1D Model of a Crystal Derivation

We are working with the periodic potential

Our states have positive energy. This potential has the symmetry that a translation by the lattice spacing leaves the problem unchanged. The probability distributions must therefore have this symmetry

which means that the wave function differs by a phase at most.

The general solution in the region
is

Now lets look at the boundary conditions at . Continuity of the wave function gives

The discontinuity in the first derivative is

Substituting from the first equation

Plugging this equation for back into the equation above for we get

We now have two pairs of equations for the coefficients in terms of the coefficients.

Using the second pair of equations to eliminate the coefficients, we have

Now we can eliminate all the coefficients.

Multiply by .

This relation puts constraints on , like the constraints that give us quantized energies for bound states. Since can only take on values between -1 and 1, there are allowed bands of and gaps between those bands.

Jim Branson 2013-04-22