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Bound states of symmetrical 1-D, 2-D and 3-D finite barrier potentials and bound state plots Ahmed Aslam V.V and Maddineni Vasu 1st year M.Tech Nanotechnology Center for Nanotechnology Research /School of Electronics Engineering, VIT University, Vellore- 632014,Tamil Nadu, India asluveeran@gmail.com, vasu.vit@gmail.comAbstractThe objective is to evaluate the behavior of Schrodinger Wave Eigen functions, Eigen values, in boundedstates of 1D, 2D and 3D through simulation and describe the wave function behavior in bounded stateswith and without perturbation. The Eigen values are found out by solving Schrodinger equation and theEigen values will be used to obtain bounded states. The program to solve for Eigen values is run onMATLAB simulation software. The bounded state plots are also obtained. 1.0 IntroductionReference 1: The description of basic excitations and their energy spectrums in 1D system is of majorinterest in modern physics. Here we deal with finding bound states in an one dimensional well. If we takemovement of an electron in such well, the potential can be expressed as: V(x) = V, x ≤ x1 0 , x1 < x <x2 V, x ≥ x2If we know the energy levels, we can easily get the wave functions using transfer matrix method. Ref 1Reference 2: When the dimensions of the transistor goes below 100 nm, the reliability of semi classicalmethods decrease, so we use a new method known as Schrodinger equation Monte Carlo method whichuses a rigorous approach.. For two dimensional (2 D) a separate program has been used to determine thevarious Eigen values.Reference 3:Cardinal spline method has been developed to find Eigen values of wells having arbitrarypotential contour. In this method we combine transfer matrix method and cubic spline elements. These
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methods are not suitable where repeated calculations are needed. This method is much simpler and alsohas high accuracy. This method has many other advantages as compared to matrix method. 2. 0 Problem identification and approach followedFor 1 dimensional, we use the finite difference method. First, assume a value for potential barrier. Afterthat we form the matrix elements. The Matlab command [V, D]= eig(A) calculates eigen values andvectors of the matrix A. Finally we plot the wave functions. For 2 dimensional calculations of bound states, spline interpolation method is used. Thespline interpolation method joins a number of polynomials to a smooth curve. The accuracy of thismethod is similar to the finite difference method. The calculations involving 3dimensional bound states uses Bessel’s function of program ofMatlab and finding zeroes using spline interpolation. An automatic numerical search is done on knownanalytical formulas involving Bessel’s function. Both first kind and second kind Bessel’s function aretaken into consideration 2.1 Particle in 1D Finite well:Bound states can exist in a potential well with finite barrier energy. Consider the case of a particleof mass m in the presence of a simple symmetric, one-dimensional, rectangular potential well of totalwidth 2L. The potential has finite barrier energy so that V = 0 for −L < x < L and V = Vo elsewhere.The value of Vo is a finite, positive constant.We know that for obtaining bound state, system should have energy E < V0. A particle with energy E >V0 will belong to part of unbound states.The time-independent Schrödinger equation for one dimension is ,
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Figure 1 symmetric one dimensional, rectangular well having width L. 2.2 Particle in 2D Finite square well:If we consider the particle in a two dimensional well, inside the well, V(x ,y) is zero. On striking walls,the wave functions will tunnel through or will be reflected. We use the separation of variables to expressthe wave function. The wave function is written as: ψ (x , y ,t) = ψ(x ,y) ɸ (t)The wave function can be expressed as separate functions of x and y.We can write, f(x) = A SinWhere C =
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The values for energy values can be found out as 2.3 Particle in 3D finite cubic or spherical wellThe 3D cubic well or 3D spherical well will have a Hamiltonian equation: Where mo is the particle mass in the well.This Hamiltonian will be operated by 3D (Cartesian or spherical) operator. Generally 3D sphericaloperator is preferred which represents a Quantum dot like structure which has more applications. While forming the quantum mechanical equation, we aim at finding solutions ofSchrödinger’s equation with V(r).The wave functions are having the expression: R(r) is radial equation.The condition for square potential is: V(r) = Vo for r<ro 0 otherwiseWe can find many bound states, i.e states which have energy less than potential barrier. Finding thesolution of problem is done by solving Schrodinger equations inside the sphere and also outside.
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3.0 Results and Discussions(i) Bounded states in 1D finite well:Program is for a free particle in a square well of size 2L with V = +V0. A finite difference approach isused and all bound state energy Eigen value is computed.
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Figure 2 Eigen function for various eigen values(ii) Bounded states in 2D finite well: The program solves for approximate Eigen values for a 2D quantum well of given area A andconstant depth D. It solves for the Eigen values from the boundary condition, where the solutions insidethe boundary are continuously joined to the exponentially decaying solutions outside. The case is solvedby first solving the 1D problem and then adding the resulting spectra to the spectrum of the 2D case. Thenumerical search for Eigen values proceeds for increasing m = 0,1m0, until no more Eigen values arefound.
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Fig 3 Bound state energies for 2D square well(iii) Bounded states in 3D finite well:The program calculates bound states in spherical potential well of finite radius. The radius can be chosen.The bound state energy levels are found from the known analytical formula involving Bessel functions byan automatic numerical search for the solutions of a transcendental equation. This search may missweakly bound states. Fig 4 bound state energies for 3D potential well as function of angular momentum
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4.0 ConclusionsIn this paper we find out how to solve the Schrödinger’s time independent equations for symmetrical 1D,2D and 3D potential wells by applying suitable boundary conditions .Bound states are then obtained fromthe Eigen energy values. The Eigen function graphs are also obtained by suitably varying Eigen values. 5.0 Acknowledgment:First of all, I would like to thank the management for giving me such an opportunity . While preparing thepaper, i got basic ideas for creating a research paper. I would like to thank our Guide Dr J.P Raina whoinstilled us with the basic idea of approaching the problem. I would like to thank our faculties forextended support. I would also like to thank our colleagues who were of great help. 6.0 References 1. “Determination of bound state energies for a one-dimensional potential field” by D.M. Sedrakian, A.Zh. Khachatrian, Physica E (2003) pp 309-315 2. “Two-dimensional quantum mechanical simulation of electron transport in nano scaled Si- based MOSFETs” by Wanqiang Chen, Leonard F. Register, Sanjay K. Banerjee., Physica E 19 (2003) pp 28 – 32. 3. “A Local Interpolatory Cardinal Spline Method for the Determination of Eigen states in Quantum-Well Structures with Arbitrary Potential Profiles” by J. Chen, A.K Chan, C.K Chui, IEEE journal of Quantum Electronics, Vol 30(1994) pp 269-274 4. “Applied Quantum Mechanics” by A.F.J. Levi. Cambridge Press
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