Wave functions of different widths were used and then the results were analyzed next to each other. As is evident from the graphs below, the wider the wave function, the more accurate the results. The wave functions below are representative of this fact and by varying the parameters it is possible to maintain an accurate picture of very localized wave functions. =.0002, NANO-CURRENTS WITHOUT A VOLTAGE VIA QUANTUM PUMPING: A WAVE PACKET APPROACH Matthew Galligan, Kunal K. Das Department of Physics, Fordham University, Bronx NY BackgroundMethod One technique used to expedite the process was to only propagate the wave packets that would actually interact with the potential barrier, which starts at x 0 =0 and +v 0. For example, the wave packet with –x 0 and –k 0 would never interact with the barrier while the packet with +x 0 and –k 0 would always interact with the potential. The wave packet with –x 0 and +k would only interact if k>v. The wave packet with +x and +k would interact only when k<v. The case where k=v was done separately. Goals Study the transport of single electrons through a quantum pump operatiing at arbitrary rates or velocities. Probe the equivalence of localized states and extended states representation in the presence of time varying potentials Reasons for Interest Current analytical formulas for the propagation of an electron in the Quantum regime through a potential barrier assumes many electrons and the position of any one individual electron is unimportant, but rather their behavior as a group is studied. As a result, description of current is always in terms of extended states (plane wave representation) of the electrons, but when the current can comprise of one electron at a time through finite-size structures, a localized picture of individual electrons seems more appropriate, particularly in the presence of time varying potentials. In this study we probe the limits of validity of those analytical formulas by modeling electrons with localized packets of varying widths instead of extended plane wave state, and look at the differences. Model Construct four Gaussian wave packets that each represents a single electron, one with x 0 and k 0, x 0 and -k 0, -x 0 and k 0, and –x 0 and –k 0. Model a simple time-varying potential by a moving quantum potential barrier initially at the origin with v 0 =0. Use the split-step operator method to model the electron propagation and interaction with the barrier as it moves Determine the current using: Repeat the process with the potential barrier back at the origin and with a different velocity of the barrier till a maximum desired value Current in Nano Structures In large scale electrical circuits, current is measured as an average of large number of collisions of electrons in the wires. This measure only works if there are enough electrons to define an average that is unchanging. In extremely small wires, the electron no longer “bounce” around in the wire but rather moves ballistically. At this point the classical approach to current no longer is valid and quantum affects become noticeable. The drive to make computers and other electronic devices smaller and smaller is leading to nanoscale (~ meters) circuit elements. It has become become necessary to understand, and better control, the way current flows in these regimes. Quantum pumps offer a potentially better alternative to generating currents in nano structures without applying a voltage bias. Conclusions 1.Confirmed that it is possible to simulate an extended plane wave with a sufficiently broad wave packet. 2.Now have the ability to model the entire regime including non-adiabatic regime. 3.Can apply these models to any arbitrarily time varying potential. 1.Use a Fast Fourier transform (FFT) to convert the position space representation of the wave- packet into a momentum-space representation 2.Time-evolve the wave function by the kinetic part of the Hamiltonian for half-time-step 3.Use the inverse FFT to recover the position space wave packet. 4.Time-evolve the wave function by the potential energy part of the Hamiltonian for the full- time-step: 5.Use the forward FFT to get the momentum-space representation. 6.Time-evolve the wave function by the kinetic part of the Hamiltonian for the remaining half- time-step 7.Reverse FFT to recover the position-space representation of the wave-packet. 8.This entire process only propagates the wave packet through one time step and as a result, is repeated many times. Quantum Pumping Quantum pumping creates a current without a voltage bias. By creating a local potential that varies in time cyclically (like a pump) it is possible to induce a current in the nano structure without needing to create a voltage bias. This is useful because it is posible to have more control over the flow of the electrons. Also quantum pumping can use much less energy than creating a bias. Nano-technology will eventually use some form of quantum pumps as computers and electronic devices shrink in size. Split-Step Operator The split-step operator method is a tool to propagate a quantum mechanical wave-packet by stepping it through a sequence of extremely small time steps, in this case.002 seconds. This is done many times to simulate the path of the wave-packet over a longer time period. Time evolution of a wave function over a time period is through the formula: where The split step operator breaks the formula into two parts, doing the kinetic energy part solely in momentum space where the operator is a simple multiplication instead of involving derivatives and the potential energy part in position space for the same reason. A Fourier Transform is capable of moving between momentum space and position space, and in this case a Fast Fourier Transform (FFT) was used. The straight line is the adiabatic approximation and the curved line is the actual results for a delta function The solid line is the analytical solution using plane waves the circles are the model with only two wave packets and the crosses are the model with all four wave packets using a square barrier. The 4-wavepacket model agrees with the plane wave model exactly. Square barriers can be used to model delta functions as when the square barrier is much thinner than the wave packet and the area under the square barrier is the same as that under the delta function Limits The numerical method used showed very good agreement with the analytical values for plane waves when the wave packet was very wide. As a result different widths of wave packet were used to test the limit of this method. For all of them we choose a Gaussian profile with initial amplitude H  Hamiltonian T  Kinetic term V  Potential term x0x0 - x 0 -k 0 k0k0 k0k0 v0v0