__Bloch Theorem__The band theory of
solid assume that electron move in a periodic potential of the period ‘a’ (lattice
constant).

i.e

The Schrodinger's equation for free-electron moving in a constant potential V_{0 }is
given as

So the Schrodinger's equation for an electron moving in periodic potential V(x) is written as

The solution of Eq. (3) are of the type

Where is known as Bloch function and which is again
periodic with the same periodicity of the lattice.

Where

Thus Bloch Theorem is a
mathematical statement regarding the form of the one-electron wave function for
a perfectly periodic potential.

*Proof - *

We know that
Schrodinger wave eq. (3) is a second-order differential eq. and hence there
exist only two real independent solutions for this equation.

Say *f(x)**g(x)* any other solution will be simply a linear
combination of independent solutions.

*f(x+a)*and

*g(x+a)*are to be solutions of the eq. (3), there can be written as ...(7)

Let be another solution of eq. (3), hence it must be represented as

By using eq. (7) and
(8), above eq. become

Now, by using eq.
(6), we can write eq. (9) as

Now, by comparing eq.
(11) and (12), we get

Now eq. (13) have
non-vanishing solutions for A and B if and only if the determinant of the coefficient is zero, i.e.,

We have written in eq. (15) and this result is proved below
this topic.

Now, eq. (15) gives
two roots of,
so there are two wave functions and. It must be noted that and is a real function of energy E.

Let us consider two
cases for the energy ranges.

In this case the two roots will be the complex conjugate of each other, we can write and Where K is real and the corresponding wave functions and can be written as

i.e., in general

**For energy range**

__Case 2.__In this case the roots are real and will be reciprocals of each other these correspond to wave functions of the type

AndWhere is real. These are not acceptable wave
functions since these are not
bounded and approaches when approaches infinity. Thus we find that the energy spectrum of an electron in a periodic potential consists of allowed and
forbidden energy regions as bands. This will further be discussed in
Kroning-Penney Model.

As *f(x)**g(x)*

Multiply *f**g**g**f*

i.e the wronskian

*For The Kronig-Penney model - Click Here*
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