Until now, solving the Schrodinger Equation for a particle in a box was relatively easy because the boundaries conditions imposed zero value on the wave function at the boundaries. But now I must find the normalized wave function of the same problem imposing just these periodic boundaries conditions:
Y(x,y,z)=Y(x+L,y,z),Y(x,y,z)=Y(x,y+L,z),Y(x,y,z)=Y(x,y,z+L).
I got stuck in the normalization process. Before, using the boundary condition (one dimension) Y(0)=Y(L)=0, I could get just one constant to solve for, in
Y(x)=Asinkx+Bcoskx.
Applying the conditions above, I get
Y(x)=Asinkxwherek=πn/L
which is easy normalize. But now, with this periodic boundary condition, Y(0)=Y(L). How could I find it?
Answer
When imposing a periodic boundary condition, the amplitude of the wavefunction at coordinate x must match that at coordinate x+L, so we have:
Ψ(x)=Ψ(x+L)
In your previous 'particle in a box' scenario, you mention that the general form of the wavefunction is given by a linear combination of sine and cosine with complex coeficients. It might be helpful to remember that this can also be expressed as an exponential with the form:
Ψ(x)∝eikx
Hopefully that should get you off the starting block.
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