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{{Quantum field theory}}
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'''Second quantization''' is a powerful procedure used in [[quantum field theory]] for describing the [[n-body problem|many-particle]] systems by [[canonical quantization|quantizing]] the fields using a basis that describes the '''number of particles''' occupying each state in a complete set of single-particle states. This differs from the [[first quantization]], which uses the single-particle states as basis.
 
==Introduction==
 
The starting point of this formalism is the notion of [[indistinguishability]] of particles that bring us to use [[determinant]]s of single-particle states as a basis of the [[Hilbert space]] of N-particles states{{Clarify|date=April 2013}}. Quantum theory can be formulated in terms of occupation numbers (number of particles occupying one determined energy state) of these [[wave function|single-particle states]]. The formalism was introduced in 1927 by [[Paul Dirac|Dirac]].<ref name="Dirac">{{cite doi|10.1098/rspa.1927.0039}}</ref>
 
===The occupation number representation===
 
Consider an ordered and complete single-particle basis 
<math> \left\{| \nu_1 \rang, | \nu_2 \rang, | \nu_3 \rang, ...\right\} </math>, where <math>| \nu_i \rang</math> is the set of all states <math>\nu</math> available for the <math>i</math>-th particle. In an N-particle system, only the occupied single-particle states play a role. So it is simpler to formulate a representation where one just counts how many particles there are in each orbital <math>| \nu \rang</math>. This simplification is achieved with the occupation number representation.
The basis states for an N-particle system in this representation are obtained simply by listing the occupation numbers of each basis state,
<math>|n_{\nu_1}, n_{\nu_2}, n_{\nu_3},\dots \rang</math>, where <math> \sum_j n_{\nu_j} = N</math> The notation means that there are <math> n_{\nu_j}</math> particles in the state <math> \nu_j</math>. It is therefore natural to define the occupation number operator <math> \hat{n}_{\nu_j}</math> which obeys
:<math> \hat{n}_{\nu_j}|n_{\nu_j} \rang=n_{\nu_j}|n_{\nu_j} \rang</math>
 
For [[fermions]] <math> n_{\nu_j}</math> can be 0 or 1, while for [[bosons]] it can be any non negative number
:<math>n_{\nu_j}= \begin{cases}
  \ 0, 1. &\text{fermions}\\
  0,1,2,...          &\text{bosons}
\end{cases}
</math>
The space spanned by the occupation number basis is denoted the [[Fock space]].
 
==Creation and annihilation operators==
The [[creation and annihilation operators]] are the way to connect the [[first quantization|first]] and second quantizations. It is fundamental for the many-body theory that every operator can be expressed in terms of annihilation and creation operators. Originally constructed in the context of the [[quantum harmonic oscillator]], these operators are the most general form to describe quantum fields.<ref name="Mahan">{{cite book|last=Mahan|first=GD|authorlink=|title=Many Particle Physics|publisher=Springer|location=New York|isbn=0306463385|year=1981}}</ref> Depending on the nature of the fields we can use two different approaches:
 
===Bosons===
Given the occupation number, we introduce the annihilation <math>b_{\nu_j}</math> and creation <math>{b^{\dagger}}_{\nu_j}</math>  operators that lowers(raises) the occupation number in the state <math>| \nu_j \rang</math> by 1,
:<math>b_{\nu_j}|\dots,n_{\nu_{j-1}}, n_{\nu_j}, n_{\nu_{j+1}},\dots \rang=\sqrt{n_{\nu_j}}|\dots,n_{\nu_{j-1}}, n_{\nu_j}-1, n_{\nu_{j+1}},\dots \rang</math>
:<math>{b^{\dagger}}_{\nu_j}|\dots,n_{\nu_{j-1}}, n_{\nu_j}, n_{\nu_{j+1}},\dots \rang=\sqrt{n_{\nu_j}+1}|\dots,n_{\nu_{j-1}}, n_{\nu_j}+1, n_{\nu_{j+1}},\dots \rang</math>
 
