Numerical Solution via Numerov Method of the 1D-Schrödinger Equation with Pseudo-Delta Barrier
Martinz Samuel D.G. 1, Ramos Rubens V. 2
1Federal Institute of Education, Science and Technology of Ceara, Fortaleza-Ce, Brazil
E-mail: samueldgm@fisica.ufc.br2Lab. of Quantum Information Technology, Department of Teleinformatic Engineering Federal University of Ceara – DETI/UFC, C.P. 6007 – Campus do Pici, 60455-970 Fortaleza-Ce, Brazil
E-mail: rubens.viana@pq.cnpq.br
Received:
Received: 09 February 2018; revised: 25 July 2018; accepted: 08 August 2018; published online: 30 September 2018
DOI: 10.12921/cmst.2018.0000017
Abstract:
In this work, aiming to solve numerically the Schrödinger equation with a Dirac delta function potential, we use the Numerov method to solve the time independent 1D-Schrödinger equation with potentials of the form V (x) + αδp(x), where δp(x) is a pseudo-delta function, a very high and thin barrier. The numerical results show good agreement with analytical results found in the literature. Furthermore, we show the numerical solutions of a system formed by three delta function potentials inside of an infinite quantum well and the harmonic potential with position dependent mass and a delta barrier in the center.
Key words:
Dirac delta function potential, Numerov method, Schrödinger equation
References:
[1] J.J. Álvarez, M. Gadella, F.J.H. Heras, L.M. Nieto, A one- dimensional model of resonances with a delta barrier and mass jump,Phys.Lett.A373,4022–4027(2009).
[2] F.M.Toyama,Y.Nogami,Transmission-reflectionproblemwith a potential of the form of the derivative of the delta function, J. Phys. A: Math. Theor. 40, F685–F690 (2007).
[3] A.V. Zolotaryuk, Y. Zolotaryuk, Controllable resonant tun- nelling through single-point potentials: A point triode, Phys. Lett. A 379, 511–517 (2015).
[4] A.V. Zolotaryuk, Boundary conditions for the states with reso- nant tunnelling across the δ’-potential, Phys. Lett. A 374, 1636– 1641 (2010).
[5] M. Belloni, R.W. Robinett, The infinite well and Dirac delta function potentials as pedagogical, mathematical and physical models in quantum mechanics, Phys. Rep. 540, 25–122 (2014).
[6] M. Pillai, J. Goglio, T.G. Walker, Matrix Numerov method for solving Schrödinger’s equation, Am. J. Phys., 80, 11, 1017– 1019, 2012.
[7] P. Pedram, M. Vahabi, Exact solutions of a particle in a box with a delta function potential: The factorization method, Am. J. Phys. 78, 8, 839–841 (2010).
[8] D.J. Griffiths, Introduction to Quantum Mechanics, 2nd Edition; Pearson Education Chapter 2, 2005. Available online in http://physicspages.com/2012/08/21/infinite-square- well-with-delta-function-barrier.
[9] O. von Roos, Position-dependent effective masses in semicon- ductor theory, Phys. Rev. B 27, 7547 (1983).
[10] A.Ganguly,S ̧.Kuru,J.Negro,L.M.Nieto,Astudyofthebound states for square potential wells with position-dependent mass, Phys. Lett. A 360, 228–233 (2006).
[11] S.D.G. Martinz, R.V. Ramos, Double quantum well triple bar- rier structures: analytical and numerical results, Can. J. Phys. 94, 11, 1180–1188 (2016).
In this work, aiming to solve numerically the Schrödinger equation with a Dirac delta function potential, we use the Numerov method to solve the time independent 1D-Schrödinger equation with potentials of the form V (x) + αδp(x), where δp(x) is a pseudo-delta function, a very high and thin barrier. The numerical results show good agreement with analytical results found in the literature. Furthermore, we show the numerical solutions of a system formed by three delta function potentials inside of an infinite quantum well and the harmonic potential with position dependent mass and a delta barrier in the center.
Key words:
Dirac delta function potential, Numerov method, Schrödinger equation
References:
[1] J.J. Álvarez, M. Gadella, F.J.H. Heras, L.M. Nieto, A one- dimensional model of resonances with a delta barrier and mass jump,Phys.Lett.A373,4022–4027(2009).
[2] F.M.Toyama,Y.Nogami,Transmission-reflectionproblemwith a potential of the form of the derivative of the delta function, J. Phys. A: Math. Theor. 40, F685–F690 (2007).
[3] A.V. Zolotaryuk, Y. Zolotaryuk, Controllable resonant tun- nelling through single-point potentials: A point triode, Phys. Lett. A 379, 511–517 (2015).
[4] A.V. Zolotaryuk, Boundary conditions for the states with reso- nant tunnelling across the δ’-potential, Phys. Lett. A 374, 1636– 1641 (2010).
[5] M. Belloni, R.W. Robinett, The infinite well and Dirac delta function potentials as pedagogical, mathematical and physical models in quantum mechanics, Phys. Rep. 540, 25–122 (2014).
[6] M. Pillai, J. Goglio, T.G. Walker, Matrix Numerov method for solving Schrödinger’s equation, Am. J. Phys., 80, 11, 1017– 1019, 2012.
[7] P. Pedram, M. Vahabi, Exact solutions of a particle in a box with a delta function potential: The factorization method, Am. J. Phys. 78, 8, 839–841 (2010).
[8] D.J. Griffiths, Introduction to Quantum Mechanics, 2nd Edition; Pearson Education Chapter 2, 2005. Available online in http://physicspages.com/2012/08/21/infinite-square- well-with-delta-function-barrier.
[9] O. von Roos, Position-dependent effective masses in semicon- ductor theory, Phys. Rev. B 27, 7547 (1983).
[10] A.Ganguly,S ̧.Kuru,J.Negro,L.M.Nieto,Astudyofthebound states for square potential wells with position-dependent mass, Phys. Lett. A 360, 228–233 (2006).
[11] S.D.G. Martinz, R.V. Ramos, Double quantum well triple bar- rier structures: analytical and numerical results, Can. J. Phys. 94, 11, 1180–1188 (2016).
