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Second quantization method in the presence of bound states of particles

We develop an approximate second quantization method for describing the many-particle systems in the presence of bound states of particles at low energies (the kinetic energy of particles is small in comparison to the binding energy of compound particles). In this approximation the compound and elementary particles are considered on an equal basis. This means that creation and annihilation operators of compound particles can be introduced. The Hamiltonians, which specify the interactions between compound and elementary particles and between compound particles themselves are found in terms of the interaction amplitudes for elementary particles. The nonrelativistic quantum electrodynamics is developed for systems containing both elementary and compound particles. Some applications of this theory are considered.


I. INTRODUCTION
The second quantization is the effective method, which is used for description of physical processes in quantum theory of many-particle systems. The key role of this method consists in introduction of creation and annihilation operators of particles in a certain quantum state. The operators of physical quantities are constructed in terms of creation and annihilation operators. Such a description of quantum many-particle systems implies the particles to be elementary (not consisting of other particles). Moreover, it is absolutely accurate despite of the possible existence of compound particles. Since the interactions between particles may lead to formation of bound states, the standard second quantization method becomes too cumbersome. For this reason the construction of approximate quantummechanical theory for many-particle systems consisting of elementary particles and their bound states represents an actual problem. In this theory it is necessary to introduce the creation and annihilation operators of bound states as the operators of elementary objects (not compound). In addition we should preserve the required information concerning internal degrees of freedom for bound states. Such an approach may be actually realized in the approximation in which the binding energy of compound particles is much greater in comparison to their kinetic energy.
The problems leading to the necessity of accounting for compound particles along with elementary ones are typical in studying of the interaction of irradiation with matter (the matter consists of neutral atoms or molecules). The atoms or molecules may be both in the ground and excited states when it is necessary to take into account their internal structure retaining the convenience and simplicity of the second quantization method. Such a situation should take place in describing the experiments on laser cooling of atoms 1−3 or experiments on observation of Bose-Einstein condensates in magnetic traps, where a condensate temperature is reached by laser cooling of gases of alkali atoms 4 . The similar problem of accounting for the bound states occurs also in quantum chemistry, for example, in studying of chemical reactions.
Thus, the construction of the quantum theory of many-particle systems in the presence of bound states of particles within the second quantization method represents not only academic but also applied interest. The formulation of the quantum theory itself is to be done so that the operators of physical quantities are constructed through the creation and annihilation operators of elementary particles and their bound states.
In this paper we demonstrate a possibility of construction of the above-mentioned quantum-mechanical theory by considering a system of two kinds of charged fermions. In this theory the compound particles (atoms) consist of two fermions of various kinds. The choice of such model is not associated with principle difficulties and makes it possible to simplify calculations and to obtain the visual results.
As applications of the developed theory we study a spontaneous radiation of two-component excited atom and obtain the expression for its probability (see, e.g., 5 ). We also investigate the attractive forces acting between neutral atoms at long distances (the van der Waals forces) and derive the expression for the potential of the van der Waals forces 6 . Finally we consider a question concerning the scattering of photons and elementary particles by bound states 7 .

II. CREATION AND ANNIHILATION OPERATORS OF BOUND STATES OF PARTICLES
Consider a system consisting of two kinds of fermions with masses m 1 and m 2 . As was mentioned in the Introduction this case is more visual in order to demonstrate the procedure for obtaining the operators of basic physical quantities within the second quantization method in the presence of bound states of particles. We shall see that the developed methodology does not contain any difficulties for its generalization (the bound states of more than two particles, the presence of bosons in the system, the bound states of bosons and boson-fermion bound states). For simplicity we do not take into account a spin variable (this can be done without any difficulties).
We assume that the particles of two different kinds form a bound state specified by the wave function where x is the space coordinate and α are the quantum numbers of the bound state (atom) (we suppose that the particles of the same kind do not form the bound states). The corresponding state vector has the form |α, x = dx 1 dx 2 ϕ α (x 1 − x 2 )δ(x − X)ψ + 1 (x 1 )ψ + 2 (x 2 )|0 .
