Editing Phonon (section)

==Lattice dynamics==
The equations in this section do not use [[axiom]]s of quantum mechanics but instead use relations for which there exists a direct [[correspondence principle|correspondence]] in classical mechanics.

For example: a rigid regular, [[crystalline]] (not [[amorphous solid|amorphous]]) lattice is composed of ''N'' particles. These particles may be atoms or molecules. ''N'' is a large number, say of the order of 10<sup>23</sup>, or on the order of the [[Avogadro number]] for a typical sample of a solid. Since the lattice is rigid, the atoms must be exerting [[force]]s on one another to keep each atom near its equilibrium position. These forces may be [[Van der Waals force]]s, [[covalent bond]]s, [[electrostatic attraction]]s, and others, all of which are ultimately due to the [[electric field|electric]] force. [[magnetism|Magnetic]] and [[gravity|gravitational]] forces are generally negligible. The forces between each pair of atoms may be characterized by a [[potential energy]] function ''V'' that depends on the distance of separation of the atoms. The potential energy of the entire lattice is the sum of all pairwise potential energies multiplied by a factor of 1/2 to compensate for double counting:<ref name=latticemechanics3>{{cite book
 | last = Krauth| first = Werner| title =Statistical mechanics: algorithms and computations 
 | publisher =Oxford University Press| date =April 2006| location =International publishing locations| pages =231–232 
 | url =https://books.google.com/books?id=EnabPPmmS4sC&q=Mechanics+of+particles+on+a+lattice&pg=RA1-PA231| isbn =978-0-19-851536-4}}</ref>

:<math>\frac12\sum_{i \neq j} V\left(r_i - r_j\right)</math>

where ''r<sub>i</sub>'' is the [[space|position]] of the ''i''th atom, and ''V'' is the potential energy between two atoms.

It is difficult to solve this [[many-body problem]] explicitly in either classical or quantum mechanics. In order to simplify the task, two important [[approximation]]s are usually imposed. First, the sum is only performed over neighboring atoms. Although the electric forces in real solids extend to infinity, this approximation is still valid because the fields produced by distant atoms are effectively [[electric field screening|screened]]. Secondly, the potentials ''V'' are treated as [[harmonic oscillator|harmonic potentials]]. This is permissible as long as the atoms remain close to their equilibrium positions. Formally, this is accomplished by [[Taylor series|Taylor expanding]] ''V'' about its equilibrium value to quadratic order, giving ''V'' proportional to the displacement ''x''<sup>2</sup> and the elastic force simply proportional to ''x''. The error in ignoring higher order terms remains small if ''x'' remains close to the equilibrium position.

The resulting lattice may be visualized as a system of balls connected by springs. The following figure shows a cubic lattice, which is a good model for many types of crystalline solid. Other lattices include a linear chain, which is a very simple lattice which we will shortly use for modeling phonons. (For other common lattices, see [[crystal structure]].)

:[[File:Cubic.svg|class=skin-invert]]
<!-- Unsourced image removed: [[File:Linear crystal shape.png]] -->

The potential energy of the lattice may now be written as

:<math>\sum_{\{ij\} (\mathrm{nn})} \tfrac12 m \omega^2 \left(R_i - R_j\right)^2.</math>

Here, ''ω'' is the [[natural frequency]] of the harmonic potentials, which are assumed to be the same since the lattice is regular. ''R<sub>i</sub>'' is the position coordinate of the ''i''th atom, which we now measure from its equilibrium position. The sum over nearest neighbors is denoted (nn).

It is important to mention that the mathematical treatment given here is highly simplified in order to make it accessible to non-experts. The simplification has been achieved by making two basic assumptions in the expression for the total potential energy of the crystal. These assumptions are that (i) the total potential energy can be written as a sum of pairwise interactions, and (ii) each atom interacts with only its nearest neighbors.  These are used only sparingly in modern lattice dynamics.<ref name=":0">Maradudin, A.; Montroll, E.; Weiss, G.; Ipatova, I. (1971). ''Theory of lattice dynamics in the harmonic approximation''. Solid State Physics. Vol. Supplement 3 (Second ed.). New York: Academic Press.</ref> A more general approach is to express the potential energy in terms of force constants.<ref name=":0" /> See, for example, the Wiki article on [[Multiscale Green's function|multiscale Green's functions.]]

