A generalized scattering theory in quantum mechanics

In quantum mechanics textbooks, a single-particle scattering theory is introduced. In the present work, a generalized scattering theory is presented, which can be in principle applied to the scattering problems of arbitrary number of particle. In laboratory frame, a generalized Lippmann-Schwinger scattering equation is derived. We emphasized that the derivation is rigorous, even for treating infinitesimals. No manual operation such as analytical continuation is allowed. In the case that before scattering N particles are plane waves and after the scattering they are new plane waves, the transition amplitude and transition probability are given and the generalized S matrix is presented. It is proved that the transition probability from a set of plane waves to a new set of plane waves of the N particles equal to that of the reciprocal process. The generalized theory is applied to the cases of one- and two-particle scattering as two examples. When applied to single-particle scattering problems, our generalized formalism degrades to that usually seen in the literature. When our generalized theory is applied to two-particle scattering problems, the formula of the transition probability of two-particle collision is given. It is shown that the transition probability of the scattering of two free particles is identical to that of the reciprocal process. This transition probability and the identity are needed in deriving Boltzmann transport equation in statistical mechanics. The case of identical particles is also discussed.


Introduction
The single-particle scattering theory in quantum mechanics (QM) is a mature theory with complete formalism.It is usually introduced in QM textbooks [16].Trace to its source, Lippmann and Schwinger did the first work and they established the famous scattering equation named after them [7].
There are three motivations for the present work.The first motivation is to generalize the single-particle scattering theory to manyparticle one, so that one is able to deal with the scattering problems concerning arbitrary number of particle.In the literature, besides the single-particle scattering [7,8], the fewparticle scattering problems have also been touched, e.g., [912].Our aim is to give a uniform formalism applicable to arbitrary number of particle, as well as to identical particles.We will start from time-dependent Hamiltonians to obtain universal formulas.Time-independent Hamiltonians are special cases.When our theory is applied to the case of single-particle, the formalism of the scattering theory introduced in QM textbooks is naturally retrieved.
The second motivation is that the formalism should be derived rigorously, even for the occurrence of infinitesimals, without manual operation such as analytical continuation or manually inserting a factor into a formula, which is explained in Appendix A.
To specify this point, let us first recall how Lippmann and Schwinger obtain their time-independent scattering equation.We start from any time-dependent Hamiltonian.Suppose that the Hamiltonian H of a system can be divided into two parts, 0 1 H H H . (1.1) Usually, the 0 H is chosen such that its eigenvalues and eigen functions are easily solved.For example, for an N-particle system, 0 H is chosen as the Hamiltonian of the N noninteractive particle, so that its eigenvalues and eigenvectors are those of plane waves, and 1 H is chosen as the interaction between the particles.The Schrödinger equation corresponding to the H and 0 H are respectively Here, the spatial arguments of the wave functions are not explicitly written.If the Hamiltonian (1.1) is time-independent, a time factor in the exponential form can be separated from the wave functions as follows., where 0    , into the integral expression of the solution () t  such that they obtained a time-independent equation [7,8,10,11]: and making analytical continuation belong to manual operation but not rigorous derivation.Analytical continuation may not be unique, e.g., Eqs. ( 6) and (7) in Ref. [35].
The aim of introducing the infinitely small imaginary part i0  in the denominator of (1.5) was to reflect the effect of time delay.One "used an infinitesimal imaginary energy part i  to obtain, respectively, the incoming and outgoing solutions of the Lippmann-Schwinger equations, and distinguished between ''states at time 0 tt   =time defined by preparation'' and ''states characteristic of the experiment,'' observed at 0 tt   ."[36] This infinitesimal imaginary part was thought as a boundary condition.
