# Stochastic Schrödinger Equation

I have a stochastic coupled Schrödinger equation to solve.

$$i\frac{\mathrm d X_k(t)}{\mathrm dt}=-\left(x_{k+1}(t)+x_{k-1}(t)\right)+V_k x_k(t)+\eta_k t x_k(t)$$

where $\left\langle\eta_k(t)\eta_j(t^\prime)\right\rangle=\delta_{k,j}\theta(\tau-|t-t^\prime|)$ and $\theta(t)$ is Heaviside theta.

It is a coupled equation of $n$ variables and $n$ could be any number. In my case $n$ is typically 100-200.So there are 100-200 coupled differential equations. $\eta_k(t)$ is a random function of $\mathrm{time}(t)$ which is, in my case a gaussian random variable corresponding to the variable $x_k$. Different $\eta_k$ are unrelated. $V_k$ is a fixed number corresponding to the variable $X_k$. I don't know how to solve this stochastic differential equation.Can anybody help me?

• Can you write down your equations?. What have you tried with Mathematica? – Enrique Pérez Herrero Oct 25 '15 at 17:49
• Well start small. Do you know how to write the code for n=1? If not, your question is actually about solving SDEs in general. – acl Oct 25 '15 at 19:20
• There was a picture OP forgot to include. I transcribed the equations in it as best as I can. – J. M.'s technical difficulties Oct 26 '15 at 0:45
• Is this even about Mathematica? – m_goldberg Oct 26 '15 at 2:35
• " I don't know how to solve this stochastic differential equation.Can anybody help me?" There is a book: Risken, H. The Fokker-Plank Equation. Methods of Solution and Applications. 2 edn, (Springer-Verlag, 1989). There it is written, how to deal with them. Have fun! – Alexei Boulbitch Oct 26 '15 at 8:42