Bloch's Theorem for Particles in a Periodic Potential

BLOCH'S THEOREM FOR PARTICLES IN A PERIODIC POTENTIAL

The motion of electron inside the lattice is not free as expected, but the electron experiences a periodic potential variation. The potential energy on the electron is maximum between adjacent ions and gradually decrease as the electron moves towards ions as shown in fig.7.20.

Bloch Theorem

It is a mathematical statement regarding the form of one electron wave function for a perfectly periodic potential.

Statement

If an electron in a linear lattice of lattice constant 'a' characterised by potential function V(x) = V(x + a) satisfies the Schrodinger equation

then the wave functions Ψ(x) of electron (with energy E) is obtained as a solution of Schrodinger equation are of the form

Here uk (x) is also periodic with lattice periodicity.

The potential V(x) is periodic as V(x) = V(x + a) where a is a lattice constant.

From the Block theorem, we can say that the free electron is modulated by the periodic function

In other words the solutions are plane waves modulated by the function uk (x) which has the same periodicity as the lattice. This theorem is known as Bloch Theorem. The functions of the type (2) are called Bloch functions.

Proof

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

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

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

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

This equation is similar to that of eqn (2) and eqn (4) i.e., If the potential is a function of 'x' and 'a', then the wave function is also a function of ‘x' and 'a'.