Bloch Theorem-Solid State Physics-Lecture Slides, Slides of Solid State Physics

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In The Name of ALLAH

Bloch Theorem

• The Schrodinger wave equation for an electron moving in a constant potential ‘Vo’ is;
• (1 and the solution is
• (2 where

• There is an important theorem known as the Bloch theorem or ‘Floquet’s theorem’
• According to this theorem, the solution of eq. 4 is plane wave, which is modulated by a function uk(x) having the same periodicity as that of lattice

• Thus the solution can be written as;
• Also uk(x) = uk(x + a) (
• The wave function of the eq. 5 is called Bloch Function

• Since V(x) = V(x + a) then the functions g(x + a) and f(x + a) must be the solutions of eq. 4 then,
• Where α 1 , α 2 , β 1 and β 2 are real functions of energy ‘E’

• Another solution of eq. 8 is given as:
• If the A and B constants are chosen such that
• Where λ is a constant, we can find and write

• If , then g(x) and f(x) are the real and independent solution of eq. 4, we get
• Multiplying eq 15 by g(x) and 16 by f(x), then

• By subtracting, we get;
• The left-hand side of eq. 18 is called Wronskian function W(x) of the solution

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