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### Abstract

The quantum mechanical transmission probability is calculated for one-dimensional finite lattices with three types of potentials: periodic, quasiperiodic, and random. When the number of lattice sites included in the computation is systematically increased, distinct features in the transmission probability vs. energy diagrams are observed for each case. The periodic lattice gives rise to allowed and forbidden transmission regions that correspond to the energy band structure of the infinitely periodic potential. In contrast, the transmission probability diagrams for both quasiperiodic and random lattices show the absence of well-defined band structures and the appearance of wave localization effects. Using the average transmissivity concept, we show the emergence of exponential (Anderson) and power-law bounded localization for the random and quasiperiodic lattices, respectively.

This work was supported by CONACYT through grants Repatriados-2009 (S-4130) and Fondos Sectoriales-SEP-2009 (S-3158). The author thanks Dr. Paul Riley, Dr. Kirk Madison, and anonymous reviewers for a careful reading of the manuscript.

I. INTRODUCTION

II. STATEMENT OF THE PROBLEM

III. TRANSMISSION IN A PERIODIC LATTICE

IV. TRANSMISSION IN A QUASIPERIODIC LATTICE

V. TRANSMISSION IN A RANDOM LATTICE

VI. WAVE LOCALIZATION

VII. CONCLUSIONS AND FINAL REMARKS

### Key Topics

- Transmission coefficient
- 17.0
- Localization effects
- 13.0
- Band structure
- 11.0
- Wave propagation
- 9.0
- Mechanical waves
- 7.0

## Figures

Schematic of the generalized impedance method used to compute quantum transmission over a finite potential barrier. (a) An electrical transmission line of characteristic impedance is terminated with a load impedance . A wave of amplitude travelling to the right will undergo reflection at the connection point. The amplitude of the reflected wave is given by Eq. (4) . (b) In the quantum mechanical problem of transmission across a potential barrier, the generalized impedance method regards the potential (thick solid line) as the load impedance . To compute transmission, the potential barrier is approximated by *N* segments of equal width (thin bars, ). Each segment ( ) has a characteristic impedance ( ). Starting with the load impedance corresponding to the *N*th segment ( ), the input impedance of the *N*th segment ( ) is computed using Eq. (7) , which in turn is taken as the load impedance of segment *N* − 1 ( ). Then, an iterative procedure is followed until the load impedance of the whole potential barrier ( ) is found. Finally, the quantum reflection coefficient can be found by substituting and the characteristic impedance into Eq. (4) . See text for details.

Schematic of the generalized impedance method used to compute quantum transmission over a finite potential barrier. (a) An electrical transmission line of characteristic impedance is terminated with a load impedance . A wave of amplitude travelling to the right will undergo reflection at the connection point. The amplitude of the reflected wave is given by Eq. (4) . (b) In the quantum mechanical problem of transmission across a potential barrier, the generalized impedance method regards the potential (thick solid line) as the load impedance . To compute transmission, the potential barrier is approximated by *N* segments of equal width (thin bars, ). Each segment ( ) has a characteristic impedance ( ). Starting with the load impedance corresponding to the *N*th segment ( ), the input impedance of the *N*th segment ( ) is computed using Eq. (7) , which in turn is taken as the load impedance of segment *N* − 1 ( ). Then, an iterative procedure is followed until the load impedance of the whole potential barrier ( ) is found. Finally, the quantum reflection coefficient can be found by substituting and the characteristic impedance into Eq. (4) . See text for details.

Quantum transmission in a finite, periodic lattice. (a) The finite, periodic lattice considered, consisting of *M* = 25 lattice sites in this case. (b) The transmission coefficient *T* as a function of the dimensionless parameter *q*, for an increasing number of lattice sites. See text for discussion of results. To compute quantum transmission, each lattice site was divided into *N* = 100 segments, and 3000 energy points were computed for each graph.

Quantum transmission in a finite, periodic lattice. (a) The finite, periodic lattice considered, consisting of *M* = 25 lattice sites in this case. (b) The transmission coefficient *T* as a function of the dimensionless parameter *q*, for an increasing number of lattice sites. See text for discussion of results. To compute quantum transmission, each lattice site was divided into *N* = 100 segments, and 3000 energy points were computed for each graph.