When (1.5) is written in the form of , (see Eq. (2.3) in [36]), then, | E   "fulfill different boundary conditions expressed by +i0 and i0." [36] Since the i0   in the denominator in the Lippmann-Schwinger equation respectively show the significance of time delay and advance, they ought to be derived naturally, but not added through a manual operation of artificially inserting a factor || e t   or of making analytical continuation.In a previous work by the author [35], it was pointed out that the time step function There was a detailed discussion of how to achieve this transformation [35].Its derivative is [35,37,38] Usually, the infinitesimal is ignored in the literature.We stress that it should not be ignored, because it is of the physical meaning of time delay.The Dirac delta function 0 () tt   itself does not reflect the meaning of time delay, but 0 () tt   does.Since in Eq. (1.7) the left hand side is of the meaning of time delay, the right hand side should also be.The infinitesimal in (1.7) comes from the expression (1.6), which correctly embodies time delay, and it is also the source of the infinitesimal imaginary part in the denominator of (1.5).It seemed that there was only one researcher [39] who was aware of that the infinitesimal in (1.5) was from the Fourier transformation (1.6).
The third motivation is to prove an identity needed in statistical mechanics.
It is well known that derivation of Boltzmann transport equation involves an important physical quantity "The function w can in principle be determined only by solving the mechanical problem of collision of particles interacting according to some given law.However, certain properties of this function can be elucidated from general arguments."After these words, it was argued that the assumption of (1.8) was reasonable [40].In [41],  [41]).In [42], the same assumption was made: "In general, the symmetry does not necessarily hold.However, let us assume this for simplicity."In [43,44], Eq. (1.8) was illustrated by graphical manner without rigorous mathematical derivation.
In short, Eq. (1.8) is assumed to be true, but no proof has been given in QM.The reason is that when discussing the collision between two particles in the literature, usually, the mass-center frame, but not laboratory frame, is adopted such that the twoparticle collision problem is reduced to be one-particle scattering [2,3,6].The formalism for one-particle scattering is unable to give the expression of The generalized scattering theory is in principle able to deal with the problems of scattering of an arbitrary number of particle in laboratory frame.In the case of two-particle collision, the expression of  2.37) below) and time-independent equation (1.5) (also see Eq. (2.41) below, which is obtained under the condition that the Hamiltonian is time-independent) are derived.The generalized S matrix is put forth, and its unitary is proved.In Section 3, the theory is applied to the case of single-particle scattering.The discrepancy between two scattering pictures appeared in textbooks (see Figs. 1 and 2 below) is conceptually clarified.In Section 4, the theory is applied to the case of two-particle collision, and Eq.(1.8) is proved.Our conclusion is in Section 5.
In this work, we merely discuss elastic scattering.We do not consider the inner structures of colliding particles, and do not consider inelastic collision.The two words scattering and collision are regarded as synonym.Laboratory frame is employed.We choose laboratory frame because all experiments are done in laboratory frame.In textbooks, when treating the problem of two-particle collision, usually, the mass-center frame is adopted as mentioned above.However, the results have to be transformed to be those in the laboratory frame so as to compare them with experimental ones.Moreover, for the problems of collisions of more than two particles, the mass-center frame is very difficult to use.

The generalized scattering theory
Suppose that a system contains N particles and its Hamiltonian is H.The set of all the spatial coordinates in this system is denoted by an italic letter R. The Schrödinger equation is In the following formulas, the spatial arguments may be dropped and only the time argument is explicitly shown.Equation (2.1) is a differential equation with the first derivative of time.In order to solve such an equation for a concrete system, an initial condition is required.The initial time is denoted by 0 t .The initial condition sets the values of the solutions at time 0 t .
The solved solutions apply for the time 0 tt  , i.e., the time-delay or time-retarded solutions, which was mentioned previously [35] It is seen that Eq. (2.5) contains Eq. (2.7) and the initial condition (2.8) simultaneously.
We stress that it is R G that is of the meaning of time delay.The G reflects neither time delay nor time advance.So, it can be called an auxiliary function [35].We simply call the G as Green's function, consistent with habit.

