http://qs1969.pair.com?node_id=11139137

Lucero has asked for the wisdom of the Perl Monks concerning the following question:

Hello. Im trying to calculate the transition probability for a constant pertubation. What I have to do is apply the RK4 method to a vectorial function
``` :fn(cm, t) =sum over m((-i/h)*<n|V|m>exp(-i(Em -En)t/h))Cm)
The values of |cm|^2 must oscillate around 1 and 0 but I run the code the values I get are bigger and dont have the oscillatory behavior.
```
use warnings;
use strict;
use PDL;
use PDL::Complex;

my \$eprim=0.001;

my \$X =pdl [
[ 0, -sqrt(1),0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0
+],
[sqrt(1),0, sqrt(2),0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0
+,0,0],
[0, sqrt(2),0,sqrt(3),0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0
+,0,0],
[0,0,sqrt(3),0,sqrt(4),0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
+0,0],
[0,0,0,sqrt(4),0,sqrt(5),0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
+0,0],
[0,0,0,0,sqrt(5),0,sqrt(6),0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
+0,0],
[0,0,0,0,0,sqrt(6),0,sqrt(7),0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
+0,0],
[0,0,0,0,0,0,sqrt(7),0,sqrt(8),0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
+0,0],
[0,0,0,0,0,0,0,sqrt(8),0,sqrt(9),0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
+0,0],
[0,0,0,0,0,0,0,0,sqrt(9),0,sqrt(10),0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0
+,0,0],
[0,0,0,0,0,0,0,0,0,sqrt(10),0,sqrt(11),0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
+0,0,0],
[0,0,0,0,0,0,0,0,0,0,sqrt(11),0,sqrt(12),0,0,0,0,0,0,0,0,0,0,0,0,0,0,
+0,0,0],
[0,0,0,0,0,0,0,0,0,0,0,sqrt(12),0,sqrt(13),0,0,0,0,0,0,0,0,0,0,0,0,0,
+0,0,0],
[0,0,0,0,0,0,0,0,0,0,0,0,sqrt(13),0,sqrt(14),0,0,0,0,0,0,0,0,0,0,0,0,
+0,0,0],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,sqrt(14),0,sqrt(15),0,0,0,0,0,0,0,0,0,0,
+0,0,0],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,sqrt(15),0,sqrt(16),0,0,0,0,0,0,0,0,0,
+0,0,0],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,sqrt(16),0,sqrt(17),0,0,0,0,0,0,0,0,
+0,0,0],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,sqrt(17),0,sqrt(18),0,0,0,0,0,0,0,
+0,0,0],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,sqrt(18),0,sqrt(19),0,0,0,0,0,0,
+0,0,0],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,sqrt(19),0,sqrt(20),0,0,0,0,0,
+0,0,0],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,sqrt(20),0,sqrt(21),0,0,0,0,
+0,0,0],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,sqrt(21),0,sqrt(22),0,0,0,
+0,0,0],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,sqrt(22),0,sqrt(23),0,0,
+0,0,0],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,sqrt(23),0,sqrt(24),0,
+0,0,0],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,sqrt(24),0,sqrt(25),
+0,0,0],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,sqrt(25),0,sqrt(26
+),0,0],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,sqrt(26),0,sqrt(
+27),0],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,sqrt(27),0,sqr
+t(27)],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,sqrt(28),0],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,sqrt(28)]

];

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]; #matrix T de los vectores del doblepozo
my \$vect= transpose(\$vec); #matrix con los vectores en la columna

# <n|V|m>
my \$V= \$vec x \$X x \$vect;
my \$step=0.1;
my \$hbar=1;
#print &Vnm(0,1);
#print&Vnm(1,0);
my @Em=(-3.49090786,-3.48609059,-0.94249804,-0.72223315,0.84486159,1.9
+8334101,3.52168993,5.19562490,7.00900149,8.96758961,11.06144444445427
+60,13.1282317263938650,15.4597663541535190,18.2923955058715430,19.575
+1133106800170,21.3254642902799140,27.3060148356095880,33.615327480770
+2470,48.2864949409823790,58.1657460989797170,83.0782032408711330,97.3
+475523518670660,137.5307666823436800,157.3985331875067300,221.5161226
+004345300,248.6116702653141900,353.3740736965299900,390.0680354316574
+500,577.9792266930371600,628.4681868621187300);

my \$w=\$Em[1]-\$Em[0];
sub Vnm {
my (\$x1,\$x2)=@_;
return \$V->range([\$x1,\$x2]);
};
my \$n;
my \$m;
my \$f;
my @c = (1, 0, 0, 0); #INITIAL CONDITIONS
#@c=F=(F0,F1,F2,F3)
sub F0 {
my (\$t,@c)=@_;
\$m=0;
\$f=0;
for (\$n=0;\$n<4;\$n++){
my \$a= &Vnm(\$n,\$m)*exp(-(i)*(\$Em[\$m]-\$Em[\$n])*(\$t/\$hbar));
if (\$n!=\$m){
\$f = \$f + \$a*(-(i)*(-\$eprim)*cos(\$w*\$t)*\$c[\$n]);
};
};
return \$f;
};
sub F1 {
my (\$t,@c)=@_;
\$m=1;
\$f=0;
for (\$n=0;\$n<4;\$n++){
my \$a= &Vnm(\$n,\$m)*exp(-(i)*(\$Em[\$m]-\$Em[\$n])*(\$t/\$hbar));
if (\$n!=\$m){
\$f = \$f + \$a*(-(i)*(-\$eprim)*cos(\$w*\$t)*\$c[\$n]);
};
};
return \$f;
};
sub F2 {
my (\$t,@c)=@_;
\$m=2;
\$f=0;
for (\$n=0;\$n<4;\$n++){
my \$a= &Vnm(\$n,\$m)*exp(-(i)*(\$Em[\$m]-\$Em[\$n])*(\$t/\$hbar));
if (\$n!=\$m){
\$f = \$f + \$a*-(i)*(-\$eprim)*cos(\$w*\$t)*\$c[\$n];
};
};
return \$f;
};
sub F3 {
my (\$t,@c)=@_;
\$m=3;
\$f=0;
for (\$n=0;\$n<4;\$n++){
my \$a= &Vnm(\$n,\$m)*exp(-(i)*(\$Em[\$m]-\$Em[\$n])*(\$t/\$hbar));
for (\$n!=\$m){
\$f = \$f +\$a*(-(i)*(-\$eprim)*cos(\$w*\$t)*\$c[\$n]);
};
};
return \$f;
};

