Editing Phonon (section)

====Classical treatment====
The forces between the atoms are assumed to be linear and nearest-neighbour, and they are represented by an elastic spring. Each atom is assumed to be a point particle and the nucleus and electrons move in step ([[adiabatic theorem]]):

::::::::''n'' − 1 {{pad|1em}} ''n'' {{pad|2em}} ''n'' + 1 {{pad|5em}} ← {{pad|1em}} ''a'' {{pad|1em}} →
···o++++++o++++++o++++++o++++++o++++++o++++++o++++++o++++++o++++++o···
::::::::→→{{pad|2em}}→{{pad|2em}}→→→
::::::::''u''<sub>''n'' − 1</sub>{{pad|2em}}''u<sub>n</sub>''{{pad|2em}}''u''<sub>''n'' + 1</sub>
where {{mvar|n}} labels the {{mvar|n}}th atom out of a total of {{mvar|N}}, {{mvar|a}} is the distance between atoms when the chain is in equilibrium, and {{math|''u<sub>n</sub>''}} the displacement of the {{mvar|n}}th atom from its equilibrium position.

If ''C'' is the elastic constant of the spring and {{mvar|m}} the mass of the atom, then the equation of motion of the {{mvar|n}}th atom is
:<math>-2Cu_n + C\left(u_{n+1} + u_{n-1}\right) = m\frac{d^2u_n}{dt^2} .</math>
This is a set of coupled equations.

Since the solutions are expected to be oscillatory, new coordinates are defined by a [[discrete Fourier transform]], in order to decouple them.<ref>{{cite book|last=Mattuck |first=R. |title=A guide to Feynman Diagrams in the many-body problem|date=1976 |publisher=McGraw-Hill |isbn=9780070409545 |url=https://archive.org/details/guidetofeynmandi0000matt |url-access=registration }}</ref>

Put
:<math>u_n = \sum_{Nak/2\pi=1}^N Q_k e^{ikna}.</math>
Here, {{math|''na''}} corresponds and devolves to the continuous variable {{mvar|x}} of scalar field theory. The {{math|''Q<sub>k</sub>''}} are known as the ''normal coordinates'' for continuum field modes <math>\phi_k = e^{ikna}</math> with <math>k = 2\pi j/(Na)</math> for <math>j=1\dots N</math>.

Substitution into the equation of motion produces the following ''decoupled equations'' (this requires a significant manipulation using the orthonormality and completeness relations of the discrete Fourier transform),<ref>{{cite book  |title= Theoretical Mechanics of Particles and Continua| publisher=Dover Books on Physics |
  first1=Alexander|last1=Fetter|first2= John|  last2=Walecka |isbn= 978-0486432618| date=2003-12-16 }}</ref>
: <math> 2C(\cos {ka-1})Q_k = m\frac{d^2Q_k}{dt^2}.</math>

These are the equations for decoupled  [[harmonic oscillators]] which have the solution

:<math>Q_k=A_ke^{i\omega_kt};\qquad \omega_k=\sqrt{ \frac{2C}{m}(1-\cos{ka})}.</math>

Each normal coordinate ''Q<sub>k</sub>'' represents an independent vibrational mode of the lattice with wavenumber {{mvar|k}}, which is known as a [[normal mode]].

The second equation, for {{math|''ω<sub>k</sub>''}}, is known as the [[dispersion relation]] between the [[angular frequency]] and the [[wavenumber]].

In the [[continuum limit]], {{mvar|a}}→0,  {{mvar|N}}→∞, with {{math|''Na''}} held fixed, {{math|''u<sub>n</sub>''}} → {{math|''φ''(''x'')}}, a scalar field, and  <math> \omega(k)  \propto k a</math>. This amounts to classical  free [[scalar field theory]], an assembly of independent oscillators.