===Lattice waves===
[[File:Lattice wave.svg|class=skin-invert-image|200px|thumb|right|Phonon propagating through a square lattice (atom displacements greatly exaggerated)]]
Due to the connections between atoms, the displacement of one or more atoms from their equilibrium positions gives rise to a set of vibration [[wave]]s propagating through the lattice. One such wave is shown in the figure to the right. The [[amplitude]] of the wave is given by the displacements of the atoms from their equilibrium positions. The [[wavelength]] ''λ'' is marked.

There is a minimum possible wavelength, given by twice the equilibrium separation ''a'' between atoms. Any wavelength shorter than this can be mapped onto a wavelength longer than 2''a'', due to the periodicity of the lattice. This can be thought of as a consequence of the [[Nyquist–Shannon sampling theorem]], the lattice points being viewed as the "sampling points" of a continuous wave.

Not every possible lattice vibration has a well-defined wavelength and frequency. However, the [[normal mode]]s do possess well-defined wavelengths and [[frequency|frequencies]].

===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}}

===Three-dimensional lattice===
This may be generalized to a three-dimensional lattice. The wavenumber ''k'' is replaced by a three-dimensional [[wavevector]] '''k'''. Furthermore, each '''k''' is now associated with three normal coordinates.

The new indices ''s'' = 1, 2, 3 label the [[polarization (waves)|polarization]] of the phonons. In the one-dimensional model, the atoms were restricted to moving along the line, so the phonons corresponded to [[longitudinal wave]]s. In three dimensions, vibration is not restricted to the direction of propagation, and can also occur in the perpendicular planes, like [[transverse wave]]s. This gives rise to the additional normal coordinates, which, as the form of the Hamiltonian indicates, we may view as independent species of phonons.

===Dispersion relation===
[[File:Diatomic phonons.png|class=skin-invert-image|thumb|Dispersion curves in linear diatomic chain]]
[[File:Optical & acoustic vibrations-en.svg|class=skin-invert-image|thumb|250px|Optical and acoustic vibrations in a linear diatomic chain.]]
[[File:Diatomic chain.gif|class=skin-invert-image|thumb|Vibrations of the diatomic chain at different frequencies.]]
[[File:Phonon dispersion relations in GaAs.png|class=skin-invert-image|thumb|250px|Dispersion relation ''ω''&nbsp;=&nbsp;''ω''('''k''') for some waves corresponding to lattice vibrations in GaAs.<ref name=Cardona/>]]
For a one-dimensional alternating array of two types of ion or atom of mass ''m''<sub>1</sub>, ''m''<sub>2</sub> repeated periodically at a distance ''a'', connected by springs of spring constant ''K'', two modes of vibration result:<ref name=Misra/>
:<math>\omega_\pm^2 = K\left(\frac{1}{m_1} +\frac{1}{m_2}\right) \pm K \sqrt{\left(\frac{1}{m_1} +\frac{1}{m_2}\right)^2-\frac{4\sin^2\frac{ka}{2}}{m_1 m_2}} ,</math>
where ''k'' is the wavevector of the vibration related to its wavelength by
<math>k = \tfrac{2 \pi}{\lambda}</math>.

The connection between frequency and wavevector, ''ω''&nbsp;=&nbsp;''ω''(''k''), is known as a [[dispersion relation]]. The plus sign results in the so-called ''optical'' mode, and the minus sign to the ''acoustic'' mode. In the optical mode two adjacent different atoms move against each other, while in the acoustic mode they move together.

The speed of propagation of an acoustic phonon, which is also the [[speed of sound]] in the lattice, is given by the slope of the acoustic dispersion relation, {{sfrac|∂''ω<sub>k</sub>''|∂''k''}} (see [[group velocity]].) At low values of ''k'' (i.e. long wavelengths), the dispersion relation is almost linear, and the speed of sound is approximately ''ωa'', independent of the phonon frequency. As a result, packets of phonons with different (but long) wavelengths can propagate for large distances across the lattice without breaking apart. This is the reason that sound propagates through solids without significant distortion. This behavior fails at large values of ''k'', i.e. short wavelengths, due to the microscopic details of the lattice.

