Engineering Physics

5.8 Bloch theorem

Suppose an electron passes along X-direction in a one-dimensional crystal having periodic potentials:

V(x) = V (x + a)

where ‘a’ is the periodicity of the potential. The Schrödinger wave equation for the moving electron is:

The solution for Equation (5.60) is of the form:

ψ(x) = e^iKx u_k(x) _____ (5.61)

where u_k(x) = u_k(x + a) _____ (5.62)

Equation (5.62) represents periodic function and e^ikx represents plane wave. The above statement is known as Bloch theorem and Equation (5.62) is called Block function. The Bloch function has the property:

ψ(x + a) = exp [ik (x + a)] u_k(x + a) = ψ(x) exp ika _____ (5.63)

or ψ(x + a) = Qψ