A simplified picture of the periodic potential is given in Fig. 3.5

It is quite similar to the finite potential analysis

periodicity = a + b

, where k is continuous

We will first consider the case: 0 < E < U 0

Inside well wave function =

Outside well wave function =

Schrödinger equation for 0 < x < a:

Schrödinger equation for -b < x < 0:

The general solutions to equations (1) and (2) are:

Now applying the continuity condition on wave function and its derivative at x = 0

Fig. 3.6

continuity requirement

Note that   x = -b is the same boundary as that of     x = a.

Now wave function and its derivative must observe Bloch Theorem

With

periodicity requirements.

Applying Boundary and Periodicity conditions

Eliminating and by using first two equations in last two equations

   2 eqns in 2 unknown constants. For non trivial values of , the determinant formed from coefficient should be equal to zero.