For this reason the operatorsφ we shall call as the creation operators of the bound states (atoms), so that If the atom has a certain momentum, then its state vector is given by where V is the volume of the system. The corresponding creation operatorφ + α (p) of the atom in the state with momentum p is defined by Taking into account that it is easy to find the following commutation relations: have an obvious physical meaning under the following conditions: we can use the Wick theorem with the following contractions: if to consider the operatorsψ 1 ,ψ 2 ,φ referred to the moment of time +0, and operatorsφ + ,ψ + 2 ,ψ + 1 to the moment of time −0. In addition, we should remember that the creation and annihilation operatorsφ α ,φ + α ′ depend onψ i ,ψ i ′ (see (3)). We also assume that the wave function (2) of the atom differs from zero for |x 1 − x 2 | < r 0 . Taking into account (3) and noting that for (the double contractions correspond to the operatorsφ α ,φ + α ) we get This relation shows that the vectors (6) form the orthonormalized basis in the subspace H a if to consider the creation and annihilation operatorsφ + α (z),φ α (z) as the Bose operators, which commute withψ i (x),ψ + i (x). The quantity P x ′ in (7) is equal to +1 if the number of permutations of the arguments x ′ 1 . . . x ′ n is even and it is equal to −1 if the number of these permutations is odd. The quantity P y ′ is defined similarly. However, the case when it is necessary to take into account a more complicated arrangement of the contractions between ϕ α , ϕ + α : can take place (the odd indices correspond to the creation operators and even to annihilation operators). Since Therefore, if a ≤ 4r 0 , then the above contractions are to be taken into account. If however, we consider the n-particle states, then the discussed arrangements of the contractions are not to be taken into account for n a/r 0 (we emphasize that r 0 /a is a small parameter in our problem, see (11)). The said above concerns to the majority of problems in non-relativistic quantum theory, where a finite number of particles is usually studied (see sections V, VI).
With the use of (7) it is easy to find the projection operator P Ha onto the subspace H a : such that the operators A acting in the subspace H a ∈ H correspond to the operators of physical quantities A that act in H (hereinafter the sums of 1, 2, . . . n-particle subspaces over the bound states are considered; n ≈ a/r 0 >> 1), Let us introduce now an auxiliary spaceH with the Fermi creation and annihilation operatorsχ + 1 (x),χ + 2 (x), χ 1 (x),χ 2 (x) and Bose creation and annihilation operatorsη + α (x),η α (x) and take the vectors as a basis of this space, where |0) is a vacuum vector inH. Then the linear span of the vectors |x 1 , . . . , y 1 , . . . , z 1 , . . .) ∈H a , x, y, z ∈ R a determines the subspaceH a of the spaceH.
Now we can easily establish the isomorphic correspondence between H a andH a : We can also establish the isomorphism between the operators A ⇐⇒Ã, acting in the spaces H a andH a according to the formula: This isomorphic correspondence is remained after the multiplication of the operator by a number, after addition of operators, and after multiplication of operators: This relation determines the operators of various physical quantitiesÃ acting inH a and, hence, transfers the quantum theory in which the compound (the bound states) and elementary particles exist on an equal basis from the space of states H onto the space of statesH a . We would like to remind here thatφ + α entering (4) is determined by (3). The relation (10) defines the operatorÃ inH a uniquely, but it does not define it uniquely inH. It is evident that the operatorÃ acting inH (continued fromH a to the whole spaceH) is determined up to the termÃ ′ , the matrix elements of which are zero in the spaceH a (Ã =Ã ′ +Ã ′′ ). If to introduce the projection operator PH a onto the subspaceH a and to require that the operatorÃ has no nonzero matrix elements in the orthogonal subspace, then the operatorÃ will be defined inH uniquely,Ã = PH aÃ ′′ PH a . We will omit the projection operator PH a when constructing the operatorÃ acting inH. The reason for this is the assumption that the matrix elements of the operators (in the position space), corresponding to a quite large external parameter R ∼ |x i − x j | give a dominant contribution to the quantum-mechanical processes. Further we will assume that R >> r 0 (usually in case of particle collisions R −1 ∼ √ mE, where E is the particle kinetic energy). The latter inequality makes it possible to choose parameter a (see (5)) as follows 10 : R >> a >> r 0 .
The matrix elements of the operators should not depend on parameter a chosen by this way. Let us mentally decrease the radius of a bound state, r 0 → 0. Then the whole written scheme, as we have already noted, does not depend on a up to the values a = r 0 . Therefore, the subspaceH a can be identified withH due to the inequality R >> a. In other words one can considers that PH a → 1 for a → 0. At the same time we do not break the quantum-mechanical description of bound states in virtue of the inequality a >> r 0 . From the physical point of view the inequality r 0 << R gives the stability domain for the bound states considered as elementary particles. The calculation of the following approximation is to be associated with accounting of the difference of the subspace H a from the space H.
