quantum well problem generated by the confined harmonic ...qis2019.las.uz/lectures/jafarov.pdf•...
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Quantum well problem generated by the confined harmonic oscillator potential
Elchin JafarovANAS Institute of Physics
Baku, [email protected]
QIS 2019 – Samarkand, Uzbekistan, 13.09.2019
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Azerbaijan
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Baku - Samarkand
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Baku
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Notable Baku residents
Lev Landau Lotfi A. Zadeh
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Optics currently dominates for long distance interconnectsIncreasingly, optics is used in local area network applications
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Quantum well problem as a base on the superconducting qubit implementation
M.H. Devoret, A. Wallraff, and J. M. Martinis, Superconducting Qubits: A Short Review, arXiv:cond-mat/0411174, 07 Nov 2004
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Theoretical computations
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Confined quantum oscillator vs. infinite potential well
-a a
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1D Schrödinger equation – infinite potential well
−ℏ2
2𝑚
𝑑2
𝑑𝑥2+ 𝑉 𝑥 𝜓 𝑥 = 𝐸𝜓 𝑥 ,
we have to solve the following second order differential equation:
𝑑2𝜓
𝑑𝑥2+ 𝜅2𝜓 = 0,
where, 𝜅 =2𝑚𝐸
ℏ2> 0.
explicit expressions of the wavefunction 𝜓𝑛(𝑥) of the stationary states and the
discrete energy spectrum 𝐸𝑛 as follows:
𝜓𝑛 𝑥 =1
𝑎cos 𝜋
2𝑎𝑛𝑥 ,
𝐸𝑛 =𝜋2ℏ2𝑛2
4𝑚𝑎, 𝑛 = 1,2,3, … .
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1D Schrödinger equation – harmonic oscillator within the canonical approach
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Confined harmonic oscillator
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Hypergeometric function
Pochhammer symbol
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The Polynomial
If
Then
With a definition
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Orthogonal Polynomials: Basic Properties
- Definition in terms of the hypergeometric function
- Orthogonality (continuous or discrete, finite or infinite measure)
- How kind of recurrence relations it satisfies
- It is a solution of some equation
- Shift operators for it (Forward & Backward)
- Rodrigues-type formula
- Possible generating functions (including bilinear generating ones)
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Askey Scheme of Hypergeometric Orthogonal Polynomials - 1
J. LABELLE, Tableau d’Askey. In: Polynomes Orthogonaux et Applications, (eds. C. Brezinski et al.). Lecture Notes in Mathematics 1171, Springer-Verlag, New York, 1985, xxxvi–xxxvii
R. ASKEY and J.A. WILSON, Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials. Memoirs of the American Mathematical Society 319, Providence, Rhode Island, 1985
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Askey Scheme of Hypergeometric Orthogonal Polynomials - 2
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Askey scheme of orthogonal polynomials
continuous or discrete
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Askey scheme of orthogonal polynomials
explicit solutions of differential
or difference equations
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Askey scheme of orthogonal polynomials
finite or infinite
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Hermite polynomials
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Jacobi polynomials
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Gegenbauer polynomials
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Gegenbauer - > Hermite
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Confined quantum oscillator vs. infinite potential well
-a a
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Confined quantum oscillator potential with the position-dependent effective mass
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Definition of the position-dependent effective mass
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Electron Tunneling in Superconductors
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BenDaniel-Duke kinetic energy operator
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BenDaniel-Duke kinetic energy operator – explicitly
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BenDaniel-Duke kinetic energy operator – solution
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BenDaniel-Duke kinetic energy operator – solution
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BenDaniel-Duke kinetic energy operator – PDEM M(x)
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Schrödinger equation with the BenDaniel-Duke kinetic energy
operator – solution
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Nikiforov-Uvarov method
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Nikiforov-Uvarov method –polynomial solution
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Nikiforov-Uvarov method –polynomial solution-2
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Nikiforov-Uvarov method –polynomial solution-3
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Nikiforov-Uvarov method –polynomial solution-4
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Nikiforov-Uvarov method –polynomial solution-5
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Nikiforov-Uvarov method –polynomial solution-6
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Wavefunction
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Energy spectrum
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Li-Kuhn kinetic energy operator
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Li-Kuhn kinetic energy operator –explicitly
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Li-Kuhn kinetic energy operator –explicitly
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Li-Kuhn kinetic energy operator –Schrödinger equation
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Wavefunction
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Energy spectrum
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Gora-Williams kinetic energy operator
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Gora-Williams kinetic energy operator – explicitly
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Gora-Williams kinetic energy operator – solution
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Gora-Williams kinetic energy operator – PDEM M(x)
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Gora-Williams kinetic energy operator – solution
