Editing Second quantization (section)

== Quantum many-body states ==

The starting point of the second quantization formalism is the notion of [[Identical particles|indistinguishability]] of particles in quantum mechanics. Unlike in classical mechanics, where each particle is labeled by a distinct position vector <math>\mathbf{r}_i</math> and different configurations of the set of <math>\mathbf{r}_i</math>s correspond to different many-body states, ''in quantum mechanics, the particles are identical, such that exchanging two particles, i.e. <math>\mathbf{r}_i\leftrightarrow\mathbf{r}_j</math>, does not lead to a different many-body quantum state''. This implies that the quantum many-body wave function must be invariant (up to a phase factor) under the exchange of two particles. According to the [[particle statistics|statistics]] of the particles, the many-body wave function can either be symmetric or antisymmetric under the particle exchange:
:<math>\Psi_{\rm B}(\cdots,\mathbf{r}_i,\cdots,\mathbf{r}_j,\cdots)=+\Psi_{\rm B}(\cdots,\mathbf{r}_j,\cdots,\mathbf{r}_i,\cdots)</math> if the particles are [[bosons]],
:<math>\Psi_{\rm F}(\cdots,\mathbf{r}_i,\cdots,\mathbf{r}_j,\cdots)=-\Psi_{\rm F}(\cdots,\mathbf{r}_j,\cdots,\mathbf{r}_i,\cdots)</math> if the particles are [[fermions]].
This exchange symmetry property imposes a constraint on the many-body wave function. Each time a particle is added or removed from the many-body system, the wave function must be properly symmetrized or anti-symmetrized to satisfy the symmetry constraint. In the first quantization formalism, this constraint is guaranteed  by representing the wave function as linear combination of  [[Permanent (mathematics)|permanent]]s (for bosons) or [[determinant]]s (for fermions) of single-particle states. In the second quantization formalism, the issue of symmetrization is automatically taken care of by the creation and annihilation operators, such that its notation can be much simpler.

=== First-quantized many-body wave function ===
Consider a complete set of single-particle wave functions <math>\psi_{\alpha}(\mathbf{r})</math> labeled by <math>\alpha</math> (which may be a combined index of a number of quantum numbers). The following wave function
:<math>\Psi[\mathbf{r}_i]=\prod_{i=1}^{N}\psi_{\alpha_i}(\mathbf{r}_i)\equiv \psi_{\alpha_1}\otimes\psi_{\alpha_2}\otimes\cdots\otimes\psi_{\alpha_N}</math>
represents an ''N''-particle state with the ''i''th particle occupying the single-particle state <math>|{\alpha_i}\rangle</math>. In the shorthanded notation, the position argument of the wave function may be omitted, and it is assumed that the ''i''th single-particle wave function describes the state of the ''i''th particle. The wave function <math>\Psi</math> has not been symmetrized or anti-symmetrized, thus in general not qualified as a many-body wave function for identical particles. However, it can be brought to the symmetrized (anti-symmetrized) form by operators <math>\mathcal{S}</math> for symmetrizer, and <math>\mathcal{A}</math> for [[antisymmetrizer]].

For bosons, the many-body wave function must be symmetrized,
:<math>\Psi_{\rm B}[\mathbf{r}_i]=\mathcal{N}\mathcal{S}\Psi[\mathbf{r}_i]=\mathcal{N}\sum_{\pi\in S_N}\prod_{i=1}^{N}\psi_{\alpha_{\pi(i)}}(\mathbf{r}_i)=\mathcal{N}\sum_{\pi\in S_N}\psi_{\alpha_{\pi(1)}}\otimes\psi_{\alpha_{\pi(2)}}\otimes\cdots\otimes\psi_{\alpha_{\pi(N)}};</math>
while for fermions, the many-body wave function must be anti-symmetrized,
:<math>\Psi_{\rm F}[\mathbf{r}_i]=\mathcal{N}\mathcal{A}\Psi[\mathbf{r}_i]=\mathcal{N}\sum_{\pi\in S_N}(-1)^\pi\prod_{i=1}^{N}\psi_{\alpha_{\pi(i)}}(\mathbf{r}_i)=\mathcal{N}\sum_{\pi\in S_N}(-1)^\pi\psi_{\alpha_{\pi(1)}}\otimes\psi_{\alpha_{\pi(2)}}\otimes\cdots\otimes\psi_{\alpha_{\pi(N)}}.</math>
Here <math>\pi</math> is an element in the ''N''-body permutation group (or [[symmetric group]]) <math>S_{N}</math>, which performs a [[permutation]] among the state labels <math>\alpha_i</math>, and <math>(-1)^\pi</math> denotes the corresponding [[parity of a permutation|permutation sign]]. <math>\mathcal{N}</math> is the normalization operator that normalizes the wave function. (It is the operator that applies a suitable numerical normalization factor to the symmetrized tensors of degree ''n''; see the next section for its value.)