Since bosons are symmetric in the single-particle state index <math>\nu_j</math> we demand that <math>b_{\nu_j}</math> and <math>{b^{\dagger}}_{\nu_j}</math> [[commutation|commute]],
So, we can obtain the mean properties of these operators:
 
:<math>\begin{matrix}
  [{b^{\dagger}}_{\nu_j},{b^{\dagger}}_{\nu_k}] = 0  & [b_{\nu_j},b_{\nu_k}]=0 & [b_{\nu_j},{b^{\dagger}}_{\nu_k}]=\delta_{\nu_j\nu_k}\\
{b^{\dagger}}_{\nu_j}|n_{\nu_j}\rang=\sqrt{n_{\nu_j}+1}|n_{\nu_j}+1 \rang & b_{\nu_j}|n_{\nu_j}\rang=\sqrt{n_{\nu_j}}|n_{\nu_j} -1\rang & b_{\nu_j}|0\rang=0\\
{b^{\dagger}}_{\nu_j}b_{\nu_j}|n_{\nu_j} \rang=n_{\nu_j}|n_{\nu_j} \rang &\left({b^{\dagger}}_{\nu_j}\right)^{n_{\nu_j}}|0 \rang=\sqrt{(n_{\nu_j})!}|n_{\nu_j} \rang & n_{\nu_j}=0,1,2,\dots\\
\end{matrix}</math>
 
and therefore identify the [[identical particles|first]] and second quantized states,
:<math> \hat{S}_+|\psi_{n_{\nu_1}}(\bold{r}_1)\rang|\psi_{n_{\nu_2}}(\bold{r}_2)\rang\dots |\psi_{n_{\nu_N}}(\bold{r}_N)\rang= {b^{\dagger}}_{n_{\nu_1}}{b^{\dagger}}_{n_{\nu_2}}\dots{b^{\dagger}}_{n_{\nu_N}}|0\rang</math>
 
with <math> \hat{S}</math> the [[symmetrization]] operator. Here, both contain N-particle state-kets completely symmetric in the single-particle state index  <math>\psi_{\nu_j}</math>. Because the creation and annihilation operators of the [[quantum harmonic oscillator]] obey these properties, one can classify the field associated to it as bosonic.
 
===Fermions===
[[Fermions]] have similar annihilation <math>c_{\nu_j}</math> and creation <math>{c^{\dagger}}_{\nu_j}</math> operators:
:<math>\begin{matrix}
c_{\nu_j}|\dots,1,\dots \rang= |\dots,0,\dots \rang & c_{\nu_j}|\dots,0,\dots \rang=0 \\
{c^{\dagger}}_{\nu_j}|\dots,0,\dots \rang=|\dots,1,\dots\rang& {c^{\dagger}}_{\nu_j}|\dots,1,\dots \rang=0
 
\end{matrix}</math>
 
(Note that the only permitted number of occupation are 0 or 1.) To maintain the fundamental fermionic antisymmetry upon exchange of orbitals, the operators must [[Anticommutativity|anticommute]], rather than commute:
 
:<math>\begin{matrix}
\{ {c^{\dagger}}_{\nu_j},{c^{\dagger}}_{\nu_k}\} = 0  & \{c_{\nu_j},c_{\nu_k}\}=0 & \{c_{\nu_j}, {c^{\dagger}}_{\nu_k}\}=\delta_{\nu_j\nu_k}\\
\left({c^{\dagger}}_{\nu_j}\right)^2=\left(c_{\nu_j}\right)^2=0 & {c^{\dagger}}_{\nu_j}c_{\nu_j}|n_{\nu_j} \rang=n_{\nu_j}|n_{\nu_j} \rang & n_{\nu_j}=0,1\\
\end{matrix}</math>
Where we have used the [[Commutator#Anticommutator|anticommutator]] <math>\{,\}</math>
Therefore one can identify the [[identical particles|first]] quantized states in terms of the second quantized:
:<math> \hat{S}_-|\psi_{n_{\nu_1}}(\bold{r}_1)\rang|\psi_{n_{\nu_2}}(\bold{r}_2)\rang\dots |\psi_{n_{\nu_N}}(\bold{r}_N)\rang= {c^{\dagger}}_{n_{\nu_1}}{c^{\dagger}}_{n_{\nu_2}}\dots{c^{\dagger}}_{n_{\nu_N}}|0\rang</math>
with <math> \hat{S}_-</math> the [[Antisymmetrizer]] operator. Here, both contain N-particle state-kets completely anti-symmetric in the single-particle state index  <math>\psi_{\nu_j}</math> in accordance with [[Pauli exclusion principle]].
 