If the R
G is worked out, it can be employed to find the retarded solution of (2.2) through the following procedure.An inverse operator This is a fundamental formula in path integral [49].
In this work, we merely discuss retarded solutions.So, hereafter, if in a formula the factor 0 () tt   or the superscript R does not appear, there is an acquiescence that 0 tt  . (2.10) We point out that it is Eq.(2.9a) that is rigorous.Equation (2.9b) is actually a simplified form of (2.9a) as (2.10) is satisfied.Often, it is hard to exactly solve the wave functions and Green's functions of a Hamiltonian H.A commonly used method is dividing a Hamiltonian H into two parts, 0 1 where the 0 H is chosen such that its eigen wave functions (2.12) )

.13)
It follows that Eq. (2.12), when (2.13) is used, contains both the Schrödinger equation Its solution should also have a time step function factor.
After (2.16) is substituted into (2.15),we obtain the equation satisfied by (0)   G , and an initial condition The retarded wave function can be expressed by the retarded Green's function.From (2.9a), it is known that When (2.10) is defaulted, the time step functions can be dropped and we have Equation ( 2.19) has the same physical significance as (2.9).The former is just for the 0 H system.Both equations can be iterated repeatedly.The time step functions in these formulas guarantee the semigroup evolution in time order [36].
It is a customary choosing a time-independent 0 H the eigenenergies {} n E and corresponding eigen wave functions of which are easily solved.Note that we are considering a system containing an arbitrary number of particle, so that here the subscript n actually represents a set of quantum numbers.The stationary equation is Then, the stationary wave function is

.21)
All the eigen wave functions constitute a complete set. ( (2.22) then, it can be proved by use of ( 21)-( 23) that The physical significance of (2.26a) is similar to that of (2.9b).
Similarly, suppose that a Hamiltonian H is time-independent and its eigen spectrum is easily solved.If a function * ( , ) Rt  can be expanded by the complete set of the eigen functions of the H, just as (2.25), then we have which corresponds to (2.9a).It is noted that (2.9) is valid for any Hamiltonian, while (2.26) is proved only for time-independent Hamiltonian.
Once the quantities of 0 H system are gained, they can be utilized, together with the interaction 1 H , to find the quantities of the H system.For instance, the R(0) G and 1 H can be used to express the R G .We substitute (2.11) into (2.5) and obtain is acted on both sides of (2.27), and then, the This equation can be iterated repeatedly.As an abbreviation, we drop the integrals in (2.28) to simplify it to be the products of the functions. where The T was called t matrix, a short form of transition matrix [6].Strictly speaking, T should also have a superscript R. Nevertheless, dropping the superscript does not affect the discussion below.Please note that all the Green's functions in ( then, the T defined by (2.31) is of the property We suppose that at time 0 t , the wave function (0) 0 Then, Eq. (2.9) is written as This equation has the same physical meaning as that of (2.9), with the initial state Repeated iteration of this equation, with the help of (2.30) and (2.31), leads to Equations (2.36) and (2.37) are formally the same as Lippmann-Schwinger equation, so that can be called generalized Lippmann-Schwinger equation, which can be used to systems composed of any number of particle.We explicitly write the step functions in (2.37).
(0) 0 0 Up to now, the derivation is rigorous, and Hamiltonian H can be an arbitrary one.
If the Hamiltonian is time-independent, the formulas above can be simplified somewhat.The time factor of a wave function can be separated, as in the form of Eq. (1.4).Then, (2.37) becomes This equation is exact, as long as Hamiltonian is time-independent.
We are considering the solution at the time satisfying (2.10 This is generalized Lippmann-Schwinger equation for time-independent systems.As a comparison, Eq. (2.39) applies to any time-dependent Hamiltonian.By the way, the formulas in Appendix B in [12] were derived from (2.41).It is easily shown that from (2.39), the same formulas can be derived.
Here, we stress the differences between Eq. (2.41) and the original Lippmann-Schwinger equation, although they are of the same form.One is that (2.41) is not simply a single-particle equation.The Hamiltonian H (2.11) can contain arbitrary number of particles.Another one is that the infinitesimal in (2.41) is not from manual analytic continuation, but from Eq. (2.15) and its solution (2.16) where time retard has been embodied.
Because of the rigorous derivation above and implicit geralized physical meaning, we are able to do the following work.Now, we further assume that the initial state (0)    in (2.37) is just one of the 0 H 's eigenstates, i.e., one satisfying (2.21), denoted by The corresponding final state is denoted as   ).i With the help of the second expression of the R(0) G in (2.23), any order term of (2.44) can be expressed uniformly, and they are presented in Appendix B.
The transition probability from the (0) Furthermore, assume that the 1 H satisfies (2.32).Then, (2.33) is also satisfied.In this case, (2.44) is simplified to be Suppose that there are N free particles.After they collide with each other, they become free again.This is the transition from a set of plane waves to a new set of plane waves.Equation (2.50) reveals that the transition probability of this process equals to that of the reciprocal process.
Because the Hamiltonian 0 () HR is time independent, its eigen functions are of the form (2.21).We let Please note that because of the time step function, the upper and lower limits of the integral are actually  .In (2.51), an operator is defined: When collisions between particles happen, usually, the time of collision is much less than the time that a free particle takes going through its mean free path.Therefore, in evaluation, the upper and lower limits of the integral in (2.52) can be extended to positive and negative infinities.H is time independent, we have The transition probability per unit of time is This expression is called "Fermi's Golden Rule" [50,51].
The summation of all the final states results in

