Abstract
A new numerical method for solving of one-dimensional stationary Schrödinger equation has been developed. The method is based on the Fourier transformation of the wave equation. As a result, an integral equation has been obtained where the integral is replaced by the summation, and problem has been transformed in the eigenvalue/eigenvector problem which corresponds to discrete energy levels as well as the Fourier transform of wave functions. The wave function has been obtained by the usage of the inverse Fourier transformation. It is shown that discrete energy levels are split and form the forbidden and permitted zones for the one-dimensional finite crystal. The all discrete energy levels and corresponding Fourier transform of the wave functions were obtained by one series of computer calculations if the number of energy levels is finite. The method is characterized by high accuracy and stability of search of the discrete levels of energy.
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West LC, Eglash SJ (1985) First observation of an extremely large-dipole infrared transition within the conduction band of a GaAs quantum well. Appl Phys Lett 46(12):1156–1158
Faist J, Capasso F, Hutchnson AL, Pfeiffer L, West KW (1993) Suppression of optical absorption by electric-field induced quantum interference in coupled potential wells. Phys Rev Lett 71(1):3573–3576
Levine BF (1993) Quantum well infrared photodetectors. J Appl Phys 74:R1–R81
Faist J, Capasso P, Sivco DL, Sirtori C, Hutchnson AL, Cho AY (1994) Quantum cascade lasers. Science 264:553–556
Noda S, Yamashita T, Ohya M, Muromoto Y, Sasaki A (1993) All-optical modulation for semiconductor lasers by using three energy levels in n-doped quantum wells. IEEE J Quantum Electron 29(6):1640–1647
Bastard G (1988) Wave mechanics applied to semiconductor heterostructures. Les editions de physique, Les Ulis
Vakarchuk IO (2004) Quantum mechanics. Ivan Franko National University of Lviv, Lviv, (in Ukrainian)
Caticha A (1995) Construction of exactly soluble double-well potentials. Phys Rev A 51(5):4264–4267
Dutra AS (1993) Conditionally exactly soluble class quantum potentials. Phys Rev A 47(4):R2435–R2437
Tkachuk VM, Fityo TV (2003) Multidimensional quasi-exactly solvable potentials with two known eigenstates. Phys Lett A 309(5–6):351–356
Gedeon A (1974) Comparison between rigorous theory and WKB-analysis of modes in graded-index waveguides. Opt Commun 12(3):329–332
Smith RE, Houde-Walter SN, Forbes GW (1990) Mode determination for planar waveguide using the four-sheeted dispersion relation. J. Quantum Electron 26(4):627–630
Anemogiannis E, Glytsis EN (1992) Multilayer waveguides: efficient numerical analysis of general structures. J Lightwave Technol 10(10):1344–1351
Chatak AK, Thyaagarajan K, Shenoy MR (1987) Numerical analysis of planar optical waveguides using matrix approach. J Lightwave Technol 5(5):660–667
Baba T, Kokubun Y (1992) Dispersion radiation loss characteristics of antiresonant reflecting optical waveguides-numerical results and analytical expressions. J Quantum Electron 28(7):1689–1700
Rganov AG, Grigas SE (2010) Numerical algorithm for waveguide and leaky modes determination in multilayer optical waveguides. Tech Phys 55(11):1614–1618
Fitio VM, Romakh VV, Bobitski YV (2014) Numerical method for analysis of waveguide modes in planar gradient waveguides. Material Science (Medžiagotyra) 20(3):256–261
Fitio VM, Romakh VV, Bartkiv LV, Bobitski YV (2016) The accuracy of computation of mode propagation constants for planar gradient waveguides in the frequency domain. Mater Sci Eng Technol (Materialwissenschaft und Werkstofftechnik) 47(2–3):237–245
Meškinis Š, Čiegis A, Vasiliauskas A, Šlapikas K, Gudaitis R, Yaremchuk I, Fitio V, Bobitski Y, Tamulevičius S (2016) Annealing effects on structure and optical properties of diamond-like carbon films containing silver. Nanoscale Res Lett 11:146. doi:10.1186/s11671-016-1362-4
Yaremchuk I, Tamulevicius T, Fitio V, Grazuleviciute I, Bobitski Y, Tamulevicius S (2013) Guide-mode resonance characteristics of periodic structure on base of diamond-like carbon film. Opt Commun 301:1–6
Smirnova N, Sakhno O, Fitio V, Gritsai Y, Stumpe J (2014) Simple and high performance DFB laser based on dye-doped nanocomposite volume gratings. Laser Phys Lett 11(5):125814–1 – 8
Smirnova TN, Sakhno OV, Stumpe J, Fitio VM (2016) Polymer distributed feedback dye laser with an external volume Bragg grating inscribed in a nanocomposite by holographic technique. JOSA B 33(2):202–210
Fitio VM, Yaremchuk IY, Romakh VV, Bobitski YV (2015) A solution of one-dimensional stationary Schrödinger equation by the Fourier transform. Appl Comput Electromagn Soc 30(5):534–539
Goodman JW (1968) Introduction to fourier optics. McGraw-Hill Book Company, San Francisco
Yariv A (1975) Quantum electronics. John Wiley and Sons, New York
Kittel C (1978) Introduction to solid state physics, 4th edn. John Wiley and Sons, Inc., New York
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Fitio, V.M., Yaremchuk, I.Y., Romakh, V.V., Bobitski, Y.V. (2017). Numerical Solution of One-Dimensional Stationary Schrödinger Equation in the Frequency Domain. In: Fesenko, O., Yatsenko, L. (eds) Nanophysics, Nanomaterials, Interface Studies, and Applications . NANO 2016. Springer Proceedings in Physics, vol 195. Springer, Cham. https://doi.org/10.1007/978-3-319-56422-7_1
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DOI: https://doi.org/10.1007/978-3-319-56422-7_1
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