Project 5

Propagation of Quantum Wave Packets in One and Two Dimensions

Introduction

This project explores tunneling of quantum wave packets in one dimension through a single potential barrier or a double barrier structure where resonant tunneling can be observed at special energies. In quasi-one-dimensional wire with the spin-orbit coupling one can observe spin precession and spin decoherence studied in current research on spintronics [1].

Part I for both PHYS460 and PHYS660 students: Quantum tunneling of spinless wave packets

By numerically solving the time-dependent Schrödinger equation

$-{\frac {\hbar ^{2}}{2m}}{\frac {\partial ^{2}\Psi }{\partial x^{2}}}+V(x)\Psi (x)=i\hbar {\frac {\partial \Psi }{\partial t}}$

via the Crank-Nicholson algorithm for partial-differential equations in one spatial dimension, study reflection and transmission of a quantum wave packet from a barrier for which the potential energy is greater than the kinetic energy of the incident wave packet. For computer simulation the units should be chosen as $m=1$ and $\hbar =1$, and your discrete time and space grid should be defined using:

$\Delta t=5\times 10^{-7};\ \Delta x=5\times 10^{-4}$.

Take the incident wave packet to be of the form

$\Psi (x,t=0)={\frac {1}{(2\pi )^{1/4}{\sqrt {\sigma }}}}e^{-(x-x_{0})^{2}/4\sigma ^{2}}e^{ik_{0}(x-x_{0})}$

with parameters

$\sigma ^{2}=0.00025;\ k_{0}=700$; \ x_0=0.3 </math>.

Let the height of the potential barrier be $V_{0}=2k_{0}^{2}$, the center of the wave packet is at $x_{0}$ and the barrier itself itself starts at $x_{b}=0.6$. Plot the time evolution of the wave packet for $x<1.0$.

References

• [1] D. D. Awschalom and M. E. Flatté, Challenges for semiconductor spintronics, Nature Physics 3, 153 (2007).
• [2] B. K. Nikolić, L. P. Zarbo, and S. Welack, Transverse spin-orbit force in the spin Hall effect in ballistic quantum wires, Phys. Rev. B 72, 075335 (2005).[PDF]
• [3] B. K. Nikolić, L. P. Zarbo, and S. Souma, Spin currents in semiconductor nanostructures: A nonequilibrium Green function approach, Chapter 24, page 814-866 in Volume I of The Oxford Handbook of Nanoscience and Technology: Frontiers and Advances, edited by A. V. Narlikar and Y. Y. Fu (Oxford University Press, Oxford, 2010); also available as arXiv:0907.4122.
• [4] S. A. Crooker and D. L. Smith, Imaging spin flows in semiconductors subject to electric, magnetic, and strain fields, Phys. Rev. Lett. 94, 236601 (2005). [PDF]