Eigen value equation

We get the following eigen value equation by simplifying the determinant:

Now, say: (system constant)

   (normalized energy)

For

Putting these values in our eigen value equation:

Now consider the 1D crystal lattice.

Fig 3.3

does not necessarily mean that the electron is outside the crystal lattice. Therefore, we can also consider the case of

here.

For

Putting these values in the eigen value equation:

Now we assumed infinitely long lattice in K-P model. Therefore, k can take any value.

RHS of eigen value eqn. can thus take any value between +1 and -1. The LHS with value between +1 and -1 gives allowed solutions for .

The plot of as a function of is shown in Fig 3.7

Note that the allowed values of    are shown by the shaded regions where . Therefore there are allowed bands of i,e. allowed bands of E(energy bands).