.60)
The above derivation starts from Eq. (2.1).The Dirac equation is also of this form, so that the above formalism is valid for both the Schrödinger equation and Dirac equation.The formalism applies to three-, two-, and one-dimensions.In the following, the above generalized scattering theory is applied to one-and two-particle systems.

Singe-particle scattering
It is seen that the forms of the formulas in the generalized scattering theory are basically the same as those of the single-particle scattering theory introduced in the literature.Now, we apply the generalized theory to the case of single-particle scattering.On one hand, this is the simplest example.On the other hand, we introduce the two scattering pictures usually presented in textbooks, clarifying the concept of the probability of the transition between the initial and final states.
The spatial coordinate set R is that of a particle in three-dimensional space, R  r .0 H is the Hamiltonian of a free particle.The  p r .Equation (3.9) is the special case of (2.43), and is the basis of evaluation of cross section.
In the following, we schematically show scattering by figures.In the figures, we use a straight line with an arrow and a bold letter to represent a wave function with a momentum.A solid line means the incident wave before scattering, and dashed lines mean outgoing waves after scattering.
We assume that the potential 1 H in (3.9) is independent of time.Equation (3.9) reflects the physical picture of scattering as in Fig. 1 where the wave functions of  Besides Fig. 1, there is another picture describing the single-particle scattering, usually seen in textbooks of relativistic QM and quantum electrodynamics, shown in Fig. 2(a).
Figure 2(a) describes the scattering process in the following language.The initial state is a plane wave of an incident particle with a momentum p.When the particle is far away from the origin, it does not feel interaction.When it is approaching the origin, the 1 H is turned on, and the particle suffers the interaction.After the scattering, the particle moves away from the origin.When it is far enough, it does not undergo the interaction anymore.That is to say, the 1 H is turned off, and the particle become a new plane wave with a momentum  p ，as shown by the arrowed dashed line in Fig.

