particle in simple multiscale systems subject to chaotic noise and clarified the reason for the particular behavior because it is under a periodic potential (Chialvo et al., 1997). They concluded the occurrence of drift in a symmetric periodic potential would be expected to be a common feature of noise-driven systems. Essentially this effect is quite similar to the problem of a particle placed in a potential well of finite barrier (Cohen-Tannoudji et al., 1977) and the present issue of the photon distribution on a crystallite surface. When the crystallite size decreases to a few nanometers, the light absorption or photoluminescence in air moves to the direction of smaller wavelength, which is known as the blueshift (Kale and Lokhande, 2000; Nanda et al., 1999; Tsunekawa et al., 2003; Von Behren et al., 2000). The basic theory for this phenomenon is the quantum confinement effect (Andersen et al., 2002; Sharma et al., 2005). According to this theory the number of photons confined in each unit cell increases with the decreasing crystallite size. It is also said to enlarge the band gap between the valence band and the conduction band. Another approach is to take crystalline unit cells as numerical elements and employ different potential energies over a unit cell (Choi, 2007; Choi and Kim, 2007), realizing light intensity fluctuations. The effect of the film thickness on the blueshift was investigated by one of the authors (Choi and Pyun, 2008) about cellular crystalline surfaces with periodic potential, getting the blueshift numerically demonstrated in good agreement with experiments. In this paper we deal with surface nanostructures and try to prove the blueshift numerically by predicting the location of absorption wavelength. One of the best ways to uphold the present numerical method would be to demonstrate the blueshift phenomenon observed widely experimentally (Lu et al., 2008; Miyake et al., 1999; Tan et al., 2005; Tsunekawa et al., 2000). They demonstrated experimentally the dependence of the blueshift on the crystallite size for CeO and CdS. One of the authors has been studying this problem recently, publishing a few papers (Choi, 2007; Choi and Kim, 2007; Choi and Pyun, 2008), and this paper may be taken as an extension of these consecutive efforts. Another publication (Choi and Choi, 2009) predicting the blueshift with respect to the shell layer thickness has been already accomplished and led the present investigation to be made. 17

We demonstrate that particles confined to two dimensions (2D) and subjected to a one-dimensional (1D) periodic potential exhibit a rich phase diagram, with both ``locked floating solids'' and smectic phases. The resulting phases and phase transitions are studied as a function of temperature and potential strength. We find reentrant melting as a function of the potential strength. Our results lead to universal predictions consistent with recent experiments on 2D colloids in the presence of a laser-induced 1D periodic potential.

CONCEPTUAL FOUNDATIONS OF QUANTUM MECHANICS: THE ROLE OF EVIDENCE THEORY, QUANTUM SETS, AND MODAL LOGIC