PDE - Boundary value problem found in QM

  • Thread starter kostas230
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  • #1
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Main Question or Discussion Point

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

Answers and Replies

  • #2
612
23
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:
[tex]\frac{\partial}{\partial t}\psi (x,t) = i\sigma \psi (x,t)[/tex]
where [itex]\sigma > 0[/itex]
The initial condition is:
[tex]\psi (x,0) = \sqrt{\frac{2}{L_0}}\sin \left(\frac{n\pi x}{L_0}\right)[/tex]
and the boundary conditions are:
[tex]\psi (0,t) = \psi(L(t),t) = 0[/tex]
where [itex]L(t)[/itex] is a smooth function with [itex]L(0)=L_0[/itex].
I don't even know how to begin :uhh:. Any ideas?
$$\frac{\partial}{\partial t}[e^{i\sigma t}]=i\sigma e^{i\sigma t}.$$
 
  • #3
96
3
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:

[tex]\frac{\partial}{\partial t}\psi(x,t) = i\sigma \frac{\partial^2}{\partial x^2}\psi(x,t)[/tex]
 
  • #4
612
23
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:

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

What might you think be the first step?
 

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