2(a). According to this description, the potential 1
H varies with time.The initial and final states do not suffer interaction so that both are plane waves.Figure 2  The author thinks that Fig. 1 describes the single-particle scattering correctly.Firstly, in Fig. 1, the dashed lines represent the wave function of (3.9), which is the solution of the Schrödinger equation with Hamiltonian H.The scattering wave spreads in all directions.This should be the correct physical picture of the scattering.In contrast, the plane wave  p in Fig. 2 does not satisfy the Schrödinger equation with Hamiltonian H. Secondly, the whole Hamiltonian of a gas in an equilibrium state is time-independent.
The gas is composed of a large amount of molecules that collide with each other frequently.The Hamiltonian of a pair of particles colliding with each other should be independent of time because the whole Hamiltonian of the gas is.
According to the description of Fig. 2, the scattering Hamiltonian varies with time, so that it is a virtual Hamiltonian.The final state is a plane wave, which does not agree with (3.9).
Nevertheless, Fig. 2(a) is still useful in understanding transition probability.In fact, the plane waves (3.4) constitute a set.The scattering wave, the second term in (3.9), can be written as the linear superposition of all the plane waves.Figure 2(a shows is actually the projection of the final state onto a plane wave with momentum  p .Besides this plane wave, the final state can also project onto other plane waves.On every plane wave, there is a projection probability. The project amplitude is expressed by Eq. (2.44).We are considering elastic scattering.Substituting (3.8) and (3.9) into (2.44),we obtain 0 (0)* (0 This is the projection amplitude of the final state in Fig. 1 onto the outgoing plane wave  p in Fig. 2. It is also the transition amplitude of a particle from plane wave with p to that with  p .Here, we have assumed that 1 H is time-independent.If the T only takes the first-order term in (2.31), the first-order term in (3.10) is For a Coulomb potential, where 1 q and q are the charges of the incident particle and scatter center, respectively.
Under this potential, the first-order transition amplitude will be  ( , ; ) The transition amplitudes of this pair of reciprocal processes are complex conjugate to each other.Consequently, their transition probabilities are equal., ; ) | ( , , ) |  ( , ; ) It is easily seen that when all the momenta above take opposite directions, Eqs. ( still stand.That is to say, ( ) In the direction of the  p , the differential cross section is proportional to the transition probability (3.15),There was a more concise formula to evaluation the differential cross section [6]: where 0 j and s j are the currents of the incident and outgoing particles, respectively.
In one-and two-dimensional spaces, Eq. (3.16) is respectively modified to be .We [54] utilized these formulas to evaluate the differential cross sections up to the first order for both the low-momentum and relativistic particles in three-, two-, and one-dimensional spaces.

Two-particle scattering
We take laboratory frame, in which both particles move before and after collision.In Fig. 3(a) illustrated is the collision process of two classical particles.We assume that the momenta 1 p and 2 p of the two particles before the scattering are known.After the scattering, their outgoing momenta are 1  p and 2  p , respective.Please note that the outgoing momenta are not unique, which was discussed in [55].Figure 3(b) illustrates the reciprocal or inverse process of Fig. 3(a).
We are investigating scattering problem in QM.The spatial coordinate set R is the coordinates of two particles in three-dimensional space, The total energy of the two free particles is

Distinguishable particles
The stationary equation of Hamiltonian (4.1) are as follows. ( The two single-particle' stationary wave functions are A wave function satisfying Eq. (2.14a) is the product of two particles' wave functions (3.4).
The corresponding Green's function is H between the two particles is real, then, the is also real.When the two particles are far away from each other, the interaction between them is weak enough so that they can be treated as free particles.When they are close to each other, the interaction 1 H takes effect.After the collision, the final state is evaluated by Eq. (2.43).Substituting (4.5) and (4.6) into (2.43),we obtain 12  We consider the inverse process of Fig. 3(a) [43,44,5659].That is to say, the As a matter of fact, in Section 2, we have represented a universal formula showing that any scattering transition and its reciprocal process have the same transition probability for any number of particle.Suppose that N free particles collide.After the collision, the final state is projected onto a new set of N plane waves with momenta different from those before the collision.The formula of the projection amplitude was given, and then, the equality of a pair of reciprocal processes of scattering was proven, as manifested by Eqs.(2.44)- (2.50).
When the interaction between two particles is the first-order transition amplitude, by (4.9), will be In the following discussion, we omit the time integrals in (4.9).
Let us consider a detector in the ( , )  direction.The particle current it detects is proportional to the transition probability by which the outgoing particles move in this direction.
Since the scattering is elastic, the total momentum and kinetic energy of the two particles are conserved.
Please note that we should not say that (4.21b) is the exchange term of (4.21a).This is because both The application of three-particle collision will be investigated later.Here, we merely mention that the triad of three-particle Lippmann-Schwinger equations [25][26][27]29] were actually from flexible divisions of 0 H and 1 H in the total three-particle Hamiltonian H.Each division lead to an equation, and the equations were combined to be solved.