Finally we note that for an arbitrary vector | ) ∈H a the following evident relations are valid: orζ (x)ξ(x ′ )H a = 0,H aζ whereζ andξ are any of the annihilation operatorsχ 1 ,χ 2 ,η.

III. THE STRUCTURE OFÃ(u, v) OPERATORS
Here we consider the method for obtaining the operatorsÃ. Let the operatorÂ represents a normal-ordered production ofψ i (v),ψ + i (u), (i = 1, 2): The operators of such type are the particle density operatorρ i (x), the momentum density operatorπ i (x), a Hamiltonian of the system, etc.
The matrix element in the right-hand side of (10) may be written as the following vacuum average: Let us note that when calculating this average by using Wick's theorem, the quantity, which is averaged over the vacuum state has the meaning of a mixed T -product if to consider the operatorsψ 1 , . . . ,ψ 2 , . . . ,φ . . . referring to the moment of time +0, the operatorsψ + 1 , . . . ,ψ + 2 , . . . ,φ + . . . to −0, and the normal-ordered operatorÂ(u, v) refereing to the moment of time 0. Thus, there is no need to place the contractions inside the expression forÂ(u, v). Let be the analytic expression that corresponds to the diagram "b". For this diagram the operators with the arguments u are related to the operators with the arguments y 1 , . . . , y 2 , . . . , y, . . .. In addition, the latter arguments are spaced apart by the distances greater than a and coincide with some of the arguments x 1 , . . . , x 2 , . . . , x, . . . ,. The similar statement should be also made concerning the arguments y ′ 1 , . . . , y ′ 2 , . . . , y ′ , . . .. Then, the operatorÃ(u, v) acting iñ H is given, according to (5), byÃ Here the summation is taken over all diagrams of the described type.
In this case the following five diagrams correspond to this operator (the indices 1 and 2 forψ andψ + can be easy restored if to take into account (3) and the definition of contractions). The analytic expressions for these diagrams have the form whence, we find the operators (15) corresponding to the diagrams (14), In obtaining the latter expression, we have taken into account the anticommutative relations forχ,χ + . Next bearing in mind (16), (18) we obtain the final expression forÃ( Similarly, one getsψ . . .
In general case the following formula is valid: To explain this formula we note that each ofψ j (v j ) (orψ + i (u i )) enteringÂ(u, v) (see (13)) is related to other operatorŝ ψ + (orψ) not enteringÂ(u, v) only by unique way, which leads to the binary relations (19), (20). Therefore, we come to (21) by sorting out all the operatorsψ i ,ψ i ′ containing inÂ(u, v).

IV. THE OPERATORS OF BASIC PHYSICAL QUANTITIES
In this section we consider the operators of basic physical quantities, which act in the Hilbert spaceH. Let us start from the density operator for particles of the first kind. The corresponding operator acting in the original Hilbert space H is of the formρ Hence, in accordance with (20), one findŝ Note that the operators with zero matrix elements in the subspaceH a appear in the right-hand side of (19) because of points u 1 and v 1 are close to each other. Sincê andφ(x, y) differs from zero only for |x − y| a, these operators according to (12) do not have the matrix elements in the subspaceH a and therefore can be omitted. Using the permutation relation {χ The matrix element of the second term is zero in the subspaceH a because of z 1 ≈ z 2 ≈ x (see (12)). For this reason this term can be omitted. Thus, with the use of the method that was described in previous section we havẽ Similarly, ifρ 2 (x) =ψ + 2 (x)ψ 2 (x) represents the density operator for particles of the second kind, thenρ 2 (x) →ρ 2 (x), whereρ Bearing in mind (16) and the assumption concerning the "small radius" of the bound state, one gets the following formulas: which allow us to obtain the densities operators for particles of the first and second kinds, Thus, the operatorsη + α (x),η α (x) can be interpreted as the creation and annihilation operators of the bound states with quantum numbers α at the point x, andη + α (x)η α (x) as the density operator of the bound states. For example, the first formula from (24) have a simple physical meaning: the density of particles of the first kind is equal to the sum of densities for free particles of the same kind and bound states (each bound state contains one particle of the first kind).
Consider a state vector Φ(X) =η + α (X)|0 , which specifies a compound particle at the point X (this state vector corresponds to continuous spectrum). Then, in accordance with (24), we have For a wave packet should be treated as the probability density to find the first particle at the point we come (as it should be) to the probability distribution for the space coordinate of the first particle in atom (the atom is at the point X 0 ).
