Occupation Number Representation
In quantum mechanics, the occupation number representation allows us to describe the states of a system of identical particles. Any state |\psi\rangle of such a system can be expressed as a linear combination of many-particle basis states:
|\psi\rangle = \sum_j \mathbf b_j |\phi_j\rangle
Here, each basis state |\phi_j\rangle is defined by the occupation numbers n_j corresponding to each member of a complete set of orthonormal single-particle states \{|\mathbf n \rangle, j = 1, 2, 3, \dots\}. The set of occupation numbers \{n_1, n_2, \dots, n_j, \dots\} contains all the information necessary to construct an appropriately many-particle state, denoted by:
|\phi\rangle = |n_1, n_2, \dots, n_j, \dots\rangle
The vector space spanned by these basis states is known as Fock space. A key feature of Fock space is that the total number of particles is not fixed but is a dynamical variable represented by the total number operator:
N = \sum_j n_j
In this representation, there exists a unique vacuum state or no-particle state, represented as:
|\mathbf 0\rangle = |0, 0, 0, \dots\rangle
Considering a single-particle state |\mathbf n \rangle can be denoted by having a single occupation number of 1 in the position corresponding to j:
|\mathbf j\rangle = |0, 0, \dots, 0, n_j = 1, 0, \dots\rangle \equiv |0_1, 0_2, \dots, 0_{j-1}, 1_j, 0_{j+1}, \dots\rangle
Bosons and fermions require distinct definitions for their quantum state operations due to fundamental differences in their characteristics, governed by their statistics.
Bosons follow Bose-Einstein statistics and are not subject to the Pauli exclusion principle, allowing multiple bosons to occupy the same quantum state. Consequently, their creation operators can increase the occupation number indefinitely.
Fermions, on the other hand, adhere to Fermi-Dirac statistics and are subject to the Pauli exclusion principle, which restricts each quantum state to be occupied by no more than one fermion. This restriction means that for fermions the creation operators can only increase the occupation number from 0 to 1, leading to significantly different mathematical treatments and physical behaviors in quantum systems.
The terms “creation operator” and “annihilation operator” reflect the functions these operators perform. The creation operator, denoted as \mathbf a^\dagger_{j}, adds one particle to the j^{th} quantum state, effectively “creating” a particle and increasing the occupation number of that state. Conversely, the annihilation operator, denoted as \mathbf a_{j}, “annihilates” or removes one particle from the j^{th} state, reducing its occupation number. The number operator N_{j} = \mathbf a^\dagger_{j}\mathbf a_{j} calculates the number of particles in the j^{th} state by multiplying the creation and annihilation operations, thereby yielding the occupation number for that state directly.
The creation operator \mathbf a^\dagger_{j} is defined as:
\mathbf a^\dagger_{j}|n_1, n_2, \dots, n_{j-1}, n_{j}, n_{j+1}, \dots\rangle = \sqrt{n_{j} +1} |n_1, n_2, \dots, n_{j-1}, n_{j} +1, n_{j+1}, \dots\ \rangle
and the corresponding annihilation operator \mathbf a_{j}:
\mathbf a_{j}|n_1, n_2, \dots, n_{j-1}, n_{j}, n_{j+1}, \dots\rangle = \sqrt{n_{j}} |n_1, n_2, \dots, n_{j-1}, n_{j} -1, n_{j+1}, \dots\rangle
These equations allow us to define the number operator N_{j} = \mathbf a^\dagger_{j}\mathbf a_{j}, such that
N_{j}|n_1, n_2, \dots, n_{j}, \dots\rangle = n_{j}|n_1, n_2, \dots, n_{j}, \dots\rangle
and
N = \sum_{j} N_{j}
The simplest application of the creation and annihilation operators involves the single-particle states:
\begin{aligned} & \mathbf a^\dagger_{j}|\mathbf 0\rangle = |\mathbf j\rangle \\ & \mathbf a_{j}|\mathbf j \rangle = |\mathbf 0\rangle \end{aligned}
When applied to multi-particle states, the properties of the creation and annihilation operators must be consistent with the symmetry of bosons states under pairwise interchange of particles.