Conclusion
In this work, a generalized scattering theory in quantum mechanics is established.The generalized Lippmann-Schwinger equation and scattering matrix are presented.Although the generalized formulas seem formally the same as those of original Lippmann-Schwinger theory, the physical implications are not the same.The generalized formalism can be applied to scattering problems of arbitrary number of particle.We stress that the derivation is rigorous, even for infinitesimals, without manual operation such as analytical continuation.
When this generalized theory is applied to single-particle scattering problem, the results usually introduced in quantum textbooks are retrieved, e.g., Lippmann-Schwinger equation and S matrix.We clarified the concepts of the final state after scattering and transition.
When this generalized theory is applied to two-particle scattering, the formula of two-particle transition probability is derived.This probability emerged in the collision term in Boltzmann transport equation.It is proved that a pair of reciprocal scattering processes have the same transition probability.For the collision between two identical particles, the formula of transition probability is also presented.
Suppose that there are N free particles described by plane waves.They collide and transit to a new set of plane waves.The transition probability of this scattering process is the same as that of the reciprocal process.
This work provides a useful means for dealing with the problems of many-particle scattering.Its realistic meaning will be shown in the author's following works.Here, we give two clues.
One clue is related to quantum electrodynamics (QED) where various scattering problems of particles are investigated.Usually, the famous Feynman propagator is used in evaluating scattering cross section.The Feynman propagator was obtained by analytic continuation, while the retarded Green's function in the present work is solved from a differential equation.There is a bit of difference between them (please compare Eqs. ( 43) and ( 53) in [35]).We will enter QED.In a previous work [54], the Green's functions were given, and the cross sections were evaluated up to the first-order term.In later works, we will revisit the topics in QED by means of the generalized scattering formulas presented in this paper, and clarify the role of the negative kinetic energy (NKE) solutions of relativistic quantum mechanics equations.Dirac explained the NKE solutions as the holes in filled electron sea.We have pointed out [61] that this explanation lead to contradictions.Feynman explained the NKE solutions as antiparticles moving counterclockwise.We will elaborate the physical picture of the NKE solutions in scattering processes.In the appendix D in [62], we listed 13 points that were topics to be studied.All of them, except point 12, have been dealt with in the author's works [61,[63][64][65][66][67][68][69][70][71].Point 12 will be touched later.
The other clue is related to the irreversibility of motion.It has been a long time that people felt puzzled by a paradox: every motion happened in reality is irreversible but the equations that reflect physical laws, say, Eq. (2.1), are time-reversible.We will figure out this problem.We will tell why all the happened motion is time-irreversible.
Here, we merely mention the discrepancy of Eqs.(2.1) and (2.2): the former is of invariance of time-inversion but the latter is not.In the present work, we have shown that the transition probabilities of a pair of reciprocal scattering processes are the same.
In terms of this result, we will tell the real meaning of the "detailed balance". (3) where the nth order term is With the help of (2.23), we obtain the first to third terms in the following forms.How to make use of this formula is still under investigation.
i0  .Since Eq. (1.5) appeared, it has often been employed as the starting point of studying scattering problems [834].Another way of obtaining (1.5) is to make the E in Green's function( the meaning of time delay.That this factor was contained in a wave function meant that we were considering an evolution of a system at any time 0

