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)*<nVm>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)]
];
my $vec= pdl [
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]; #matrix T de los vectores del doblepozo
my $vect= transpose($vec); #matrix con los vectores en la columna
# <nVm>
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);