Let us find now the momentum density operator in the spaceH. In original Hilbert space, the momentum density operatorπ 1 (x) for particles of the first kind is defined aŝ Then according to (20)π Following the derivation of (22), (23) forρ 1 (x) andρ 2 (x) we obtaiñ It is convenient for our further consideration to rewrite (25), (26) in terms of the center of mass variables y = y 1 − y 2 and Y = (m 1 y 1 + m 2 y 2 )/(m 1 + m 2 ): whereφ(y 1 , y 2 ) ≡φ(y, Y). Note that in terms of these variables the operatorsρ 1 (x) andρ 2 (x) have the form It is clear that (27), (28) can be expressed through the creation and annihilation operatorsη + α (x),η α (x) of atoms if to employ (17). Taking into account (27) it is easy to find the operatorπ =π 1 (x) +π 2 (x) of the total momentum density of the system in the approximation, where the radius of bound state is small, The third term in this formula is in accordance with interpretation ofη + α (x),η α (x) as the creation and annihilation operators of the bound state with quantum numbers α at the point x.
Finally let us consider a Hamiltonian in the spaceH. We suppose that this Hamiltonian has a standard form in the Hilbert space H and can be written asĤ =Ĥ 0 +V , whereĤ 0 andV are the operators of kinetic energy and potential energy given bŷ and ν ij (x − x ′ ) is a potential energy of interaction for particles of the kinds i and j. After the similar calculations that led us to the expressions forρ,π, we obtain Next changing to the center of mass variables (see (17)) and noting that one getŝ Let us find nowṼ (V →Ṽ ). According to (20) we havẽ (20)). Thus,Ṽ can be represented in the form where index i ′ is defined as 1 ′ = 2, 2 ′ = 1. ThenṼ 0 is of the form Note that the operatorsφ andφ + in (33) are normally ordered, whereasχ andχ + are not normally ordered. Therefore, we put them in order using Wick's theorem: : (y 4 ) : The operatorφ(x, y) ≡ ϕ α (x − y)η α ( m1x+m2y m1+m2 ) differs from zero only for x ≈ y, (|x − y| < a). Thus, only those of ϕ i , for which |y 1 − y 4 | < a contribute to the integral over y in (33). This means that the first term in (34) in virtue of (12) does not contribute to the matrix element ofṼ 0 taken between the states belonging toH a because ofφ i (x 1 , y 4 )χ i (y 1 )Φ = 0. Similarly, one can prove that the terms, which contain the single contractions in (34) do not give a contribution to the matrix element ofṼ 0 taken between the states inH a . The penultimate term in (34) containing the double contractions does not also contribute to the above mentioned matrix element. Indeed, the penultimate term in (34) equals to δ(y 1 − y 3 )δ(y 2 − y 4 ). In this case the nonzero matrix element exists for x 1 ≈ x 2 andφ j (x 2 , y 3 )φ i (x 1 , y 4 )Φ = 0 in virtue of (12). Thus, only the latter term in (34) equal δ(y 2 − y 3 )δ(y 1 − y 4 ) can give the contribution to the matrix element ofṼ 0 . Therefore, not changing the matrix elements inH a , the operator V 0 can be represented in the form or, according to (32), Similarly, noting that and performing the same derivation as forṼ 0 , it is easy to verify (using the anticommutative relations forχ,χ + and (34)) that we can considerṼ 1 =Ṽ 3 = 0 not changing the matrix elements ofṼ 1 andṼ 3 in the subspaceH a . Next it is evident that the following formula is valid: Finally let us findṼ 2 , It can be easily seen that the first two terms and the corresponding Hermitian conjugate terms do not give a contribution to the matrix element ofṼ 2 in the subspaceH a . Therefore, we can consider The first and second terms in this expression contribute to the matrix element ofṼ 2 inH a under the following arrangements of contractions: Thus, we havẽ The first term in this formula quadratic in field operators can be combined with the latter term in (31). As a result we obtain Since ϕ β (x) satisfies the Schroedinger equation where ε β are the atomic energy levels, the latter formula takes the form Hence, taking into account (30), (31), (36), the Hamiltonian of the systemH is of the form is the Hamiltonian for free particles and bound states, and are the Hamiltonians of interaction. The HamiltonianH 1 int corresponds to scattering of particles of the first and second kinds by bound states; the HamiltonianH 2 int corresponds to scattering of bound states by bound states; finally, the HamiltonianH 3 int corresponds to scattering of particles of the first and second kinds by particles of the same kinds. The Hamiltonians of interaction (38), (39) may be written through the creationη + α (x) and annihilation η α (x) operators of atoms by using of (17). We want to emphasize that the obtained Hamiltonians of interaction do not lead to decay processes and formation of compound particles as it should be in the low-energy approximation.
