Abstract
This paper includes a MAPLE® code giving numerical solution of two dimensional Schrödinger equation in a functional space. The Galerkin method has been used to get the approximate solution. The results have been examined with numerical examples.
Keywords: Time dependent schrödinger equation, Weak solution, Numerical approximation, Galerkin method, Maple.
Received:22 November 2016/ Revised:17 December 2016/ Accepted:28 December 2016/ Published:9 January 2017
Contribution/ Originality
This study is one of very few studies which have investigated to obtain an efficient computation tool for numerical examinations of two dimensional Schrödinger Equation.
1. INTRODUCTION
As it is known the Schrödinger equation appears in underwater acoustics, in electromagnetic wave propagation, in optics and in optoelectronic devices [1-4] . The analytical and numerical solutions have been a subject of considerable interest [5-8] .
The numerical investigations on two dimensional Schrödinger equation have drawn considerable interest in different aspects [9-13] .
We consider one single electron of an atomic system that could be extended for several electron problems and assume that external potential is time dependent only. Besides, we assume that the system has a perturbing term. Depending on the nature of perturbation the system can set out quite complicated features.
The purpose of this study is to obtain an operable and efficient code for the solution of a general two dimensional Schrödinger equation in the form of;
2. THE APPLICATION OF GALERKIN METHOD
It is known that, in numerical analysis, Galerkin method is a kind of method for converting a continuous operator problem to a discrete problem. Actually, it is the equivalent of applying the method of variation of parameters to a function space, by converting the equation to a weak formulation. So, some constraints can be applied on the function space to characterize the space with a finite set of basis functions.
According to the Faedo-Galerkin Method or mostly briefly as Galerkin Method, the approximations for the solution of the problem (1)-(3) are such as:
3. NUMERICAL ILLUSTRATIONS
In this section we have generated three test problems and obtained numerical examples reflecting the results of the method. These examples are produced by using the code given in the Appendix.
Figure-1. Visualizations of Solutions
Source: These figures are plotted by Maple 15.
Figure-2. Visualization of Solutions
Source: These figures are plotted by Maple 15.
Figure-3. Visualization of Solutions
Source: These figures are plotted by Maple 15.
4. RESULT AND DISCUSSION
| > TDSE:=proc(l1,l2,N,v,phi,f) > u:=array(1..N,1..N):du:=array(1..N,1..N):A:=array(1..N,1..N):C:=array(1..N,1..N): > F:=array(1..N,1..N):IC:=array(1..N,1..N):S:=array(1..N,1..N): > for k from 1 to N do > for m from 1 to N do > u[k,m]:=2/sqrt(l1*l2)*sin(k*Pi/l1*x)*sin(m*Pi/l2*y): > du[k,m]:=simplify(diff(u[k,m],x$2)+diff(u[k,m],y$2)): > od; > od; > for i from 1 to N do > for j from 1 to N do > C[i,j]:=c[i,j](t): > A[i,j]:=evalf(int((du[i,j]-v*u[i,j])*u[i,j],x=0..l1,y=0..l2)); > F[i,j]:=evalf(int(f*u[i,j],x=0..l1,y=0..l2)); > IC[i,j]:=evalf(int(phi*u[i,j],x=0..l1,y=0..l2)); > od; > od; > for i from 1 to N do > for j from 1 to N do > d[i,j]:=I*diff(C[i,j],t)+A[i,j]*C[i,j]=F[i,j]; > S[i,j]:=rhs(dsolve({d[i,j],c[i,j](0)=IC[i,j}],C[i,j])); > od; > od; > psi:=sum(sum(S[r,s]*u[r,s],s=1..N),r=1..N); > end: |
Funding: This study received no specific financial support. |
Competing Interests: The authors declare that they have no competing interests. |
Contributors/Acknowledgement: We are very grateful to the referee for his/her appropriate and constructive suggestions and for his/her proposed corrections to improve the paper |
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