- #1
Marin
- 193
- 0
Hi all!
I'm trying to solve the following system of ODE's, but somewhat unsuccessful...
[tex]\dot \vec x = [-i\omega(t)\sigma_z - \nu(t)\sigma_y]\vec x[/tex]
with sigma_i the Pauli matrices and w(t) and v(t) well-behaved functions of t (actually I also have that w = 1+v). Nevertheless, v(t+T) = v(t) and the same holds of course for w(t).
I know such systems are hardly ever solvable, however here we have some nice properties (e.g. the time dependence is outside the matrices, the matrices themselves have nice properties, the coefficients are periodic) which make me think it's doable..
The problem comes from the fact that the Pauli matrices do not commute and so an exponential ansatz fails (unlike if one of the matrices was the identity).
The periodicity reminds me of the Floquet thm./analysis, but I haven't found any statement towards solvability in there so far. It only says that the solution can be written in the form
[tex]\vec x(t) = Q(t)e^{tR}\vec x_0[/tex]
for some matrix-valued function Q(t) and a constant matrix R. However, when I plug it in the equation above it doesn't help much, mainly because of the a priori non-commutativity of Q and R (even if I assume they are both invertible for all t).
I tried some ways to rewrite the system in a different way, but it only made it more complicated, since matrix-valued functions appeared, where the time dependence cannot be separated.
Also tried to use the fact that w = 1+v but the non-commutativity problem remains and so I doubt it could be useful.
Any ideas or hints are well appreciated. If someone had this or similar problem before and would like to share their experience would be perfect :)
Thanks a lot,
marin
I'm trying to solve the following system of ODE's, but somewhat unsuccessful...
[tex]\dot \vec x = [-i\omega(t)\sigma_z - \nu(t)\sigma_y]\vec x[/tex]
with sigma_i the Pauli matrices and w(t) and v(t) well-behaved functions of t (actually I also have that w = 1+v). Nevertheless, v(t+T) = v(t) and the same holds of course for w(t).
I know such systems are hardly ever solvable, however here we have some nice properties (e.g. the time dependence is outside the matrices, the matrices themselves have nice properties, the coefficients are periodic) which make me think it's doable..
The problem comes from the fact that the Pauli matrices do not commute and so an exponential ansatz fails (unlike if one of the matrices was the identity).
The periodicity reminds me of the Floquet thm./analysis, but I haven't found any statement towards solvability in there so far. It only says that the solution can be written in the form
[tex]\vec x(t) = Q(t)e^{tR}\vec x_0[/tex]
for some matrix-valued function Q(t) and a constant matrix R. However, when I plug it in the equation above it doesn't help much, mainly because of the a priori non-commutativity of Q and R (even if I assume they are both invertible for all t).
I tried some ways to rewrite the system in a different way, but it only made it more complicated, since matrix-valued functions appeared, where the time dependence cannot be separated.
Also tried to use the fact that w = 1+v but the non-commutativity problem remains and so I doubt it could be useful.
Any ideas or hints are well appreciated. If someone had this or similar problem before and would like to share their experience would be perfect :)
Thanks a lot,
marin