This fact reflects that atoms are absolutely stable in the main approximation.
In conclusion of this section we address to the Galilean invariance of the developed theory. We suppose that the initial theory based only on the field operatorsψ i (x),ψ + i (x), (i = 1, 2) is Galilean invariant. This means that under the following transformations:ψ the operators for particles numberN i , momentumP k and energyĤ transform as follows: Sinceψ i (x) andψ ′ i (x) meet the same commutation relations, they are related to each other by the unitary transformationψ where the unitary operator U v , is defined by 8 :

Let us show thatŨ
defines the Galilei transformations in the presence of bound states of particles (compare to (41)). To this end we note that this operator can be written, according to (28), in the form changing to the center of mass variables one finds, Therefore, just as we expected The employment of this formula and the canonic permutation relations result iñ Since the quantityφ + (x, X)φ(x, X) is not changed under such transformation, the Hamiltonians of interactionH 1 int , H 2 int ,H 3 int (see (38) -(40)) are invariant with respect to (42). It is easy to verify that the HamiltonianH 0 for free particles and bound states transforms according to the law whereP k ,Ñ 1 ,Ñ 2 ,Ñ b are the momentum operator and the particle number operators (for free particles of the first and second kinds and their bound states). These formulas prove the Galilean invariance of the developed theory.
Finally we note the validity of the following formula: which follows from (27), (28) by using (42).

V. ELECTROMAGNETIC INTERACTION
Here we consider the electromagnetic interaction assuming that the formation of bound states of particles is caused by Coulomb's (electromagnetic) forces. Therefore, the potential energy ν ij (x−x ′ ) entering (38)-(40) should be written as We also introduce the additional interactions of particles (see below) with an external electromagnetic field A (e) (x, t), ϕ (e) (x, t) and quantized electromagnetic field specified by a potentialâ(x) (Coulomb's gauge) (Ĉ kλ is the annihilation operator of a photon with momentum k and polarization e kλ ). As in section II, we identify the subspaceH a toH assuming that the matrix elements of the operators in coordinate representation give the main contribution to the quantum electrodynamics processes and correspond to the space scale ∆x a (a → 0, a >> r 0 ).
The density ε(x) of a Hamiltonian in the second quantization representation has the well-known form whereε f (x) is the energy density of a free electromagnetic field,ε C (x) is the energy density of Coulomb's interaction, the third term describes the interaction between the particles and external scalar potential. This formula leads to the following Hamiltonian of the system:Ĥ whereĤ are the Hamiltonians for free photons, free particles, and Coulomb's interaction respectively. The operatorV (t) represents a Hamiltonian that describes the interaction of particles with the electromagnetic fieldsÂ(x, t), ϕ (e) (x, t) whereσ i (x) = e iρi (x) is the charge density for particles of the kind i,σ =σ 1 +σ 2 . This Hamiltonian is exact and describes (in this sense) the processes in which the bound states of particles are involved. Sinceâ(x) commute among themselves and withψ i (x),ψ + i (x), in order to find the effective HamiltonianH(t) that describes electrodynamic processes at low-energies in the presence of bound states of particles we need, in accordance with section IV, to replace the operatorsψ i ,ψ + i ,â byψ i ,ψ + i ,ã: (ã(x) coincides withâ(x) and acts in the spaceH). As a result the effective Hamiltonian is defined by (44)-(49), in whereĥ (x) = rotâ(x),Ĥ (e) (x, t) = rot A (e) (x, t), and the operator of electric fieldÊ The transverseê t (x) and longitudinalê l (x) components of the electric field created by particles are defined as follows: Hereâ 0 (x) represents the operator of the scalar potential, Next it is easy to verify that lead to the following commutation relation: whereφ(x, X) ≡φ(x 1 , x 2 ) is determined by (17). Hereinafter we assume that a set of wave functions ϕ α (x ′ ) is complete. Therefore this set accounts for the wave functions of continuous spectrum. In addition, if we introduce the Then in accordance with the commutation relations for the field operatorsχ i (x), Next according to (30) we have

Bearing in mind