If one arranges the single-particle wave functions in a matrix <math>U</math>, such that the row-''i'' column-''j'' matrix element is <math>U_{ij}=\psi_{\alpha_{j}}(\mathbf{r}_i)\equiv \langle\mathbf{r}_i|\alpha_j\rangle</math>, then the boson many-body wave function can be simply written as a [[Permanent (mathematics)|permanent]] <math>\Psi_{\rm B}=\mathcal{N}\operatorname{perm} U</math>, and the fermion many-body wave function as a [[determinant]] <math>\Psi_{\rm F}=\mathcal{N}\det U</math> (also known as the [[Slater determinant]]).<ref name="Koch2013">{{cite book
 | author = Koch, Erik
 | chapter = Many-electron states
 | title = Emergent Phenomena in Correlated Matter
 |editor=Pavarini, Eva |editor2=Koch, Erik |editor3=Schollwöck, Ulrich
 | series = Modeling and Simulation
 | volume = 3
 | publisher = Verlag des Forschungszentrum Jülich
 | location = Jülich
 | pages = 2.1–2.26
 | url = http://hdl.handle.net/2128/5389
 | year = 2013
 | hdl = 2128/5389
 | isbn = 978-3-89336-884-6
}}</ref>

=== Second-quantized Fock states ===
First quantized wave functions involve complicated symmetrization procedures to describe physically realizable many-body states because the language of first quantization is redundant for indistinguishable particles. In the first quantization language, the many-body state is described by answering a series of questions like ''"Which particle is in which state?"''. However these are not physical questions, because the particles are identical, and it is impossible to tell which particle is which in the first place. The seemingly different states <math>\psi_1\otimes\psi_2</math> and <math>\psi_2\otimes\psi_1</math> are actually redundant names of the same quantum many-body state. So the symmetrization (or anti-symmetrization) must be introduced to eliminate this redundancy in the first quantization description.

In the second quantization language, instead of asking "each particle on which state", one asks ''"How many particles are there in each state?"''. Because this description does not refer to the labeling of particles, it contains no redundant information, and hence leads to a precise and simpler description of the quantum many-body state. In this approach, the many-body state is represented in the occupation number basis, and the basis state is labeled by the set of occupation numbers, denoted
:<math>|[n_{\alpha}]\rang\equiv|n_1,n_2,\cdots, n_{\alpha}, \cdots \rang,</math>
meaning that there are <math> n_{\alpha}</math> particles in the single-particle state <math>|\alpha\rangle</math> (or as <math>\psi_\alpha</math>). The occupation numbers sum to the total number of particles, i.e. <math display="inline"> \sum_\alpha n_{\alpha} = N</math>. For [[fermions]], the occupation number <math> n_{\alpha}</math> can only be 0 or 1, due to the [[Pauli exclusion principle]]; while for [[bosons]] it can be any non-negative integer 
:<math>n_{\alpha}= \begin{cases}
  0, 1 &\text{fermions,}\\
  0,1,2,3,...           &\text{bosons.}
\end{cases}
</math>
The occupation number states <math>|[n_{\alpha}]\rang</math> are also known as Fock states. All the Fock states form a complete basis of the many-body Hilbert space, or [[Fock space]]. Any generic quantum many-body state can be expressed as a linear combination of Fock states.

Note that besides providing a more efficient language, Fock space allows for a variable number of particles. As a [[Hilbert space]], it is isomorphic to the sum of the ''n''-particle bosonic or fermionic tensor spaces described in the previous section, including a one-dimensional zero-particle space '''C'''.

The Fock state with all occupation numbers equal to zero is called the [[vacuum state]], denoted <math>|0\rangle\equiv|\cdots,0_\alpha,\cdots\rangle</math>. The Fock state with only one non-zero occupation number is a single-mode Fock state, denoted <math>|n_\alpha\rangle\equiv|\cdots,0,n_\alpha,0,\cdots\rangle</math>. In terms of the first quantized wave function, the vacuum state is the unit tensor product and can be denoted <math>|0\rangle=1</math>. The single-particle state is reduced to its wave function <math>|1_\alpha\rangle=\psi_\alpha</math>. Other single-mode many-body (boson) states are just the tensor product of the wave function of that mode, such as <math>|2_\alpha\rangle=\psi_\alpha\otimes\psi_\alpha</math> and
<math>|n_\alpha\rangle=\psi_\alpha^{\otimes n}</math>. For multi-mode Fock states (meaning more than one single-particle state <math>|\alpha\rangle</math> is involved), the corresponding first-quantized wave function will require proper symmetrization according to the particle statistics, e.g. <math>|1_1,1_2\rangle=(\psi_1\psi_2+\psi_2\psi_1)/\sqrt{2}</math> for a boson state, and <math>|1_1,1_2\rangle=(\psi_1\psi_2-\psi_2\psi_1)/\sqrt{2}</math> for a fermion state (the symbol <math>\otimes</math> between <math>\psi_1</math> and <math>\psi_2</math> is omitted for simplicity). In general, the normalization is found to be <math display="inline">\sqrt{ \frac{1} {N!{\prod_{\alpha}{n_\alpha!}}}}</math>, where ''N'' is the total number of particles. For fermion, this expression reduces to <math>\tfrac{1}{\sqrt{N!}}</math> as <math>n_\alpha</math> can only be either zero or one. So the first-quantized wave function corresponding to the Fock state reads
:<math>|[n_\alpha]\rangle_{\rm B}=\left(\frac{1}{N!\prod_{\alpha}n_\alpha!}\right)^{1/2}\mathcal{S}\bigotimes\limits_\alpha\psi_\alpha^{\otimes n_\alpha}</math>

for bosons and
:<math>|[n_\alpha]\rangle_{\rm F}=\frac{1}{\sqrt{N!}}\mathcal{A}\bigotimes\limits_\alpha\psi_\alpha^{\otimes n_\alpha}</math> 
for fermions. Note that for fermions, <math>n_\alpha=0,1</math> only, so the tensor product above is effectively just a product over all occupied single-particle states.