1. 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

2. 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
,

3. 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) ɸ (t)
The wave function can be expressed as separate functions of x and y.
We can write, f(x) = A Sin
Where C =

4. 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.

5. 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.

6. 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.

7. 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

8. 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