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## Main Question or Discussion Point

If we have an infinite square well, I can follow the usual solution in Griffiths but I now want to impose periodic boundary conditions. I have

[tex]\psi(x) = A\sin(kx) + B\cos(kx)[/tex]

with boundary conditions [tex]\psi(x) = \psi(x+L)[/tex]

In the fixed boundary case, we had [tex]\psi(0) = 0[/tex] which meant [tex]B=0[/tex] and [tex]\psi(L)=0[/tex] which allows discrete values of k. I'm a little stuck with how to proceed with this in the periodic boundary condition case.

I think that [tex]k = n\pi/L[/tex] must still be true to satisfy the boundary condition (though I'm unable prove it). But now, I think that negative n also matter and they're different to the positive n case.

What is the general wavefunction solution in this case?

[tex]\psi(x) = A\sin(kx) + B\cos(kx)[/tex]

with boundary conditions [tex]\psi(x) = \psi(x+L)[/tex]

In the fixed boundary case, we had [tex]\psi(0) = 0[/tex] which meant [tex]B=0[/tex] and [tex]\psi(L)=0[/tex] which allows discrete values of k. I'm a little stuck with how to proceed with this in the periodic boundary condition case.

I think that [tex]k = n\pi/L[/tex] must still be true to satisfy the boundary condition (though I'm unable prove it). But now, I think that negative n also matter and they're different to the positive n case.

What is the general wavefunction solution in this case?