and taking into account the definitions ofρ i ,π i (i = 1, 2), one finds Similarly, Let us find now a commutator [ρ i (x ′ ),π k (x)], (i, k = 1, 2). With the use of (27) we have Noting also that we get finally These commutation relations along with (49) result in Thus, according to (52), (55), (56) we havė The definitions (52), (53) of the longitudinal component of the electric field along with the above formula givė Let us find the equation of motion forê t . The use of commutation relations for creationĈ + kλ and annihilationĈ kλ operators of photons, and also This formula and the definition ofṼ (t) allows us to find the commutator: Therefore, it follows from (57), (58) thaṫ Upon calculating this commutator with the use of (51), one obtainṡ In addition, (51), (52) give divĥ(x) = 0, divê(x) = 4πσ(x). These equations along with (59), (60) represent the Maxwell-Lorentz equations for the operators of electromagnetic field Notice that (46), (51), (58) result in where we neglect the unessential term kλ 1 2 ω k . Up to now we used the Schroedinger representation, in which the Schroedinger equation has the form In this representation a dot above the certain operatorḃ(x) (see (52)) does not mean the differentiation with respect to time. It means the following:ḃ Here the HamiltonianH depends on dynamic variablesχ(x ′ ) (these variables uniteχ andη) and time t. In the Heisenberg representation the vector Φ(0) = Ψ is taken as a time-independent state vector. Then, the time dependent Heisenberg operatorsχ Since the external fields H (e) , E (e) satisfy the Maxwell equations, the operators of total fieldsÊ,Ĥ in the Heisenberg representation satisfy, in accordance with (62), the equations where J (e) and σ (e) are the extrinsic current and charge densities.
Let us write out the expressions for the charge density operatorσ and current density operatorJ in terms of creation and annihilation operators of particles and bound states. Bearing in mind (50) we havẽ where the particle density operatorsρ i (x) are defined by (28) and the momentum density operatorsπ i (x) by (27).
The first terms in expressions forρ i (x) andπ i (x) define the contribution of particles, whereas the second terms define the contribution of bound states (see (27), (28)).
Let us find now the equations of motion for the field operators of particlesχ i (x) and bound statesφ(x, y). In the Heisenberg representation these equations are obtained from the general formulas (62). Indeed, it follows from These formulas allow to find the equations of motion (62) forχ 1 (x),χ 2 (x) in the Heisenberg representation, It is easy to find also the equation of motion for the field operatorsφ(x, y) of bound states, Hereυ The operatorsη α (X, t) related toφ( The equations of motion (63), (65) are gauge-invariant. Indeed, under the following transformations of fieldsχ i ,φ: ϕ(x, y, t) →φ ′ (x, y, t) = e ie1α(x,t)+ie2α(y,t)φ (x, y, t), and electromagnetic field, whereasπ i (x) transform (see (65)), by the laŵ As a result we have (see (64))J that provides a gauge invariance of the Maxwell-Lorentz equations (63). Next noting that we make sure of gauge-invariance of the equations of motion for the field operators of particlesχ i (x). Taking into account the transformation law for the field operators of bound statesφ(x, y), it is easy to prove a gauge invariance of the equation of motion forφ(x, y).
Sinceφ(x 1 , x 2 , t) = ϕ α (x)η α (X, t), we have under the gauge transformation (66), , Hence, we come to the transformation law for the operatorsη α (X): In conclusion of this section let us note thatψ i (x) transform under the gauge transformation as well asχ i (x):

VI. THE PROCESS OF SPONTANEOUS RADIATION BY ATOM
In this section we employ the obtained formulas in order to find the probability distribution of spontaneous radiation of an excited atom. To this end we use the general formulas for the scattering matrixŜ in terms of the T-operator: whereT (z) is found from the equationT In the first order of the perturbative approach over charge we have Here |i is the initial state (in which atom is in the state α, p) and |f is the final state (in which atom is in the state α ′ , p ′ and photon is in the state λ, k). According to (48), (50) we can takeV in the following form: where the current of bound states is defined bỹ By using of (48), (50) we find The formulas (69), (27) lead tõ and, consequently, where ∆(p) = δ p0 , and g αβ (k) = dy e 1 e iky(m2/M) + e 2 e −iky(m1/M) ϕ * α (y)ϕ β (y), The matrix element ofŜ-matrix is defined, in accordance with (67), by Next using the standard method we obtain the following expression for the probability of transition per unit time from the initial stateη + αp |0 into the final stateη + βp ′Ĉ + kλ |0 : In the domain of small k (kr 0 << 1, see section II) the Fourier-components of g αβ (x), g αβ (x) can be written in the If e 2 = −e 1 = −e (this case corresponds to the hydrogen atom), then g αβ (0) = 0, g αβ (0) = (e/µ) α|p|β . The Schroedinger equation for the wave function ϕ α (y) gives Therefore, whered = eŷ is the dipole moment of the atom. Hence, Since ε α (p) = ε α + p 2 /2M , we have for the atom of infinite mass (m 2 → ∞, M → ∞) 5 : (m 1 is the mass of light fermion).