==Quantum field operators==
 
Defining <math>{a^{(\dagger)}}_{\nu}</math> as a general annihilation(creation) operator that could be either fermionic <math>({c^{(\dagger)}}_{\nu})</math> or bosonic <math>({b^{(\dagger)}}_{\nu})</math>, the [[Position and momentum space|real space representation]] of the operators defines the [[quantum]] field [[operators]] <math> \Psi(\bold{r})</math> and <math>\Psi^{\dagger}(\bold{r})</math> by
:<math> \Psi(\bold{r})=\sum_{\nu} \psi_{\nu} \left( \bold{r} \right) a_{\nu}</math>
:<math> \Psi^{\dagger}(\bold{r})=\sum_{\nu} {\psi^*}_{\nu} \left( \bold{r} \right) {a^{\dagger}}_{\nu}</math>
 
Second quantization operators, while the coefficients <math>\psi_{\nu} \left( \bold{r} \right)</math> and <math> {\psi^*}_{\nu} \left( \bold{r} \right)</math> are the ordinary [[first quantization]] [[wavefunctions]]. Loosely speaking, <math>\Psi^{\dagger}(\bold{r})</math> is the sum of all possible ways to add a particle to the system at position '''r''' through any of the basis states <math>\psi_{\nu}\left(\bold{r}\right)</math>. Since <math> \Psi(\bold{r})</math> and <math>\Psi^{\dagger}(\bold{r})</math> are second quantization operators defined in every point in space they are called [[quantum field]] operators. They obey the following fundamental commutator and anti-commutator,
 
:<math>\begin{align}
\left[\Psi(\bold{r}_1),\Psi^\dagger(\bold{r}_2)\right]=\delta (\bold{r}_1-\bold{r}_2) &\text{    boson fields,}\\
\{\Psi(\bold{r}_1),\Psi^\dagger(\bold{r}_2)\}=\delta (\bold{r}_1-\bold{r}_2)&\text{    fermion fields.}
\end{align}</math>
 
In homogeneous systems it is often desirable to transform between real space and the momentum representations, hence, the quantum fields operators in [[Fourier transform|Fourier basis]] yields:
:<math> \Psi(\bold{r})={1\over \sqrt {V}} \sum_{\bold{k}} e^{i\bold{k\cdot r}}a_{\bold{k}}</math>
:<math> \Psi^{\dagger}(\bold{r})={ 1\over \sqrt{V}} \sum_{\bold{k}} e^{-i\bold{k\cdot r}}{a^{\dagger}}_{\bold{k}}</math>
 
== See also ==
 
* [[Canonical quantization]]
 
==References==
{{Reflist|2}}
 
==External links==
* [http://www.cond-mat.de/events/correl13/manuscripts/koch.pdf Many-Electron States] in E.  Pavarini, E. Koch, and U. Schollwöck: Emergent Phenomena in Correlated Matter, Jülich 2013, ISBN 978-3-89336-884-6
 
[[Category:Quantum field theory]]
[[Category:Mathematical quantization]]
 
[[es:segunda cuantización]]
[[ja:第二量子化]]
[[uk:Вторинне квантування ферміонів]]

Latest revision as of 19:38, 7 November 2014

Person who wrote the statement is called Roberto Ledbetter and his wife isn't really like it at all. In his professional life he can be a people manager. He's always loved living to Guam and he has everything that he conditions there. The most-loved hobby for him and his kids is growing plants but he's been claiming on new things as of late. He's been working on this website for some era now. Check it out here: http://circuspartypanama.com

Stop by my blog post; clash of clans hack