Editing Phonon (section)

===One-dimensional lattice===
[[File:1D normal modes (280 kB).gif|class=skin-invert-image|thumb|upright=1.5|Animation showing 6 normal modes of a one-dimensional lattice: a linear chain of particles.  The shortest wavelength is at top, with progressively longer wavelengths below. In the lowest lines the motion of the waves to the right can be seen.]]

In order to simplify the analysis needed for a 3-dimensional lattice of atoms, it is convenient to model a 1-dimensional lattice or linear chain. This model is complex enough to display the salient features of phonons.

====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.

====Quantum treatment====
A one-dimensional quantum mechanical harmonic chain consists of ''N'' identical atoms. This is the simplest quantum mechanical model of a lattice that allows phonons to arise from it. The formalism for this model is readily generalizable to two and three dimensions.

In contrast to the previous section, the positions of the masses are not denoted by <math>u_i</math>, but instead by <math>x_1,x_2,\dots</math> as measured from their equilibrium positions. (I.e. <math>x_i=0</math> if particle <math>i</math> is at its equilibrium position.) In two or more dimensions, the <math>x_i</math> are vector quantities. The [[Hamiltonian (quantum mechanics)|Hamiltonian]] for this system is

:<math>\mathcal{H} = \sum_{i=1}^N \frac{p_i^2}{2m} + \frac{1}{2} m\omega^2 \sum_{\{ij\} (\mathrm{nn})} \left(x_i - x_j\right)^2</math>

where ''m'' is the mass of each atom (assuming it is equal for all), and ''x<sub>i</sub>'' and ''p<sub>i</sub>'' are the position and [[momentum]] operators, respectively, for the ''i''th atom and the sum is made over the nearest neighbors (nn). However one expects that in a lattice there could also appear waves that behave like particles. It is customary to deal with [[wave]]s in [[Fourier space]] which uses [[normal modes]] of the [[wavevector]] as variables instead of coordinates of particles. The number of normal modes is the same as the number of particles. Still, the Fourier space is very useful given the [[Fourier series|periodicity]] of the system.

A set of ''N'' "normal coordinates" ''Q<sub>k</sub>'' may be introduced, defined as the [[discrete Fourier transform]]s of the ''x<sub>k</sub>'' and ''N'' "conjugate momenta" ''Π<sub>k</sub>'' defined as the Fourier transforms of the ''p<sub>k</sub>'':

:<math>\begin{align} Q_k &= \frac{1}\sqrt{N} \sum_{l} e^{ikal} x_l \\ \Pi_{k} &= \frac{1}\sqrt{N} \sum_{l} e^{-ikal} p_l. \end{align}</math>

The quantity ''k'' turns out to be the [[wavenumber]] of the phonon, i.e. 2{{pi}} divided by the [[wavelength]].

This choice retains the desired commutation relations in either real space or wavevector space

: <math> \begin{align} 
\left[x_l , p_m \right]&=i\hbar\delta_{l,m} \\ 
\left[ Q_k , \Pi_{k'} \right] &=\frac{1}N \sum_{l,m} e^{ikal} e^{-ik'am} \left[x_l , p_m \right] \\
 &= \frac{i \hbar}N \sum_{l} e^{ial\left(k-k'\right)} = i\hbar\delta_{k,k'} \\
\left[ Q_k , Q_{k'} \right] &= \left[ \Pi_k , \Pi_{k'} \right] = 0
\end{align}</math>

From the general result

: <math> \begin{align} 
\sum_{l}x_l x_{l+m}&=\frac{1}N\sum_{kk'}Q_k Q_{k'}\sum_{l} e^{ial\left(k+k'\right)}e^{iamk'}= \sum_{k}Q_k Q_{-k}e^{iamk} \\ 
\sum_{l}{p_l}^2 &= \sum_{k}\Pi_k \Pi_{-k}
\end{align}</math>

The potential energy term is
: <math> \tfrac12 m \omega^2 \sum_{j} \left(x_j - x_{j+1}\right)^2= \tfrac12 m\omega^2\sum_{k}Q_k Q_{-k}(2-e^{ika}-e^{-ika})= \tfrac12 \sum_{k}m{\omega_k}^2Q_k Q_{-k}</math>
where

:<math>\omega_k = \sqrt{2 \omega^2 \left( 1 - \cos{ka} \right)} = 2\omega\left|\sin\frac{ka}2\right|</math>

The Hamiltonian may be written in wavevector space as
:<math>\mathcal{H} = \frac{1}{2m}\sum_k \left( \Pi_k\Pi_{-k} + m^2 \omega_k^2 Q_k Q_{-k} \right)</math>

The couplings between the position variables have been transformed away; if the ''Q'' and ''Π'' were [[Hermitian operator|Hermitian]] (which they are not), the transformed Hamiltonian would describe ''N'' uncoupled harmonic oscillators.

The form of the quantization depends on the choice of boundary conditions; for simplicity, ''periodic'' boundary conditions are imposed, defining the (''N''&nbsp;+&nbsp;1)th atom as equivalent to the first atom. Physically, this corresponds to joining the chain at its ends. The resulting quantization is

:<math>k=k_n = \frac{2\pi n}{Na} \quad \mbox{for } n = 0, \pm1, \pm2, \ldots \pm \frac{N}2 .\ </math>

The upper bound to ''n'' comes from the minimum wavelength, which is twice the lattice spacing ''a'', as discussed above.

The harmonic oscillator eigenvalues or energy levels for the mode ''ω<sub>k</sub>'' are:

:<math>E_n = \left(\tfrac12+n\right)\hbar\omega_k \qquad n=0,1,2,3 \ldots</math>

The levels are evenly spaced at:
 
:<math>\tfrac12\hbar\omega , \ \tfrac32\hbar\omega ,\ \tfrac52\hbar\omega \ \cdots</math>

where {{sfrac|1|2}}''ħω'' is the [[zero-point energy]] of a [[quantum harmonic oscillator]].

An '''exact''' amount of [[energy]] ''ħω'' must be supplied to the harmonic oscillator lattice to push it to the next energy level.  By analogy to the [[photon]] case when the [[electromagnetic field]] is quantized, the quantum of vibrational energy is called a phonon.

All quantum systems show wavelike and particlelike properties simultaneously. The particle-like properties of the phonon are best understood using the methods of [[second quantization]] and operator techniques described later.<ref name="Mahan">{{cite book|last=Mahan|first=G. D.|title=Many-Particle Physics|publisher=Springer|location=New York|isbn=978-0-306-46338-9|year=1981}}</ref>

{{see also|Canonical quantization#Real scalar field}}