VII. SCATTERING OF PHOTONS AND FERMIONS BY ATOMS
Here we study the scattering process of long-wave photons by atoms. For this purpose let us use the general formulas (67), (68), where the Hamiltonian of free particles is defined bỹ and the Hamiltonian associated with electromagnetic interaction bỹ In accordance with (48), (50), (further we assume e 1 = −e 2 = e). SinceṼ ′ ∼ e 2 ,Ṽ ′′ ∼ e, in the second order in charge e, the T -matrix can be written asT (within the considered approximation it is not necessary to take into account the fieldsχ i (x)). Bearing in mind (72), (28), the operatorṼ ′ can be written aŝ where q αβ (k) = dy e 2 m 1 e iky(m2/M) + e 2 m 2 e −iky(m1/M) ϕ * α (y)ϕ β (y). Similarly,Ṽ moreover, g αβ (k), g αβ (k) are defined by (70).
be the initial and final states of the system, Then the matrix element f |T (z)|i can be represented in the form Here we have taken into account that ω + ε α (p) = ω ′ + ε β (p ′ ) is a consequence of (67) and assumed E = ω + ε α (p).
According to our approximation kr 0 << 1 (λ >> r 0 ) we have q αβ (k) ≈ e 2 µ δ αβ , g αβ ≈ 0, wherep = −i∇ is the momentum operator. Noting that ε α (p) = ε α + (p 2 /2M ) we neglect the kinetic energy in denominators of R ′′ α ′ α (the kinetic energy is small in comparison to binding energy of bound states; see section II). The result is Let us express this formula through the matrix elements of the dipole moment of atom. It is easy to verify (by using of (71) that Therefore, the quantity R ′′ α ′ α is of the form whered = eŷ is the dipole moment of atom, and Next, using that ŷe, one finds According to (73), the quantity R ′ α ′ α is equal to R ′ α ′ α = e 2 µ e ′ * eδ αα ′ . Thus, the matrix R = R ′ + R ′′ takes the form and the amplitude of transition from the initial state into the final state is given by (see 5 ) The differential scattering cross-section is of the form Let us consider the scattering of an elementary particle by compound particle (the scattering of an electron by atom). Due to the structure of the Hamiltonian of interaction (38) it is sufficient to study the scattering of particle of the first kind by bound state. The matrix element ofŜ-matrix in the first nonvanishing approximation over interaction is defined by whereH 1 int is given by (38) and consequently, Here f | = 0|η α ′ (p ′ )χ 1 (k ′ 1 ) and |i =χ + 1 (k 1 )η + α (p)|0 . The simple calculations lead to For Coulomb's interaction of particles we have Therefore, in the long-wave approximation (qr 0 << 1), one gets whered = ex is the dipole moment of atom (d α ′ α = dxϕ * α ′ x(x)ϕ α (x)). Note that the Hamiltoniañ which describes the interaction between particles and bound states leads to the same result. Hereχ,η α are the field operators of particles and bound states respectively, and v α ′ α (x) = (e/x 3 )xd α ′ α . The relation (74) defines the interaction energy of the dipole moment d and electric charge e, which are at the distance x from each other.

