The Schrodinger equation for a time-dependent Hamiltonian is:
I try to implement a solver for the Schrodinger equation for a time-dependent Hamiltonian in
H
is just an identity matrix in your case, so we can just multiply it with the psi
vector to get back the psi
vector itself. Then, we bring i*hbar
to the right-hand-side of the equation so that the final equation is in a form that ode45
accepts. Finally, we use the following code to solve for psi
:
function schrodinger_equation
psi0 = [0;1];
hbar = 1;
t = [0 100];
[T,psi] = ode45(@(t,psi)dpsi(t,psi,hbar),t,psi0);
for i = 1:length(psi0)
figure
plot(T,real(psi(:,i)),T,imag(psi(:,i)))
xlabel('t')
ylabel('Re(\psi) or Im(\psi)')
title(['\psi_0 = ' num2str(psi0(i))])
legend('Re(\psi)','Im(\psi)','Location','best')
end
end
function rhs = dpsi(t,psi,hbar)
rhs = 1/(1i*hbar)*cos(t).*ones(2,1);
end
Note that I have plotted the two components of psi
separately and for each such plot, I have also plotted the real and imaginary components separately. Here are the plots for two different values of psi0
: