How does one solve the Schroedinger equation for a 2D, time-dependent harmonic potential?

This is the Schroedinger equation with a particular 2D harmonic potential:

$$\begin{multline}i\hbar\frac{\partial}{\partial t}\Psi(x_1,x_2,t) = \\ \Biggl[-\frac{\hbar^2}{2m}\nabla^2 + \frac{1}{2} m\omega^2\Biggl(\biggl(x_1 - \frac{x_0(t)}{2}\biggr)^2 + \biggl(x_2 + \frac{x_0(t)}{2}\biggr)^2\Biggr)\Biggr]\Psi(x_1,x_2,t)\end{multline}$$

Can anyone please tell me what the upside down triangle means? I know its the second derivative, but since the problem I have is of variables $x_1$, $x_2$, and $t$, do I take the second derivative of time? Or does that upside down triangle only does it for $x_1$ and $x_2$?

$x_0$ is the separation distance between $x_1$ and $x_2$ and its a function of time because the two double HO potential wells are moving apart from each other.

Also, how would I go about doing separation of variables here? Would I move all the time pieces to one side and all the x pieces to the other side?

Won't I have to do separation of variables twice? Where the second time is when I have to separate $x_1$ and $x_2$?

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Tip: you can use LaTeX markup for math. – Emilio Pisanty Sep 13 '12 at 16:31
I think this would be more appropriate on Mathematics - any thoughts from anyone else? – David Z Sep 13 '12 at 17:50
I think it's fine here. We'd like undergrads with this kinds of questions on the Schrödinger equation to come here, right? – Emilio Pisanty Sep 14 '12 at 9:08

The symbol $\nabla^2$ usually means the sum of second derivatives if you are in a cartesian-like system. Thus for your case $$\nabla^2=\frac{\partial^2}{\partial x^2_1}+\frac{\partial^2}{\partial x^2_2}.$$ To do separation of variables, you usually postulate separate dependences on each of the variables, $$\Psi(x_1,x_2,t)=T(t)\psi_1(x_1)\psi_2(x_2).$$ However, the time-dependence of the well centres will make matters rather more difficult, and as it stands all you can hope to do is a separation of the type $$\Psi(x_1,x_2,t)=\psi_1(x_1,t)\psi_2(x_2,t).$$
If $x_0=x_2-x_1$, then it is time-independent. If it's the well centres that move apart from each other, then all you can hope for is the above.
The real question is whether $x_0$ is a number or a coordinate. What are $v_1$ and $v_2$? Since the $x_i$ are coordinates, it is not usually the case that $\dot{x}_i=v_i$. – Emilio Pisanty Sep 14 '12 at 9:11