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 . 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 . 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.

What might you think be the first step?