Time dependent Schrödinger equation queer solution - fortran

Question

I am solving the evolution of time dependent schrodinger equation under a harmonic potential and an initial gaussian wavefunction . Treating hcut=1 and 2m=1, and seperating the wavefunction in real and imaginary parts, two coupled equations are obtained in terms of the real and imaginary part, termed as yr and yc respectively.
xrange is [xi,xf]
trange is [0,tf]
method I used is :
first seperating the wavefunction in real and imaginary part, namely yr(x,t) and yc(x,t) .
then treating hcut=2m=1, and writing the wavefunction as yr[x,t]+i*yc[x,t],we get two coupled equations from the TDSE .
1.D[yr[x,t],t]=-D[yc[x,t],x,x]+V[x]*yc[x,t]
2.D[yc[x,t],t]=D[yc[x,t],x,x]-V[x]*yc[x,t]
Then I specified the inital wavefunction as
yr[x,0]=exp[-x^2]
yc[x,0]=0
After that, using finite difference scheme, I tried to find y[x,t] from y[x,0]
i.e,
yr[x,t+d]=yr[x,t]+d*D[yr[x,t],t]
=yr[x,t]+d*(D[yc[x,t],x,x]+V[x]*yc[x,t])
indexed as "a" in code and "b" for the complex part .
The values of y[x[i],t[j]] are stored as y[i,j] as array cannot have real index .
The code I used is given below:
function v(x) result(s)
real::s,x
s=x**2
end function v
real::t(10000),x(10000),yc(10000,10000),yr(10000,10000),tf,xi,xf,d
integer::i,j,k,l,m,n,time
write(*,*) "time of plot divided by step size"
read(*,*) time
tf=50
xi=-10
xf=10
d=0.01
x(1)=xi
t(1)=0
i=1
do while(x(i).lt.xf) !input all values of x in x(i) array
x(i+1)=x(i)+d
i=i+1
end do
n=1
do while(t(n).lt.tf) !input all values of t in t(i) array
t(n+1)=t(n)+d
n=n+1
end do
do j=1,i
yr(j,1)=exp(-(x(j)-5)**2) !input of initial wavefunction
yc(j,1)=0
end do
!calculation of wavefunction at higher time using finite element technique[y[x,t+d]=y[x]+d*D[y[x,t],t] and then replacing the partial derivative with time
!using equation 1 and 2 .
l=1
do while(l.le.i-2)
k=1
do while(t(k).lt.tf)
yr(l,k+1)=yr(l,k)-(yc(l+2,k)-2*yc(l+1,k)+yc(l,k))/d&
+v(x(l))*yc(l,k)*d
yc(l,k+1)=yc(l,k)+(yr(l+2,k)-2*yr(l+1,k)+yr(l,k))/d&
-v(x(l))*yr(l,k)*d
k=k+1
end do
l=l+1
end do
open(1,file="q.dat")
do m=1,i-2
write(1,*) x(m),(yr(m,time))**2+(yc(m,time))**2
end do
close(1)
end
Expected result: y(x,t)^2=yr(x,t)^2+yc(x,t)^2
Error: The wavefunction is not staying regular, after only t=0.05 or 0.06, the wavefunction is turning huge and the maxima is becoming of the order of e30, in spite of that the gaussian shape is remaining almost unchanged, as expected, as only 0.05 seconds has passsed.

Answer 1

This is not really a coding problem but a numerical mathematical one and I suggest you to first study some tutorials about numerical methods for the Schrödinger equation and similar PDEs and then ask for more at sites that are devoted to scientific computation.
First, the method you chose does not conserve the norm of the solution. That can be saved by normalizing by each time step, but it is quite ugly.
Second, the method you chose is explicit (actually, it is probably the forward Euler method that is unconditionally unstable). That means the allovable time step is going to be severly limited by diffusive-like term (even though it is complex here). A simple implicit method that also conserves the norm quite well is the Crank-Nicolson method. It is an implicit method and you will have to solve a system of linear equations at each timestep. It is quite simple in 1D as the system is tridiagonal. There are also some explicit schemes that may work, but they are more involved, not the naive scheme you tried.
Third, you do not have to split the system into the real part and the imaginary part, it can be done using complex numbers.
Fourth, you shoould keep your spatial grid step dx and time-step dt as separate variables with known values. You can compute the final coefficient d as their ratio, but you should at least know your values od dx and dt.
Fifth, you do not have to store all values of the solution in all time-steps. That will quickly become prohibitevely expenses in times of memory required for larger problems. It is enough to store the last time-step and the current time-step. More for multistep methods and some auxiliary steps for Runge-Kutta methods.