Considering any pair of single-particle states |\mathbf j\rangle and |\mathbf k\rangle, and for any vector |\psi\rangle in the Fock space, \mathbf a^\dagger_{j}\mathbf a^\dagger_{k} |\psi\rangle = \mathbf a^\dagger_{k}\mathbf a^\dagger_{j}|\psi\rangle, \mathbf a_{j}\mathbf a_{k} |\psi\rangle = \mathbf a_{k}\mathbf a_{j}|\psi\rangle and \mathbf a^\dagger_{j}\mathbf a_{k} |\psi\rangle = \mathbf a_{k}\mathbf a^\dagger_{j}|\psi\rangle for j \neq k. However,
\mathbf a^\dagger_{j}\mathbf a_{j}|\phi\rangle = n_{j}|\phi\rangle, \quad \mathbf a_{j}\mathbf a^\dagger_{j}|\phi\rangle = (n_{j} + 1)|\phi\rangle
This means that for any |\psi\rangle in the Fock space:
\mathbf a_{j}\mathbf a^\dagger_{j}|\psi\rangle - \mathbf a^\dagger_{j}\mathbf a_{j}|\psi\rangle = (N_{j} +1)|\psi\rangle - N_{j}|\psi\rangle = |\psi\rangle
This property can be summarized in the commutation relations:
[\mathbf a_{j}, \mathbf a^\dagger_{k}] = \delta_{jk}
One consequence of these commutation relations is that any multi-particle basis state can be written as a series of application to the empty state, with a factor to keep the state normalized.
|n_1, n_2, \dots, n_{j}, \dots\rangle = \frac{1}{\sqrt{n_1!n_2!\dots n_j! \dots}}(\mathbf a^\dagger_{1})^{n1} (\mathbf a^\dagger_{2})^{n2} \dots (\mathbf a^\dagger_{j})^{n_{j}} \dots |\mathbf 0\rangle
For example:
|0, 0, 0, \dots, 2_j,0_k, 1_m, \dots \rangle = \frac{1}{\sqrt{2!1!}}\left(\mathbf a^\dagger_{j}\right)^2 \mathbf a^\dagger_{m}|\mathbf 0\rangle = \frac{1}{\sqrt{2!1!}} \mathbf a^\dagger_{j} \mathbf a^\dagger_{j} \mathbf a^\dagger_{m}|\mathbf 0\rangle
Ladder operators for fermions need to enforce anti-symmetry across all pairwise particle exchanges. We define the creation operator \mathbf b^\dagger_{k} as follows:
For a state where the j^{th} position is unoccupied:
\mathbf b^\dagger_{j}|n_1, n_2, \dots, n_{j-1}, 0_{j}, n_{j+1}, \dots\rangle = (-1)^{S_{j}} |n_1, n_2, \dots, n_{j-1}, 1_{j}, n_{j+1}, \dots\rangle
If the j^{th} position is already occupied, the result is zero:
\mathbf b^\dagger_{j}|n_1, n_2, \dots, n_{j-1}, 1_{j}, n_{j+1}, \dots\rangle = 0
The annihilation operator \mathbf b_{j} is defined by:
\mathbf b_{j}|n_1, n_2, \dots, n_{j-1}, 1_{j}, n_{j+1}, \dots\rangle = (-1)^{S_{j}} |n_1, n_2, \dots, n_{j-1}, 0_{j}, n_{j+1}, \dots\rangle
and similarly gives zero for an unoccupied j^{th} position:
\mathbf b_{j}|n_1, n_2, \dots, n_{j-1}, 0_{j}, n_{j+1}, \dots\rangle = 0
S_{j} = \sum_{k < j} N_{k}, where N_{k} = \mathbf b^\dagger_{k} \mathbf b_{k}, measures the total number of particles in states indexed less than j. These definitions ensure self-consistency and maintain the necessary anti-symmetry, as described by the phase factor (-1)^{S_{j}}. The anti-symmetry relations are:
