Bloch's Theorem with proof - Engineering Physics

Bloch Theorem

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

i.e 

 ...(1)

The Schrodinger's equation for free-electron moving in a constant potential V₀ is given as  

 ...(2)

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

 ...(3)

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

 ...(4)

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

 i.e. 

 ...(5)

Hence 

 i.e.,  

or 

 ...(6)

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

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

 ...(8)

Whereandare functions of E.

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

 ...(9)

 ...(10)

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

 ...(11)

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

 ...(12)

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

 ...(13a)

 ...(13b)

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

Or  ...(14)

Or  ...(15)

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.

Case 1. For energy range 

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 

and 

i.e., in general 

 ...(16)

Case 2. For energy range

In this case the roots are real and will be reciprocals of each other these correspond to wave functions of the type 

And 

Where 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.

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How?

As f(x) and g(x) are solutions of Schrodinger eq. (3), we have 

Multiply f eq. by g and eq. g by f and then substracting, we get 

Or 

i.e the wronskian

Now from eq. (7) and (8), we may write 

Hence by combining above two equations, we get 

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