# Oscillator applications

by Mr confusion
Tags: applications, oscillator
 P: 45 Almost any perturbation of a physical system can be represented as a superposition of so-called natural modes. Each of these modes (in the linear approximation) behaves like an independent linear oscillator. In fact the Hamiltonian of the perturbation is equal to a sum of oscillator quadratic Hamiltonians: $$\hat{H} = A\sum_\alpha \left(\frac{\hat{p}^2}{2} + \frac{\omega_\alpha^2 q^2}{2}\right).$$ The energy levels of an oscillator are equidistant: $$E_n = \hbar\omega_\alpha(n+1/2),$$ so we can consider it to be a set of some "particles". If the oscillator is in it's ground state (E=E0) it contains no "particles". If E=E1 there is one "particle" and so on. When we consider electromagnetic field in a cavity the "particles" are called photons. This is the application in electrodynamics. The "particles" of acoustic oscillations in solids are called phonons. This is the application in solid-state physics. For more detailed information you can refer to 1) R. P. Feynman, Statistical Mechanics 2) Any other book where the problems of phonons in solids or electromagnetic field quantization are discussed.