For a crystal that has at least two atoms in its [[Wigner-Seitz cell#Primitive cell|primitive cell]], the dispersion relations exhibit two types of phonons, namely, optical and acoustic modes corresponding to the upper blue and lower red curve in the diagram, respectively. The vertical axis is the energy or frequency of phonon, while the horizontal axis is the [[wavevector]]. The boundaries at −{{sfrac|{{pi}}|''a''}} and {{sfrac|{{pi}}|''a''}} are those of the first [[Brillouin zone]].<ref name=Misra>{{cite book |title=Physics of Condensed Matter |first=Prasanta Kumar |last=Misra |chapter-url=https://books.google.com/books?id=J6rMISLVCmcC&pg=PA44 |pages=44 |chapter=§2.1.3 Normal modes of a one-dimensional chain with a basis |publisher=Academic Press |isbn=978-0-12-384954-0 |year=2010}}</ref> A crystal with ''N''&nbsp;≥&nbsp;2 different atoms in the [[primitive cell]] exhibits three acoustic modes: one [[Longitudinal wave|longitudinal acoustic mode]] and two [[Transverse wave|transverse acoustic modes]]. The number of optical modes is 3''N''&nbsp;–&nbsp;3. The lower figure shows the dispersion relations for several phonon modes in [[GaAs]] as a function of wavevector '''k''' in the [[Brillouin zone#Critical points|principal directions]] of its Brillouin zone.<ref name=Cardona>
{{cite book |title=Fundamentals of Semiconductors |series=Physics and Materials Properties |first1=Peter Y. |last1=Yu |first2=Manuel |last2=Cardona |chapter-url=https://books.google.com/books?id=5aBuKYBT_hsC&pg=PA111 |page=111 |chapter=Fig. 3.2: Phonon dispersion curves in GaAs along high-symmetry axes |isbn=978-3-642-00709-5 |year=2010 |edition=4th |publisher=Springer}}</ref>

The modes are also referred to as the branches of phonon dispersion. In general, if there are p atoms (denoted by N earlier) in the primitive unit cell, there will be 3p branches of phonon dispersion in a 3-dimensional crystal.  Out of these, 3 branches correspond to acoustic modes and the remaining 3p-3 branches will correspond to  optical modes. In some special directions, some branches coincide due to symmetry.  These branches are called degenerate. In acoustic modes, all the p atoms vibrate in phase. So there is no change in the relative displacements of these atoms during the wave propagation.

Study of phonon dispersion is useful for modeling propagation of sound waves in solids, which is characterized by phonons. The energy of each phonon, as given earlier, is ''ħω.'' The velocity of the wave also is given in terms of ''ω'' and k ''.'' The direction of the wave vector is the direction of the wave propagation and the phonon polarization vector gives the direction in which the atoms vibrate. Actually, in general, the wave velocity in a crystal is different for different directions of k. In other words, most crystals are anisotropic for phonon propagation.

A wave is longitudinal if the atoms vibrate in the same direction as the wave propagation. In a transverse wave, the atoms vibrate perpendicular to the wave propagation.  However, except for isotropic crystals, waves in a crystal are not exactly longitudinal or transverse. For general anisotropic crystals, the phonon waves are longitudinal or transverse only in certain special symmetry directions. In other directions, they can be  nearly longitudinal or nearly transverse. It is only for labeling convenience, that they are often called longitudinal or transverse but are actually quasi-longitudinal or quasi-transverse.   Note that in the  three-dimensional case, there are two directions perpendicular to a straight line at each point on the line. Hence, there are always two (quasi) transverse waves for each (quasi) longitudinal wave.

Many phonon dispersion curves have been measured by [[inelastic neutron scattering]].

The physics of sound in [[fluid]]s differs from the physics of sound in solids, although both are density waves: sound waves in fluids only have longitudinal components, whereas sound waves in solids have longitudinal and transverse components. This is because fluids cannot support [[shear stress]]es (but see [[viscoelastic]] fluids, which only apply to high frequencies).

