# PDE - Boundary value problem found in QM

kostas230
This is a quantum mechanics problem, but the problem itself is reduced (naturally) to a differential equations problem.

I have to solve the following equation:
$$\frac{\partial}{\partial t}\psi (x,t) = i\sigma \psi (x,t)$$
where $\sigma > 0$
The initial condition is:
$$\psi (x,0) = \sqrt{\frac{2}{L_0}}\sin \left(\frac{n\pi x}{L_0}\right)$$
and the boundary conditions are:
$$\psi (0,t) = \psi(L(t),t) = 0$$
where $L(t)$ is a smooth function with $L(0)=L_0$.
I don't even know how to begin :uhh:. Any ideas?

Mandelbroth
This is a quantum mechanics problem, but the problem itself is reduced (naturally) to a differential equations problem.

I have to solve the following equation:
$$\frac{\partial}{\partial t}\psi (x,t) = i\sigma \psi (x,t)$$
where $\sigma > 0$
The initial condition is:
$$\psi (x,0) = \sqrt{\frac{2}{L_0}}\sin \left(\frac{n\pi x}{L_0}\right)$$
and the boundary conditions are:
$$\psi (0,t) = \psi(L(t),t) = 0$$
where $L(t)$ is a smooth function with $L(0)=L_0$.
I don't even know how to begin :uhh:. Any ideas?
$$\frac{\partial}{\partial t}[e^{i\sigma t}]=i\sigma e^{i\sigma t}.$$

kostas230
Oh, wait I forgot to put the space derivative (it was 1-2 AM so my cognitive functions were down to minimum) :shy:

The correct equation is:

$$\frac{\partial}{\partial t}\psi(x,t) = i\sigma \frac{\partial^2}{\partial x^2}\psi(x,t)$$

Mandelbroth
Oh, wait I forgot to put the space derivative (it was 1-2 AM so my cognitive functions were down to minimum) :shy:

The correct equation is:

$$\frac{\partial}{\partial t}\psi(x,t) = i\sigma \frac{\partial^2}{\partial x^2}\psi(x,t)$$
My hint still helps in this case. :tongue:

What might you think be the first step?