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American Journal of Physics

Transmission resonances and Bloch states for a periodic array of delta function potentials

P. R. Berman

Published Feb 19, 2013• 11 citations
Publisher

Abstract

The relationship between transmission resonances and Bloch states for a periodic array consisting of N delta function potentials is discussed. The transmission resonances are derived for matter waves incident on the periodic array, while the Bloch states are calculated using periodic boundary conditions for the array. It is shown that approximately half of the transmission resonances map into pairs of degenerate Bloch states. Wave functions are shown for both the transmission resonances and the Bloch states for arrays of five and six delta function potentials. The origin of the band structure of the Bloch states is interpreted in terms of the wave functions and eigenenergies for a particle confined to move on a ring, subjected to a periodic array of delta function potentials on the ring.

DOI
10.1119/1.4769113

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The relationship between transmission resonances and Bloch states for a periodic array consisting of N delta function potentials is discussed. The transmission resonances are derived for matter waves incident on the periodic array, while the Bloch states are calculated using periodic boundary conditions for the array. It is shown that approximately half of the transmission resonances map into pairs of degenerate Bloch states. Wave functions are shown for both the transmission resonances and the Bloch states for arrays of five and six delta function potentials. The origin of the band structure of the Bloch states is interpreted in terms of the wave functions and eigenenergies for a particle confined to move on a ring, subjected to a periodic array of delta function potentials on the ring.
The problem of one-dimensional quantum wire along which a moving particle interacts with a linear array of N delta-function potentials is studied. Using a quantum waveguide approach, the transfer matrix is calculated to obtain the transmission probability of the particle. Results for arbitrary N and for specific regular arrays are presented. Some particular symmetries and invariances of the delta-function potential array for the N = 2 case are analyzed in detail. It is shown that perfect transmission can take place in a variety of situations.
The quantum description of a free particle subject to a delta circular ring and a delta spherical shell is studied. We determine the bound eigenstates and energy spectra for different angular momentum states. The normalization constants of the bound states and the transmission and reflection coefficients of the scattering states are calculated in closed form. The ground state of the delta ring is always present regardless of the potential strength. In contrast, the delta shell requires a threshold of the delta strength for the first eigenstate to appear. For the ring and the shell, the wave functions tend to be degenerate at high energies, and the effect of angular momentum on the wave functions becomes negligible. This work generalizes the standard problem of the one-dimensional delta potential studied in textbooks.
Abstract We introduce a new exactly solvable model in quantum mechanics that describes the propagation of particles through a potential field created by regularly spaced δ ′ -type point interactions, which model the localized dipoles often observed in crystal structures. We refer to the corresponding potentials as δ θ ′ -combs, where the parameter θ represents the contrast of the resonant wave at zero energy and determines the interface conditions in the Hamiltonians. We explicitly calculate the scattering matrix for these systems and prove that the transmission probability exhibits sharp resonance peaks while rapidly decaying at other frequencies. Consequently, Hamiltonians with δ θ ′ -comb potentials act as quantum filters, permitting tunneling only for specific wave frequencies. Furthermore, for each θ > 0, we construct a family of regularized Hamiltonians approximating the ideal model and prove that their transmission probabilities have a similar structure, thereby confirming the physical realizability of the band-pass filtering effect.
In this paper, we calculate the transmission and reflection amplitudes of wave functions for different potentials such as the delta function, the rectangular barrier, the Eckart potential, and the Hulthen potential. We describe the relationship between these amplitudes and compute the reflection resonances between each potential. We describe the transmission and reflection probabilities using the WKB formula and compare the results with ones obtained from matching the boundary conditions. Furthermore, we use a two by two transfer matrix to calculate a rigorous bound on the transmission and reflection probabilities.
We have solved exactly the two-component Dirac equation in the presence of a spatially one-dimensional Hulthén potential, and presented the Dirac spinors of scattering states in terms of hypergeometric functions. We have calculated the reflection and transmission coefficients by the matching conditions on the wavefunctions, and investigated the condition for the existence of transmission resonances. Furthermore, we have demonstrated how the transmission-resonance condition depends on the shape of the potential.