- #1
evamaster
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I want to solve the Schrodinger via the Numerov Method but I had some troubles. I'm programing in C++, so here is my code:
The code compiles perfectly but the problem arises when I want to plot the odd solutions. I obtain two satisfactory solutions but another two that it's function is continuous but not it's derivative.
I would be very thankful if somebody can help me with this problem.
Code:
#include<cstdlib>
#include<iostream>
#include<cmath>
using namespace std;
double x_min=-4.0 , x_max=4.0;
int N=2000;
double r=(x_max-x_min)/(1.0*N);
double d=2.0;
double p=0.4829; // 2m/(hbar^2)
double Vo=20.0; // Altura del pozo
double x_m=0.1; //Matching point
int i_x_m=(x_m-x_min)/r;
double Control=-123456789;
double SlopeLeft,SlopeRight;
double PAR;
double K2(double x, double E);
double NumerovL(int i, double k21, double k22, double k23, double Y[]);
double NumerovR(int i, double k21, double k22, double k23, double Y[]);
double FuncLeft(double E, double Y[]);
double FuncRight(double E, double Y[]);
void PrintFunc(double Y[]);
void Normalizar(double Y[]);
double f(double E, double Y[]);
double Biseccion(double a, double b, double Y[]);
//=========================MAIN===============================
int main(int argc, char **argv)
{
double Y[N+1]; // Función de Onda
double paso=0.02; // Escala en la que se varia la energía
double Eo=0;
for(double E=0 ; E<=Vo ; E+=paso) //Cálculo de las funciones IMPARES
{
PAR=-1;
Eo=Biseccion(E,E+paso,Y);
if(Eo != Control && SlopeRight*SlopeLeft<0.)
{
Y[i_x_m]=FuncRight(Eo,Y);
Y[i_x_m]=FuncLeft(Eo,Y);
Normalizar(Y);
PrintFunc(Y);
}
}
for(double E=0 ; E<=Vo ; E+=paso) //Cálculo de las funciones PARES
{
PAR=1;
Eo=Biseccion(E,E+paso,Y);
if(Eo != Control && SlopeRight*SlopeLeft>0.)
{
Y[i_x_m]=FuncRight(Eo,Y);
Y[i_x_m]=FuncLeft(Eo,Y);
Normalizar(Y);
PrintFunc(Y);
}
}
return 0;
}
//=========================FUNCIONES===============================
double K2(double x, double E)
{
double k2;
if(fabs(x)<=d)
{
k2=p*E;
return k2;
}
else
{
k2=p*(E-Vo);
return k2;
}
}
double NumerovL(int i, double k21, double k22, double k23, double Y[])
{ // Para la función de Onda Izquierda
double A1,B1,C1,N;
A1=2.0*(1.0-(5.0/12.0)*r*r*k21)*Y[i-1];
B1=(1.0+(1.0/12.0)*r*r*k22)*Y[i-2];
C1=1.0+(1.0/12.0)*r*r*k23;
N=(A1-B1)/(C1);
return N;
}
double NumerovR(int i, double k21, double k22, double k23, double Y[])
{ // Para la función de Onda Derecha
double A1,B1,C1,N;
A1=2.0*(1.0-(5.0/12.0)*r*r*k21)*Y[i+1];
B1=(1.0+(1.0/12.0)*r*r*k22)*Y[i+2];
C1=1.0+(1.0/12.0)*r*r*k23;
N=PAR*(A1-B1)/(C1);
return N;
}
double FuncLeft(double E, double Y[])
{
double k21,k22,k23,Yleft,b;
b=sqrt(p*(Vo-E));
Y[0]=exp(b*x_min);
Y[1]=exp(b*(x_min+r));
for(int i=2 ; i<i_x_m ; i++) // Se calcula la función de Onda Izquierda
{
k21=K2(x_min+(i-1)*r,E);
k22=K2(x_min+(i-2)*r,E);
k23=K2(x_min+i*r,E);
Y[i]=NumerovL(i,k21,k22,k23,Y);
if(i==i_x_m-1) //Función de Onda Izquierda en el Matching point
{
k21=K2(x_min+(i)*r,E);
k22=K2(x_min+(i-1)*r,E);
k23=K2(x_min+(i+1)*r,E);
Yleft=NumerovL(i+1,k21,k22,k23,Y);
}
}
SlopeLeft=(Yleft-Y[i_x_m-1])/r;
return Yleft;
}
double FuncRight(double E, double Y[])
{
double k21,k22,k23,Yright,b;
b=sqrt(p*(Vo-E));
Y[N]=PAR*exp(-b*(x_min+N*r));
Y[N-1]=PAR*exp(-b*(x_min+(N-1)*r));
for(int i=N-2 ; i>i_x_m; i--) // Se calcula la función de Onda Derecha
{
k21=K2(x_min+(i+1)*r,E);
k22=K2(x_min+(i+2)*r,E);
k23=K2(x_min+i*r,E);
Y[i]=PAR*NumerovR(i,k21,k22,k23,Y);
if(i==i_x_m+1) //Función de Onda Derecha en el Matching point
{
k21=K2(x_min+(i)*r,E);
k22=K2(x_min+(i+1)*r,E);
k23=K2(x_min+(i-1)*r,E);
Yright=NumerovR(i-1,k21,k22,k23,Y);
}
}
SlopeRight=PAR*(Y[i_x_m+1]-Yright)/r;
return Yright;
}
void PrintFunc(double Y[])
{
for(int i=0 ; i<=N+1 ; i++)
{
cout << x_min+i*r << "\t" << Y[i] << endl;
}
}
void Normalizar(double Y[])
{
double S=0;
for(int i=0 ; i<=N+1 ; i++)
{
S += Y[i]*Y[i]*r;
}
S=sqrt(S);
for (int i=0 ; i<=N+1 ; i++)
{
Y[i]=Y[i]/S;
}
}
double f(double E, double Y[])
{
double F;
F=FuncLeft(E,Y)-PAR*FuncRight(E,Y);
return F;
}
double Biseccion(double a, double b, double Y[])
{
double Tol=0.00001; //Tolerancia para encontrar la raiz
double RET=-123456789;
if(f(a,Y)*f(b,Y)<0)
{
while(fabs(a-b)>Tol)
{
double x_m,fa,fm;
fa=f(a,Y);
x_m=(a+b)/2.0;
fm=f(x_m,Y);
//fb=f(b);
if(fa*fm<0)
{
b=x_m;
//RET=b;
}
else
{
a=x_m;
//RET=a;
}
}
RET=a;
}
return RET;
}
The code compiles perfectly but the problem arises when I want to plot the odd solutions. I obtain two satisfactory solutions but another two that it's function is continuous but not it's derivative.
I would be very thankful if somebody can help me with this problem.