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.com

Abstract

The objective is to evaluate the behavior of Schrodinger Wave Eigen functions, Eigen values, in bounded

states of 1D, 2D and 3D through simulation and describe the wave function behavior in bounded states

with and without perturbation. The Eigen values are found out by solving Schrodinger equation and the

Eigen values will be used to obtain bounded states. The program to solve for Eigen values is run on

MATLAB simulation software. The bounded state plots are also obtained.

1.0 Introduction

Reference 1: The description of basic excitations and their energy spectrums in 1D system is of major

interest in modern physics. Here we deal with finding bound states in an one dimensional well. If we take

movement of an electron in such well, the potential can be expressed as:

V(x) = V, x ≤ x1

0 , x1 < x <x2

V, x ≥ x2

If we know the energy levels, we can easily get the wave functions using transfer matrix method. Ref 1

Reference 2: When the dimensions of the transistor goes below 100 nm, the reliability of semi classical

methods decrease, so we use a new method known as Schrodinger equation Monte Carlo method which

uses a rigorous approach.. For two dimensional (2 D) a separate program has been used to determine the

various Eigen values.

Reference 3:Cardinal spline method has been developed to find Eigen values of wells having arbitrary

potential contour. In this method we combine transfer matrix method and cubic spline elements. These

methods are not suitable where repeated calculations are needed. This method is much simpler and also

has high accuracy. This method has many other advantages as compared to matrix method.

2. 0 Problem identification and approach followed

For 1 dimensional, we use the finite difference method. First, assume a value for potential barrier. After

that we form the matrix elements. The Matlab command [V, D]= eig(A) calculates eigen values and

vectors of the matrix A. Finally we plot the wave functions.

For 2 dimensional calculations of bound states, spline interpolation method is used. The

spline interpolation method joins a number of polynomials to a smooth curve. The accuracy of this

method is similar to the finite difference method.

The calculations involving 3dimensional bound states uses Bessel’s function of program of

Matlab and finding zeroes using spline interpolation. An automatic numerical search is done on known

analytical formulas involving Bessel’s function. Both first kind and second kind Bessel’s function are

taken 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 particle

of mass m in the presence of a simple symmetric, one-dimensional, rectangular potential well of total

width 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

,

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 express

the wave function. The wave function is written as:

ψ (x , y ,t) = ψ(x ,y) (t) ɸ

The wave function can be expressed as separate functions of x and y.

We can write, f(x) = A Sin

Where C =

The values for energy values can be found out as

2.3 Particle in 3D finite cubic or spherical well

The 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 spherical

operator is preferred which represents a Quantum dot like structure which has more applications.

While forming the quantum mechanical equation, we aim at finding solutions of

Schrödinger’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 otherwise

We can find many bound states, i.e states which have energy less than potential barrier. Finding the

solution of problem is done by solving Schrodinger equations inside the sphere and also outside.

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 is

used and all bound state energy Eigen value is computed.

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 and

constant depth D. It solves for the Eigen values from the boundary condition, where the solutions inside

the boundary are continuously joined to the exponentially decaying solutions outside. The case is solved

by first solving the 1D problem and then adding the resulting spectra to the spectrum of the 2D case. The

numerical search for Eigen values proceeds for increasing m = 0,1m0, until no more Eigen values are

found.

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 by

an automatic numerical search for the solutions of a transcendental equation. This search may miss

weakly bound states.

Fig 4 bound state energies for 3D potential well as function of angular momentum

4.0 Conclusions

In 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 from

the 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 the

paper, i got basic ideas for creating a research paper. I would like to thank our Guide Dr J.P Raina who

instilled us with the basic idea of approaching the problem. I would like to thank our faculties for

extended 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