Actual vibration

If we associate a vibration mode of wave number q

quantum mechanical wave function

Schrödinger equ. For a harmonic oscillator each mode of vibration different energy state with energy give by

particles Picture of lattice vibrations

Each mode of vibration number of particles each with energy

State of vibration changes by creation or annihilation of such particles

q = Pseudo momentum

because of periodicity

Thus the vibration of the lattice with a wave vector collection of particles called phonons which associated by the energy and pseudo momentum . At a particular temp. Lattice atoms execute random vibrations and phonons of various energies & characteristics would be present.

Hamiltonian of harmonic oscillator

Hamiltonian of lattice system is

and normalized is

Kronig penney model with stationary lattice atoms is not entirely accurate as lattice vibration Hamiltonian of vibrating lattice + electrons

individual electrons and lattice atoms

The eigen value equation is

eigenvalues are modified energy instant due to changing position of lattice atoms

is the periodic component of Block function out of perturbed lattice.