\begin{aligned} & \mathbf b^\dagger_{j}\mathbf b^\dagger_{k} |\psi\rangle = -\mathbf b^\dagger_{k}\mathbf b^\dagger_{j}|\psi\rangle \quad j \neq k\\ & \mathbf b^\dagger_{j}\mathbf b^\dagger_{j}|\psi\rangle = 0 = -\mathbf b^\dagger_{j}\mathbf b^\dagger_{j}|\psi\rangle \\ & \mathbf b_{j}\mathbf b_{k} |\psi\rangle = -\mathbf b_{k}\mathbf b_{j}|\psi\rangle \quad j \neq k \\ & \mathbf b_{j}\mathbf b_{j}|\psi\rangle = 0 \\ & \mathbf b^\dagger_{j}\mathbf b_{k} |\psi\rangle = -\mathbf b_{k}\mathbf b^\dagger_{j}|\psi\rangle \quad j \neq k\\ & \mathbf b^\dagger_{j}\mathbf b_{j}|\phi\rangle = n_{j}|\phi\rangle \\ & \mathbf b_{j}\mathbf b^\dagger_{j}|\phi\rangle = (1 - n_{j})|\phi\rangle \\ & (\mathbf b_{j}\mathbf b^\dagger_{j} + \mathbf b^\dagger_{j}\mathbf b_{j})|\psi\rangle = |\psi\rangle \end{aligned}
These properties are guaranteed when applying the anticommutator:
\{\mathbf b_{j}, \mathbf b^\dagger_{k}\} = \delta_{jk}
where \{A, B\} \equiv AB + BA.
These fundamental anticommutation properties distinguish fermions operators from their bosons counterparts, which commute instead.
Given these anticommutation relations, any multi-particle basis state can be expressed in terms of creation operators applied to the vacuum state, allowing for permutations with sign changes due to pairwise operator interchanges:
|n_1, n_2, \dots, n_{j}, \dots\rangle = (\mathbf b^\dagger_{1})^{n1} (\mathbf b^\dagger_{2})^{n2} \dots (\mathbf b^\dagger_{j})^{n_{j}} \dots |\mathbf 0\rangle
Many important phenomena involve interactions of different kind of particles, for example the interaction of photons with electrons and phenomenon of absorption or emission.
Some additions are necessary to the framework, in particular it is necessary to include the description of the occupied single-state particle for each different particle and commutation relations between operators.
The system can be simply identified with a notation listing the particle in the fermion and in the boson state separated by a semicolon “;”, for example, for a system with two kind of particles (a set of identical electron and a set of identical photons), considering a state with one fermion in the states i,l,n , four photons in mode p, two in mode s and one in mode t, they can be written as:
|0_a,\dots,1_i,0_j,0_k,1_l,0_m,1_n,0_o,\dots;0_a,\dots,4_p,0_q,0_r,2_s,t_1,0_u,\dots \rangle Equivalently, using the creation operators applied to the vacuum state:
|N_{fj};N_{bk}\rangle \equiv \frac{1}{\sqrt{4!3!}}\mathbf b^\dag_i \mathbf b^\dag_l \mathbf b^\dag_n \left(\mathbf a^\dag_p\right)^4 \left(\mathbf a^\dag_s\right)^2 \mathbf a^\dag_t | \mathbf 0 \rangle
where N_{fj} is the j^{th} list of occupied fermion states and N_{bk} the k^{th} list of boson states over the vacuum state of the particular type of fermions and bosons.
As for the commutation relations, the postulate is that creation and annihilation operators corresponding to nonidentical particle commute under all conditions:
\begin{aligned} & \mathbf b^\dag_j \mathbf a^\dag_k - \mathbf a^\dag_k \mathbf b^\dag_j = 0 \\ & \mathbf b_j \mathbf a_k - \mathbf a_k \mathbf b_j = 0 \\ & \mathbf b^\dag_j \mathbf a_k - \mathbf a_k \mathbf b^\dag_j = 0 \\ & \mathbf b_j \mathbf a^\dag_k - \mathbf a^\dag_k \mathbf b_j = 0 \\ \end{aligned} These relation hold also if the particle are nonidentical fermions or bosons (such neutrons and electrons for example).