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:
[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 . 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:
[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 . 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:

[tex]\frac{\partial}{\partial t}\psi(x,t) = i\sigma \frac{\partial^2}{\partial x^2}\psi(x,t)[/tex]

 

[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:

[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.

What might you think be the first step?

 

[blue_raver22]

I can understand your frustration with this problem. However, as you mentioned, this is a common type of problem in quantum mechanics known as a boundary value problem (BVP). BVPs involve finding a solution to a differential equation that satisfies certain boundary conditions at the edges of the domain. In this case, the boundary conditions specify that the wave function, represented by \psi(x,t), must equal zero at both x=0 and x=L(t). This is because in quantum mechanics, the wave function represents the probability amplitude of finding a particle at a certain position, and it must go to zero at the boundaries to ensure that the particle cannot exist outside of the specified domain.

To begin solving this problem, you can use standard techniques for solving partial differential equations (PDEs), such as separation of variables or the method of characteristics. These methods involve breaking down the equation into simpler parts and solving them individually, then combining the solutions to get the overall solution. In this case, you can start by separating the time and space variables in the PDE and solving for \psi(x,t) using the given initial condition. Then, you can use the boundary conditions to determine the unknown function L(t) and complete the solution.

It is important to note that BVPs in quantum mechanics are often more complex than standard PDEs due to the presence of complex numbers and the need for the wave function to be normalized (i.e. have a total probability of 1). Therefore, it may be helpful to seek guidance from a professor or textbook that specializes in quantum mechanics or PDEs. With persistence and the right tools, you will be able to successfully solve this problem and gain a deeper understanding of quantum mechanics.