Bloch Theorem [for Reference Purpose]

Bloch Theorem [For Reference Purpose]

Bloch Theorem

Bloch theorem is a mathematical statement of an electron wave function moving in a perfectly periodic potential. These functions are called bloch functions.

Explanation

Let us consider an electron moving in a periodic potential.

The one dimensional Schroedinger wave equation for an electron moving in a periodic potential shall be written as

............(1)

Suppose the electron moves along X-direction in a one dimensional crystal, then the potential energy of the electron should satisfy the condition

V (x) = V (x+a) ............(2)

Where a is the periodicity of the potential.

The solution for equation (1) is

Ψ (x) = e^ikx u_k (x) .....(3)

where

u_k (x) = u_k (x + a) .....(4)

Here e^ikx represents the plane wave and u_k(x) represents the periodic function.

Equation (3) is called Bloch theorem and Equation (4) is called Bloch function.

Proof

If equation (1) has the solutions with the property of equation (2), we can write the property of the Bloch functions i.e., equation (3) as

Ψ (x + a) = e^{ik (x + a)} u_k (x + a)

(or) Ψ (x+ a) = e^ikx  e^ika. u_k (x + a)

Since u_k(x+ a) = u_k(x), we can write the above equation as

Ψ (x + a) = e^ikx e^ika. u_k (x)

Since Ψ (x) = e^ikx u_k(x), we can write the above equation as

Ψ (x+ a) = e^ika Ψ (x)

(or) Ψ  (x+ a) =QΨ (x)

Where Q = e^ika

If Ψ (x) is a single-valued function, then

we can write Ψ  (x) = Ψ  (x+ a) Thus Bloch theorem is proved.