Editing Quasiparticle (section)

===Relation to many-body quantum mechanics===
[[Image:Energy levels.svg|thumb|right|Any system, no matter how complicated, has a [[ground state]] along with an infinite series of higher-energy [[excited state]]s.]]
The principal motivation for quasiparticles is that it is almost impossible to ''directly'' describe every particle in a macroscopic system. For example, a barely-visible (0.1mm) grain of sand contains around 10<sup>17</sup> nuclei and 10<sup>18</sup> electrons. Each of these attracts or repels every other by [[Coulomb's law]]. In principle, the [[Schrödinger equation]] predicts exactly how this system will behave. But the Schrödinger equation in this case is a [[partial differential equation]] (PDE) on a 3×10<sup>18</sup>-dimensional vector space—one dimension for each coordinate (x, y, z) of each particle. Directly and straightforwardly trying to solve such a PDE is impossible in practice. Solving a PDE on a 2-dimensional space is typically much harder than solving a PDE on a 1-dimensional space (whether analytically or numerically); solving a PDE on a 3-dimensional space is significantly harder still; and thus solving a PDE on a 3×10<sup>18</sup>-dimensional space is quite impossible by straightforward methods.

One simplifying factor is that the system as a whole, like any quantum system, has a [[ground state]] and various [[excited state]]s with higher and higher energy above the ground state. In many contexts, only the "low-lying" excited states, with energy reasonably close to the ground state, are relevant. This occurs because of the [[Boltzmann distribution]], which implies that very-high-energy [[thermal fluctuations]] are unlikely to occur at any given temperature.

Quasiparticles and collective excitations are a type of low-lying excited state. For example, a crystal at [[absolute zero]] is in the [[ground state]], but if one [[phonon]] is added to the crystal (in other words, if the crystal is made to vibrate slightly at a particular frequency) then the crystal is now in a low-lying excited state. The single phonon is called an ''elementary excitation''. More generally, low-lying excited states may contain any number of elementary excitations (for example, many phonons, along with other quasiparticles and collective excitations).<ref>{{cite book |last1=Ohtsu |first1=Motoichi |last2=Kobayashi |first2=Kiyoshi |last3=Kawazoe |first3=Tadashi |last4=Yatsui |first4=Takashi |last5=Naruse |first5=Makoto |title=Principles of Nanophotonics |date=2008 |publisher=CRC Press |isbn=9781584889731 |page=205 |url=https://books.google.com/books?id=3za2u8FnCgUC&pg=PA205 |language=en}}</ref>

When the material is characterized as having "several elementary excitations", this statement presupposes that the different excitations can be combined. In other words, it presupposes that the excitations can coexist simultaneously and independently. This is never ''exactly'' true. For example, a solid with two identical phonons does not have exactly twice the excitation energy of a solid with just one phonon, because the crystal vibration is slightly [[anharmonic]]. However, in many materials, the elementary excitations are very ''close'' to being independent. Therefore, as a ''starting point'', they are treated as free, independent entities, and then corrections are included via interactions between the elementary excitations, such as "phonon-[[phonon scattering]]".

Therefore, using quasiparticles / collective excitations, instead of analyzing 10<sup>18</sup> particles, one needs to deal with only a handful of somewhat-independent elementary excitations. It is, therefore, an effective approach to simplify the [[many-body problem]] in quantum mechanics. This approach is not useful for ''all'' systems, however. For example, in [[strongly correlated material]]s, the elementary excitations are so far from being independent that it is not even useful as a starting point to treat them as independent.