Physics for Information Science: Unit I: Electrical Properties of Materials
Bloch Theorem [for Reference Purpose]
Definition, Explanation, Proof, Equation | Electrical Properties of Materials
Bloch theorem is a mathematical statement of an electron wave function moving in a perfectly periodic potential. These functions are called bloch functions.
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)
Where a is the periodicity of the potential.
The
solution for equation (1) is
Ψ
(x) = eikx uk (x) .....(3)
where
uk
(x) = uk (x + a) .....(4)
Here eikx represents the plane wave and uk(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) = eik (x + a) uk (x + a)
(or) Ψ (x+ a) = eikx eika. uk (x + a)
Since
uk(x+ a) = uk(x), we can write the
above equation as
Ψ (x + a) = eikx eika. uk (x)
Since
Ψ (x) = eikx uk(x), we can write the above equation as
Ψ
(x+ a)
= eika Ψ (x)
(or) Ψ (x+ a) =QΨ (x)
Where Q = eika
If
Ψ (x) is a single-valued function, then
we
can write Ψ (x) = Ψ (x+ a) Thus Bloch theorem
is proved.
Physics for Information Science: Unit I: Electrical Properties of Materials : Tag: : Definition, Explanation, Proof, Equation | Electrical Properties of Materials - Bloch Theorem [for Reference Purpose]
Physics for Information Science: Unit I: Electrical Properties of Materials
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