G 1 H
can be easily solved (see Eqs. (2.12) and (2.15) below), and is the remaining part of the H. Here, a superscript (0) marks the quantities belonging to 0 H system.The above formulas (2.1)-(2.9)are valid to any Hamiltonian H, and valid to 0 H too. We simply make the following substitutions in (2.1)-(2.9):H is replaced by 0

. 43 )HH
The physical meaning of this equation is more explicit.The initial state(0)   .When acted by the Hamiltonian H, it evolves into a final state n  that belongs to the H. constitute a complete set.Any function can be expanded by this set, and so can the n  in (2.43).The projection of the n mn

. 45 )
On the other hand, we can let the initial state be corresponding final state is obtained by replacing n in (2.43) by m.This final state can also be expanded by the compete set (0) {} n  , and its projection onto the (0) n  is denoted by nm C , which can be put down simply by exchanging the subscripts n and m in (2.44).The corresponding transition probability is 2 illustrated.In Fig.1, the arrowed solid line is the initial state, a plane wave with a momentum p, which is the function(3.4).The dashed lines represent the final state, the eigen function of 01 HH  .The final state actually contains two parts, represented by thicker and thinner lines, respectively.The thicker dashed line is an outgoing plane wave, which is the first term in (3.9).The thinner lines are the second term in (3.9), which spread in all directions.

Fig. 1 .
Fig. 1.The picture of single particle scattering [5,6,46,5153].The solid line means the incident wave (a) is conceptually different from Fig.1in that the former means the Hamiltonian depends on time while the latter does not and in that the two final states are different.

Fig. 2 .
Fig. 2.An illustration of one-particle transition process and its reciprocal process.(a) A particle with momentum p suffers an interaction at the origin, and then, transits to a ) only shows one outgoing plane wave with a momentum  p .Hence, what Fig.2(a) the inverse process of Fig. 2(a).When the p and  p are exchanged, the resultant is shown in Fig. 2(b).In Fig. 2(b), the plane wave  p is the incident wave, and the outgoing plane has a momentum p.The Hamiltonians of Figs.2(a) and (b) are the same.We want to calculate the transition amplitude of this inverse process.This is easily done because we merely need to exchange the p and  p in (3.10), and the result is * ( , ; )

p and 2 p
Fig.3(a), two arrowed solid lines with letters 1 represent the plane waves before the collision.After the collision, the form of the final state because the scattering waves of the two particles spread in all directions, as may be imagined referring to Fig.1.We merely consider the amplitude that the two particles transit, after the collision, to two plane waves with momenta 1 transition is sketched in Fig.3(a).That is to say, what we want to compute is the projection of the final state Eqs.(4.7) and (4.8) in (2.44), the transition amplitude is 0

Fig. 3 . p and 2 
Fig. 3. (a) Illustration of two-particle collision.In classical mechanics, two particles with momenta 1 p and 2 p collide, and after that, their momenta become 1  p and other hand, if 2  p is in the ( , )  direction, the 2  p and 1  p in this case are denoted by 2 ( , ) 20) which itself has contained exchange effect already.
This paper is arranged as follows.In Section 2, the generalized scattering theory is presented.A general time-dependent Lippmann-Schwinger equation (see Eqs. (2.36) and ( is Eq.(3.5).Equation (4.6) is simply the product of the two particles' Green's functions.It is of the property (2.24).The wave function (4.5) and Green's function (4.6) satisfy (2.19) and (2.26). g is used in establishing Boltzmann transport equation in statistical mechanics.In the literature, Eq. (1.8) is postulation.Here, we prove it.The key is that we give the formula evaluating the transition amplitude of two-particle scattering in laboratory frame.Equation (1.8) is hardly proved if the problem of two-particle scattering is reduced to one-particle scattering in mass-center frame.