===Interpretation of phonons using second quantization techniques===
The above-derived Hamiltonian may look like a classical Hamiltonian function, but if it is interpreted as an [[Operator (physics)|operator]], then it describes a [[quantum field theory]] of non-interacting [[boson]]s.<ref name=girvinYang />
The [[second quantization]] technique, similar to the [[ladder operator]] method used for [[Quantum harmonic oscillator#Ladder operator method|quantum harmonic oscillators]], is a means of extracting energy [[eigenvalues]] without directly solving the differential equations. Given the Hamiltonian, <math>\mathcal{H}</math>, as well as the conjugate position, <math>Q_k</math>, and conjugate momentum <math>\Pi_{k}</math> defined in the quantum treatment section above, we can define [[creation and annihilation operators]]:<ref name=ashcrroftMermin>{{cite book |last1=Ashcroft |first1=Neil W. |last2=Mermin |first2=N. David |title=Solid State Physics |date=1976 |publisher=Saunders College Publishing |isbn=0-03-083993-9 |pages=780–783}}</ref>
:<math>b_k=\sqrt\frac{m\omega_k}{2\hbar}\left(Q_k+\frac{i}{m\omega_k}\Pi_{-k}\right)</math> &nbsp; and &nbsp; <math>{b_k}^\dagger=\sqrt\frac{m\omega_k}{2\hbar}\left(Q_{-k}-\frac{i}{m\omega_k}\Pi_{k}\right)</math>
The following commutators can be easily obtained by substituting in the [[canonical commutation relation]]:
:<math>\left[b_k , {b_{k'}}^\dagger \right] = \delta_{k,k'} ,\quad \Big[b_k , b_{k'} \Big] = \left[{b_k}^\dagger , {b_{k'}}^\dagger \right] = 0</math>
Using this, the operators ''b<sub>k</sub>''<sup>†</sup> and ''b<sub>k</sub>'' can be inverted to redefine the conjugate position and momentum as:
:<math>Q_k=\sqrt{\frac{\hbar}{2m\omega_k}}\left({b_k}^\dagger+b_{-k}\right)</math> &nbsp; and &nbsp; <math>\Pi_k=i\sqrt{\frac{\hbar m\omega_k}{2}}\left({b_k}^\dagger-b_{-k}\right)</math>
Directly substituting these definitions for <math>Q_k</math> and <math>\Pi_k</math> into the wavevector space Hamiltonian, as it is defined above, and simplifying then results in the Hamiltonian taking the form:<ref name=girvinYang />
:<math>\mathcal{H} =\sum_k \hbar\omega_k \left({b_k}^\dagger b_k+\tfrac12\right)</math>

This is known as the second quantization technique, also known as the occupation number formulation, where ''n<sub>k</sub>'' = ''b<sub>k</sub>''<sup>†</sup>''b<sub>k</sub>'' is the occupation number. This can be seen to be a sum of N independent oscillator Hamiltonians, each with a unique wave vector, and compatible with the methods used for the quantum harmonic oscillator (note that ''n<sub>k</sub>'' is [[Hermitian matrix|hermitian]]).<ref name=ashcrroftMermin /> When a Hamiltonian can be written as a sum of commuting sub-Hamiltonians, the energy eigenstates will be given by the products of eigenstates of each of the separate sub-Hamiltonians. The corresponding [[energy]] [[Spectrum of an operator|spectrum]] is then given by the sum of the individual eigenvalues of the sub-Hamiltonians.<ref name=ashcrroftMermin />

As with the quantum harmonic oscillator, one can show that ''b<sub>k</sub>''<sup>†</sup> and ''b<sub>k</sub>'' respectively create and destroy a single field excitation, a phonon, with an energy of ''ħω<sub>k</sub>''.<ref name=ashcrroftMermin /><ref name=girvinYang />

Three important properties of phonons may be deduced from this technique. First, phonons are [[boson]]s, since any number of identical excitations can be created by repeated application of the creation operator ''b<sub>k</sub>''<sup>†</sup>. Second, each phonon is a "collective mode" caused by the motion of every atom in the lattice. This may be seen from the fact that the creation and annihilation operators, defined here in momentum space, contain sums over the position and momentum operators of every atom when written in position space. (See [[position and momentum space]].)<ref name=ashcrroftMermin /> Finally, using the ''position–position [[correlation function]]'', it can be shown that phonons act as waves of lattice displacement.{{citation needed|date=August 2020}}

This technique is readily generalized to three dimensions, where the Hamiltonian takes the form:<ref name=ashcrroftMermin /><ref name=girvinYang />
:<math>\mathcal{H} = \sum_k \sum_{s = 1}^3 \hbar \, \omega_{k,s} \left( {b_{k,s}}^\dagger b_{k,s} + \tfrac12 \right).</math>
This can be interpreted as the sum of 3N independent oscillator Hamiltonians, one for each wave vector and polarization.<ref name=ashcrroftMermin />