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Engineering Physics by S. Mani Naidu

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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:

images

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

ψ(x) = eiKx uk(x) _____ (5.61)

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

Equation (5.62) represents periodic function and eikx 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)] uk(x + a) = ψ(x) exp ika _____ (5.63)

or ψ(x + a) =

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