Energy band diagram corresponding to an infinite, periodic potential. The characteristic values of the Mathieu equation (*q*, thick solid lines) are displayed as a function of the quasimomentum (*k*) in the reduced-zone scheme. The resulting energy levels define the familiar band structure. See text for details.

Energy band diagram corresponding to an infinite, periodic potential. The characteristic values of the Mathieu equation (*q*, thick solid lines) are displayed as a function of the quasimomentum (*k*) in the reduced-zone scheme. The resulting energy levels define the familiar band structure. See text for details.

Comparison of quantum transmission for finite and infinite periodic lattices. The quantum transmission coefficient *T* is displayed as a function of the normalized energy *q* for finite (black, solid line) and infinite (gray, dashed line) periodic potentials. The finite potential consisted of *M* = 30 lattice sites, with each lattice site divided into 100 segments, and 3000 energy points were computed within the interval shown; the infinite potential result was derived from the energy band diagram of Fig. 3 .

Comparison of quantum transmission for finite and infinite periodic lattices. The quantum transmission coefficient *T* is displayed as a function of the normalized energy *q* for finite (black, solid line) and infinite (gray, dashed line) periodic potentials. The finite potential consisted of *M* = 30 lattice sites, with each lattice site divided into 100 segments, and 3000 energy points were computed within the interval shown; the infinite potential result was derived from the energy band diagram of Fig. 3 .

Quantum transmission in a finite, quasiperiodic lattice. (a) The finite, quasiperiodic lattice considered, consisting of the first *M* = 25 lattice sites. (b) The transmission coefficient *T* as a function of the dimensionless parameter *q*, for an increasing number of lattice sites. See text for discussion of results. To compute quantum transmission, each lattice site was divided into 100 segments, and 3000 energy points were computed for each graph.

Quantum transmission in a finite, quasiperiodic lattice. (a) The finite, quasiperiodic lattice considered, consisting of the first *M* = 25 lattice sites. (b) The transmission coefficient *T* as a function of the dimensionless parameter *q*, for an increasing number of lattice sites. See text for discussion of results. To compute quantum transmission, each lattice site was divided into 100 segments, and 3000 energy points were computed for each graph.

Distribution of random numbers used to generate the random potential. To generate the set of pseudo-random numbers that are used to define the amplitudes of the lattice sites in the random potential, the IgorPro command SetRandomSeed was set to 0.038 and the built-in routine enoise() was used. A histogram corresponding to the first 2500 numbers generated shows a uniform distribution.

Distribution of random numbers used to generate the random potential. To generate the set of pseudo-random numbers that are used to define the amplitudes of the lattice sites in the random potential, the IgorPro command SetRandomSeed was set to 0.038 and the built-in routine enoise() was used. A histogram corresponding to the first 2500 numbers generated shows a uniform distribution.

Quantum transmission in a finite, random lattice. (a) The finite, random lattice considered, consisting of the first *M* = 25 lattice sites. (b) The transmission coefficient *T* as a function of the dimensionless parameter *q*, for an increasing number of lattice sites. See text for discussion of results. To compute quantum transmission, each lattice site was divided into 100 segments, and 3000 energy points were computed for each graph.

Quantum transmission in a finite, random lattice. (a) The finite, random lattice considered, consisting of the first *M* = 25 lattice sites. (b) The transmission coefficient *T* as a function of the dimensionless parameter *q*, for an increasing number of lattice sites. See text for discussion of results. To compute quantum transmission, each lattice site was divided into 100 segments, and 3000 energy points were computed for each graph.

Fourier transform of the periodic, quasiperiodic, and random potentials. A clear transition from a single frequency peak to a floor of noise is observed as the degree of disorder is increased. For the quasiperiodic lattice, the ratio *a/b* = *b/c* = … is equal to the golden mean, reflecting the fractal nature of this structure. To compute the corresponding Fourier transforms, a total of *M* = 2500 lattice sites were considered in each case.

