Free particle density operators

B. Binary Collision Approximation for the Two-Particle Density Operator— Kinetic Equations for Free Particles and Atoms... 

In order to construct a collision integral for a bound-state kinetic equation (kinetic equation for atoms, consisting of elementary particles), which accounts for the scattering between atoms and between atoms and free particles, it is necessary to determine the three-particle density operator in four-particle approximation. Four-particle collision approximation means that in the formal solution, for example, (1.30), for F 234 the integral term is neglected. Then we obtain the expression... 

The operators W, A, occurring above, should be taken in the second-quantization form, free of explicit dependence on particle number, and Tr means the trace in Fock space (see e.g. [10] for details). Problems of existence and functional differentiability of generalized functionals F [n] and r [n] are discussed in [28] the functional F [n] is denoted there as Fi,[n] or Ffrac[n] or FfraoM (depending on the scope of 3), similarly for F [n]. Note that DMs can be viewed as the coordinate representation of the density operators. 

C. Spin-Free Excitation Operators and -Particle Density Matrices... 

The spin-free two-particle excitation operators and density matrices are symmetric with respect to simultaneous exchange of the upper and lower indices, but neither symmetric nor antisymmetric with respect to exchange of either upper or lower indices separately ... 

For spin-free fe-particle excitation operators and density matrices, linear combinations that transform as irrep of the symmetric group Sk can be defined in an analogous way [15]. 

Now we want to generalize the kinetic equation for free (unbound) particles that is, we want to derive a kinetic equation for free particles that takes into account collisions between free and bound particles as well. For this purpose it is necessary to determine the binary density operator, occurring in the collision integral of the single-particle kinetic equation, at least in the three-particle collision approximation. An approximation of such type was given in Section II.2 for systems without bound states. Thus we have to generalize, for example, the approximation for/12 given by Eq. (2.40), to systems with bound states. 

With Eq. (3.37) for Fn it is possible to write a kinetic equation for F, that describes the formation and the decay of two-particle bound states in three-particle collisions. Introducing (3.37) into the first equation of the hierarchy (1-29), we obtain in a similar way as in Section III.2 a kinetic equation for the density operator of free particles. This equation may be written in the following form ... 

In this equation, V2 = d2/dx2 + d2/dy2 + d2/dz2 denotes the Laplacian operator of cartesian second derivatives, p(r) is the charge density in a spherical shell of radius r and infinitesimal thickness dr centered at the particle of interest (see diagram), k is the effective dielectric constant, and e0 is the permittivity of free space (8.854 x 10 12 in SI units). The energy of interaction / , of ions of charge z,c with their surroundings,... 

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