Scalar particles in a narrow-band periodic potential

P. C. W. Davies, David S. Betts

Boris I. Ivlev

Quantum tunneling between two potential wells in a magnetic field can be strongly increased when the potential barrier varies in the direction perpendicular to the line connecting the two wells and remains constant along this line. A periodic structure of the wave function is formed in the direction joining the wells. The resulting motion can be coherent like motion in a conventional narrow band periodic structure. A particle penetrates the barrier over a long distance which strongly contrasts to WKB-like tunneling. The whole problem is stationary. The coherent process can be influenced by dissipation.

Wlodek Zawadzki

Herein, relations between particle and wave properties for charge carriers in periodic potentials of crystalline metals and semiconductors are derived. The particle aspects of electrons in periodic potentials are considered using properties of quasimomentum (QM), while the wave aspects are described using wave packets of Bloch waves. The two aspects are combined in the derivation of QM–wavelength relations for energy bands of arbitrary nonparabolicity and nonsphericity. An effective mass relating electron QM to its average velocity for spherical energy bands is defined and used to calculate energy dependences of wavelengths for electrons in narrow‐gap semiconductors, graphene, and surface states of topological insulators. An uncertainty relation between electron QM and its spatial coordinate in periodic potentials is derived. It is emphasized that the described properties apply to the average (not instantaneous) electron behavior. Analogies and differences between the wave and uncertainty properties of electrons in crystalline solids and those in vacuum are traced. An interference experiment with semiconductor nanostructures is proposed to verify the theoretical predictions and demonstrate to what extent the wave properties of electrons in solids are distinct from those in vacuum.

D. Baye, M. Dufour, Benjamin Fuks E

Carl M. Bender, Dorje C. Brody, Daniel W. Hook

On the basis of extensive numerical studies it is argued that there are strong analogies between the probabilistic behavior of quantum systems defined by Hermitian Hamiltonians and the deterministic behavior of classical mechanical systems extended into the complex domain. Three models are examined: the quartic double-well potential V ( x ) = x 4 − 5 x 2 , the cubic potential , and the periodic potential V ( x ) = −cos x . For the quartic potential a wave packet that is initially localized in one side of the double-well can tunnel to the other side. Complex solutions to the classical equations of motion exhibit a remarkably analogous behavior. Furthermore, classical solutions come in two varieties, which resemble the even-parity and odd-parity quantum-mechanical bound states. For the cubic potential, a quantum wave packet that is initially in the quadratic portion of the potential near the origin will tunnel through the barrier and give rise to a probability current that flows out to infinity. The complex solutions to the corresponding classical equations of motion exhibit strongly analogous behavior. For the periodic potential a quantum particle whose energy lies between −1 and 1 can tunnel repeatedly between adjacent classically allowed regions and thus execute a localized random walk as it hops from region to region. Moreover, if the energy of the quantum particle lies in a conduction band, then the particle delocalizes and drifts freely through the periodic potential. A classical particle having complex energy executes a qualitatively analogous local random walk, and there exists a narrow energy band for which the classical particle becomes delocalized and moves freely through the potential.

Zafar Ahmed