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Schrödinger equation with the Gora-Williams kinetic energy
operator – solution
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Nikiforov-Uvarov method
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Nikiforov-Uvarov method –polynomial solution
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Nikiforov-Uvarov method –polynomial solution-2
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Wavefunction
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Energy spectrum
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Zhu-Kroemer kinetic energy operator
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Zhu-Kroemer kinetic energy operator – explicitly
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Zhu-Kroemer kinetic energy operator – solution
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Zhu-Kroemer kinetic energy operator – PDEM M(x)
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Zhu-Kroemer kinetic energy operator – solution
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Schrödinger equation with the Zhu-Kroemer kinetic energy
operator – solution
![Page 72: Quantum well problem generated by the confined harmonic ...qis2019.las.uz/lectures/Jafarov.pdf• Paul Harrison, Quantum Wells, Wires and Dots (Wiley-Interscience, 2005) • G.M.Nikolopoulos](https://reader035.vdocuments.net/reader035/viewer/2022071415/611161c0effefe45af50ac27/html5/thumbnails/72.jpg)
Nikiforov-Uvarov method
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Nikiforov-Uvarov method –polynomial solution
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Nikiforov-Uvarov method –polynomial solution-2
![Page 75: Quantum well problem generated by the confined harmonic ...qis2019.las.uz/lectures/Jafarov.pdf• Paul Harrison, Quantum Wells, Wires and Dots (Wiley-Interscience, 2005) • G.M.Nikolopoulos](https://reader035.vdocuments.net/reader035/viewer/2022071415/611161c0effefe45af50ac27/html5/thumbnails/75.jpg)
Wavefunction
![Page 76: Quantum well problem generated by the confined harmonic ...qis2019.las.uz/lectures/Jafarov.pdf• Paul Harrison, Quantum Wells, Wires and Dots (Wiley-Interscience, 2005) • G.M.Nikolopoulos](https://reader035.vdocuments.net/reader035/viewer/2022071415/611161c0effefe45af50ac27/html5/thumbnails/76.jpg)
Energy spectrum
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Morrow-Brownstein kinetic energy operator
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Morrow-Brownstein kinetic energy operator – explicitly
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Morrow-Brownstein kinetic energy operator – solution
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Morrow-Brownstein kinetic energy operator – PDEM M(x)
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Morrow-Brownstein kinetic energy operator – solution
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Schrödinger equation with the Morrow-Brownstein kinetic energy
operator – solution
![Page 85: Quantum well problem generated by the confined harmonic ...qis2019.las.uz/lectures/Jafarov.pdf• Paul Harrison, Quantum Wells, Wires and Dots (Wiley-Interscience, 2005) • G.M.Nikolopoulos](https://reader035.vdocuments.net/reader035/viewer/2022071415/611161c0effefe45af50ac27/html5/thumbnails/85.jpg)
Nikiforov-Uvarov method
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Nikiforov-Uvarov method –polynomial solution
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Nikiforov-Uvarov method –polynomial solution-2
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Wavefunction
![Page 89: Quantum well problem generated by the confined harmonic ...qis2019.las.uz/lectures/Jafarov.pdf• Paul Harrison, Quantum Wells, Wires and Dots (Wiley-Interscience, 2005) • G.M.Nikolopoulos](https://reader035.vdocuments.net/reader035/viewer/2022071415/611161c0effefe45af50ac27/html5/thumbnails/89.jpg)
Energy spectrum
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von Roos kinetic energy operator
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von Roos kinetic energy operator –explicitly
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von Roos kinetic energy operator –solution
![Page 95: Quantum well problem generated by the confined harmonic ...qis2019.las.uz/lectures/Jafarov.pdf• Paul Harrison, Quantum Wells, Wires and Dots (Wiley-Interscience, 2005) • G.M.Nikolopoulos](https://reader035.vdocuments.net/reader035/viewer/2022071415/611161c0effefe45af50ac27/html5/thumbnails/95.jpg)
von Roos kinetic energy operator –PDEM M(x)
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von Roos kinetic energy operator –solution
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Schrödinger equation with the von Roos kinetic energy operator –
solution
![Page 98: Quantum well problem generated by the confined harmonic ...qis2019.las.uz/lectures/Jafarov.pdf• Paul Harrison, Quantum Wells, Wires and Dots (Wiley-Interscience, 2005) • G.M.Nikolopoulos](https://reader035.vdocuments.net/reader035/viewer/2022071415/611161c0effefe45af50ac27/html5/thumbnails/98.jpg)
Nikiforov-Uvarov method
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Nikiforov-Uvarov method –polynomial solution
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Nikiforov-Uvarov method –polynomial solution-2
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Wavefunction
![Page 102: Quantum well problem generated by the confined harmonic ...qis2019.las.uz/lectures/Jafarov.pdf• Paul Harrison, Quantum Wells, Wires and Dots (Wiley-Interscience, 2005) • G.M.Nikolopoulos](https://reader035.vdocuments.net/reader035/viewer/2022071415/611161c0effefe45af50ac27/html5/thumbnails/102.jpg)
Energy spectrum
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Further reading• Mark Fox, Optical Properties of Solids (Oxford University Press,
2001)• Paul Harrison, Quantum Wells, Wires and Dots (Wiley-
Interscience, 2005)• G.M.Nikolopoulos and I. Jex (Editors), Quantum State Transfer and
Network Engineering (Springer-Verlag Berlin Heidelberg 2014)• Y. Ohnuki and S. Kamefuchi, Quantum Field Theory and
Parastatistics (Springer Verslag, New-York, 1982)• G. Bastard, Wave mechanics applied to semiconductor
heterostructures (Les Éditions de Physique, 1988)• Nicolas Gisin, Quantum Chance: Nonlocality, Teleportation and
Other Quantum Marvels (Springer, 2012)• R. Koekoek, P.A. Lesky and R.F. Swarttouw, Hypergeometric
orthogonal polynomials and their q-analogues (Springer Verslag, Berlin, 2010)
• A.F. Nikiforov and V.B. Uvarov, Special Functions of Mathematical Physics (Birkhӓuser, Basel, 1988)
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Thank you for your attention!