quantum mechanics - Periodic boundary condition on a Wave Function of a Particle in a Box

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

$$Y(x)= A\sin{kx} + B\cos{kx}.$$

Applying the conditions above, I get

$$Y(x)= A\sin{kx}\quad \text{where}\quad k=\pi n/L$$

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:

$$\Psi(x)=\Psi(x+L)$$

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:

$$\Psi(x)\propto e^{ikx}$$

Hopefully that should get you off the starting block.