#Método  RK4
my \$k1;
my \$k2;
my \$k3;
my \$k4;

my \$k1_1;
my \$k2_1;
my \$k3_1;
my \$k4_1;

my \$k1_2;
my \$k2_2;
my \$k3_2;
my \$k4_2;

my \$k1_3;
my \$k2_3;
my \$k3_3;
my \$k4_3;

sub RK4_c0 {
my (\$t,@c)=@_;
sub k1 { my (\$t1,@c)=@_;return \$step*&F0(\$t1,@c); };
sub k2 { my (\$t2,@c)=@_;return \$step*&F0((\$t2+\$step/2),( @c + (&k1(\$t2
+)/2)));};
sub k3 { my (\$t3,@c)=@_;return \$step*&F0((\$t3+\$step/2),( @c + (&k2(\$t3
+)/2))) ;};
sub k4 { my (\$t4,@c)=@_;return \$step*&F0((\$t4+\$step),( @c + &k3(\$t4)))
+;};
return  \$c[0] + (1/6)*(&k1(\$t,@c)+ 2*(&k2(\$t,@c) + &k3(\$t,@c)) + &k4(\$
+t,@c));
};

sub RK4_c1 {
my (\$t,@c)=@_;
sub k1_1 { my (\$t1,@c)=@_;return \$step*&F1(\$t1,@c); };
sub k2_1 { my (\$t2,@c)=@_;return \$step*&F1((\$t2+\$step/2),( @c + &k1(\$t
+2/2)));};
sub k3_1 { my (\$t3,@c)=@_;return \$step*&F1((\$t3+\$step/2),( @c + &k2(\$t
+3)/2)) ;};
sub k4_1 { my (\$t4,@c)=@_;return \$step*&F1((\$t4+\$step),( @c + &k3(\$t4)
+));};
return  \$c[1] + (1/6)*(&k1_1(\$t,@c)+ 2*(&k2_1(\$t,@c) + &k3_1(\$t,@c)) +
+ &k4_1(\$t,@c));
};

sub RK4_c2 {

my (\$t,@c)=@_;
sub k1_2 { my (\$t1,@c)=@_;return \$step*&F2(\$t1,@c); };
sub k2_2 { my (\$t2,@c)=@_;return \$step*&F2((\$t2+\$step/2),( @c + &k1(\$t
+2)/2));};
sub k3_2 { my (\$t3,@c)=@_;return \$step*&F2((\$t3+\$step/2),( @c + &k2(\$t
+3)/2)) ;};
sub k4_2 { my (\$t4,@c)=@_;return \$step*&F2((\$t4+\$step),( @c + &k3(\$t4)
+));};
return  \$c[2] + (1/6)*(&k1_2(\$t,@c)+ 2*(&k2_2(\$t,@c) + &k3_2(\$t,@c)) +
+ &k4_2(\$t,@c));
};

sub RK4_c3 {
my (\$t,@c)=@_;
sub k1_3 { my (\$t1,@c)=@_;return \$step*&F3(\$t1,@c); };
sub k2_3 { my (\$t2,@c)=@_;return \$step*&F3((\$t2+\$step/2),( @c + &k1(\$t
+2)/2));};
sub k3_3 { my (\$t3,@c)=@_;return \$step*&F3((\$t3+\$step/2),( @c + &k2(\$t
+3)/2)) ;};
sub k4_3 { my (\$t4,@c)=@_;return \$step*&F3((\$t4+\$step),( @c + &k3(\$t4)
+));};
return  \$c[3] + (1/6)*(&k1_3(\$t,@c)+ 2*(&k2_3(\$t,@c) + &k3_3(\$t,@c)) +
+ &k4_3(\$t,@c));
};

my \$t=0;

open (FILE , ">C0_V1W01p2.dat");
for (\$t=0; \$t<2500;\$t+=\$step){
#print  " @c \$t \n";
#print abs(\$c[0])*abs(\$c[0]), "\n";
\$c[0]= &RK4_c0(\$t,@c);
\$c[1]= &RK4_c1(\$t,@c);
\$c[2]= &RK4_c2(\$t,@c);
\$c[3]= &RK4_c3(\$t,@c);
my \$x= abs(\$c[0])**2;
my \$b= abs(\$c[1])**2;
my \$s= abs(\$c[2])**2;
my \$d= abs(\$c[3])**2;

print FILE "\$x \$b \$s \$d \$t\n";

};
close (FILE);