Fourier transform of the periodic, quasiperiodic, and random potentials. A clear transition from a single frequency peak to a floor of noise is observed as the degree of disorder is increased. For the quasiperiodic lattice, the ratio *a/b* = *b/c* = … is equal to the golden mean, reflecting the fractal nature of this structure. To compute the corresponding Fourier transforms, a total of *M* = 2500 lattice sites were considered in each case.

Quantum transmission coefficient behavior for the periodic, quasiperiodic, and random finite lattices in the case of large numbers of lattice sites. The periodic potential case shows a clear energy band structure that is well-defined in all cases. In contrast, the quasiperiodic case does display energy band gaps, but they tend to disappear when *M* is increased. Although weak resonances can be appreciated for the random case, there is no evidence of an energy band structure. These characteristics help understand localization effects (see text). To compute quantum transmission, each lattice site was divided into 100 segments, and 3000 energy points were computed for each graph.

Quantum transmission coefficient behavior for the periodic, quasiperiodic, and random finite lattices in the case of large numbers of lattice sites. The periodic potential case shows a clear energy band structure that is well-defined in all cases. In contrast, the quasiperiodic case does display energy band gaps, but they tend to disappear when *M* is increased. Although weak resonances can be appreciated for the random case, there is no evidence of an energy band structure. These characteristics help understand localization effects (see text). To compute quantum transmission, each lattice site was divided into 100 segments, and 3000 energy points were computed for each graph.

(Color online) Emergence of localization effects for the quasiperiodic and random lattices. Transmission coefficients averaged over the energy interval *q* = [1, 2] are shown with *M* varied between 1 and 5000, for the periodic (black, dashed line), quasiperiodic (blue, solid line), and random (red, dashed-point line) finite lattices. In the cases of the quasiperiodic and random potentials the transmissivity decreases monotonically as the number of lattice sites is increased. Decay is consistent with power-law (quasiperiodic potential) and exponential (random potential) dependence, as evidenced by the same results displayed in log-linear (top graph) and log-log (bottom graph) scales. The average transmissivity for the periodic lattice remains constant beyond —identifying a point at which the finite periodic lattice can be regarded as a sufficient approximation to its infinite counterpart. In all cases was used. To compute quantum transmission in all cases, each lattice site was divided into 100 segments, and 4000 equally spaced energy points within the interval were used to determine the average transmissivity.

(Color online) Emergence of localization effects for the quasiperiodic and random lattices. Transmission coefficients averaged over the energy interval *q* = [1, 2] are shown with *M* varied between 1 and 5000, for the periodic (black, dashed line), quasiperiodic (blue, solid line), and random (red, dashed-point line) finite lattices. In the cases of the quasiperiodic and random potentials the transmissivity decreases monotonically as the number of lattice sites is increased. Decay is consistent with power-law (quasiperiodic potential) and exponential (random potential) dependence, as evidenced by the same results displayed in log-linear (top graph) and log-log (bottom graph) scales. The average transmissivity for the periodic lattice remains constant beyond —identifying a point at which the finite periodic lattice can be regarded as a sufficient approximation to its infinite counterpart. In all cases was used. To compute quantum transmission in all cases, each lattice site was divided into 100 segments, and 4000 equally spaced energy points within the interval were used to determine the average transmissivity.

Self-similar resonances in the average transmissivity for the quasiperiodic potential. The average transmissivity (black line) shows a number of resonance peaks located wherever the number of lattice sites is equal to a Fibonacci number (gray vertical lines, only lines corresponding to *M* = 89, 144, 233, 377, 610, 987 are shown). Additionally, smaller peaks can be distinguished between Fibonacci lines. All of the resonances locate at positions that define distances (arrowed, horizontal gray lines) that are related through the self-similarity property characteristic of the Fibonacci sequence.

Self-similar resonances in the average transmissivity for the quasiperiodic potential. The average transmissivity (black line) shows a number of resonance peaks located wherever the number of lattice sites is equal to a Fibonacci number (gray vertical lines, only lines corresponding to *M* = 89, 144, 233, 377, 610, 987 are shown). Additionally, smaller peaks can be distinguished between Fibonacci lines. All of the resonances locate at positions that define distances (arrowed, horizontal gray lines) that are related through the self-similarity property characteristic of the Fibonacci sequence.

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