VIII. THE VAN DER WAALS FORCES
In this section we investigate the forces acting between neutral atoms being in the ground state (the van der Waals forces) on the basis of the developed formalism. In order to solve this problem let us address to the Schroedinger equation that determines the energy spectrum of the system, HereH 0 andṼ =H 1 int +H 2 int +H 3 int are defined by (37), (38)-(40). Since the system in question consists of two atoms, we should seek the solution of (75) in the form The Hamiltonian of interactionṼ is equal, in accordance with (38)-(40), tõ where We suppose that the kinetic energy of atoms is small in comparison to the energy levels |ε α | (ε α < 0). In this case, according to (37), the operatorH 0 can be represented in the form It can be easily seen thatH These formulas show that we can seek the solution of (75) in a more simple, rather than (76), form so that the coordinates of atoms have the definite values in the state Φ αβ (X, X ′ ). Upon substituting this expression into (75) and using (78), one finds A perturbative approach for this equation can be easily developed in the domain of great |Z − Z ′ | when the quantity G λρ;γδ (Z − Z ′ ) becomes small (see (77)). Expanding K αβ;γδ in G, K αβ;γδ = K 1 αβ;γδ + K 1 αβ;γδ + K 2 αβ;γδ + . . . , one obtains in the zeroth order Taking into account this result we have in the first approximation, By setting here α = γ, β = δ, one gets and for α, β = γ, δ, The second order of the perturbative approach give Taking here α = γ, β = δ, one finds whence, according to (79), (80), we have (d βα are the matrix elements of the dipole moment of atom). Thus, the potential energy of interaction between atoms, Since ε β , ε λ > ε α , V (Z − Z ′ ) < 0. Therefore, the attractive forces (the van der Waals forces) act between the neutral atoms at long distances.
Consider now the scattering of a compound particle by compound particle, which are in the ground state. We start from the scattering matrix representation in terms of T -operators (67). Accurate to the terms, which are of the second order in interactionṼ between the atoms (see (68)), one getŝ Neglecting the kinetic energy of atoms we replace in the resolvent (E −H 0 + iη) −1 the Hamiltonian of free particleŝ H 0 by the HamiltonianH 0 of atoms in the rest (see (37)) (in the low-energy limit we can not do this under the argument of δ-function).
Since we study the collision of two atoms and the number of atoms is conserved during the process of collision, we can restrict ourselves by considering of this process in the two-particle subspace with the following completeness condition: In this case formulas (78) are true. Since the atoms are in the ground state, (83) is also valid and, therefore, the matrix element ofŜ-matrix turns to zero in the first approximation of the perturbation theory.
The second approximation of the perturbation theory is defined, according to (84), (85), by the formulã In the two-particle subspace this formula is equivalent tõ Since the initial and final states of a particle are the ground states, the scattering matrix in the two-particle subspace can be represented in the formŜ We can see that the second order inṼ of perturbation theory forŜ-matrix is equivalent to the first order of perturbation theory over effective interactionV ef f . This effective interaction is determined by van der Waals forces (see formula (82)).
This Hamiltonian corresponds to the dipole-dipole interaction of atoms.

IX. CONCLUSION
The goal of this paper was to develop a microscopic approach for describing the physical processes in many-particle systems in the presence of bound states of particles. To achieve this goal we developed an approximate second quantization method for systems with bound states of particles.
The basic results obtained in this paper are the following: 1. The Fock space was introduced in the second quantization formalism. In this space the creation and annihilation operators of elementary particlesχ + ,χ and their bound statesη + ,η were introduced on an equal status.
2. The operators of basic physical quantities acting in this space were constructed. These operators include the Hamiltonians of interaction between elementary particles and their bound states.
3. It was shown that in the approximation when the radius of interaction is small the above mentioned Hamiltonians transform into the well-known Hamiltonians for Coulomb's and dipole interactions between the particles of various kinds.
4. The non-relativistic quantum electrodynamics of charged and neutral (the bound states) particles was constructed.
5. The various physical effects including the theory of the van der Waals forces acting between atoms were considered as the approbation of the developed formalism. The description of such effects within the usual formalism requires more considerable efforts associated with introduction of the interaction between neutral currents of bound states and electromagnetic field.
6. More detailed calculations, which fall out the limits of our approximation should result in appearance of three-and many-body interactions between unbound particles and two-body bound states. These many-body interactions should be consistent with pair interaction of original particles. The appearance of these many-body terms is analogous to the origin of the terms with photon-photon scattering in transition from the usual standard Lagrangian to the effective low-energy Euler-Heisenberg Lagrangian of quantum electrodynamics.
Especially we would like to emphasize a role of the obtained Hamiltonians, which describe the interaction of quasineutral particles (the bound states of charged fermions) with electromagnetic field, the elementary particles with bound states, and also the bound states with bound states. On the basis of these Hamiltonians one can study such phenomena as Bose-Einstein condensation in a gas of excited atoms, the interaction of condensates with an electromagnetic field in Bose and Fermi systems. These Hamiltonians can be also the basis of the kinetic theory for systems with bound states of particles.
Finally we would like to stress that the developed method can be easily generalized to the case of bound states consisting of more than two particles. The generalization of the offered method for describing the systems with bound states of bosons and also of bosons and fermions taking into account the spin of particles can be also performed without principal difficulties.