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

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

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 $\hslash =1$, and your discrete time and space grid should be defined using:

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

Take the incident wave packet to be of the form

$\mathrm{\Psi }(x,t=0)=\frac{1}{(2\pi {)}^{1/4}\sqrt{\sigma }}{e}^{-(x-x_0{)}^2/4{\sigma }^2}{e}^{i{k}_0(x-x_0)}$

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]