Particle density operator

Some Theorems Concerning Particle Density Operators. Applying the theorem of Eq. (8-103) to the last factor of that same equation, we have ... 

Second quantization transforms the Schrodinger particle density into a particle density operator,... 

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

Because of the interaction this equation is not closed it is coupled with the two-particle density operator. Therefore, it is necessary to find an... 

Thus the binary density operator is given only by the single-particle density operator Fl, which depends on the earlier time t0. Therefore, retardation effects appear. In order to eliminate this time we use the formal solution (1.30) for F, ... 

It is essential to note that the contribution to the potential energy comes from the retardation correction of the two-particle density operator, which describes three-particle interactions of the self-energy type. 

Let us now consider the Bogolyubov condition for the three-particle density operator. In generalizing (2.4), the Bogolyubov condition now takes the form (if three-particle bound states are absent)... 

It turns out that it is impossible now to get a closed equation for the single-particle density operator alone. We also have to consider a kinetic... 

We will carry out our program in two steps. In this section we will derive the two-particle density operator Fn in a three-particle collision approximation for the application in the collision integral of Fl. As compared with Section II.2, the main difference will be the occurrence of bound states and, especially, the generalization of the asymptotic condition, which now has to account for bound states too. For the purpose of the application in the kinetic equation of the atoms (bound states) we need an approximation of the next-higher-order density matrix, that is, F 23 This quantity will be determined under inclusion of certain four-particle interaction. 

As in Section II, the initial values F12(t0) and F123(/0) for the binary and the three-particle density operators must be determined. For this purpose we have to generalize the Bogolyubov condition of the weakening of initial correlations given by (2.4) for systems that do not support the formation of bound states. 

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 next problem is the determination of the initial value of the three-particle density operator. Using the generalized Bogolyubov asymptotic condition in the approximation (3.39), we obtain finally... 

Obviously we may expect that the simple two- and three-particle collision approximation discussed in the previous sections is not appropriate, because a large number of particles always interact simultaneously. Formally this approximation leads to divergencies. In the previous sections we used in a systematic way cluster expansions for the two- and three-particle density operator in order to include two-particle bound states and their relevant interaction in three- and four-particle clusters. In the framework of that consideration we started with the elementary particles (e, p) and their interactions. The bound states turned out to be special states, and, especially, scattering states were dealt with in a consistent manner. 

Let us assume that, at time t = f,, the density operator of the global particle-plus-bath system is factorized in the form ppart where pbath denotes the thermal equilibrium density operator of the unperturbed bath and ppart, the particle density operator. 

Here, v(r) denotes the external potential, j/ r) is the usual fermion field operator, and p r) is the particle density operator. 

Here, is the normalization volume, p(r) is the particle-density operator, and [p(r)] / are matrix elements taken between the exact many-electron states ip n and i of energy and Ei.ground state and energy, respectively, m i = E - Ei, (J = -(w-l- a> ), and 17 is a positive infinitesimal. 

In these expressions, h, J and K are the usual monoelectronic coulombic, bielectronic coulombic and exchange operators, respectively, and R are the one-particle density operators expressed in the and 6 orbital basis ... 

Neutron wavelength Macroscopic number density Microscopic particle density operator Nucleus cross-section Nucleus coherent cross-section Nucleus incoherent cross-section... 

Zd 0.04 nm for typical metals in which interatomic distances are 0.2 nm. Below XD individual electron rather than collective behavior is emphasized. A quantum mechanical version of Eq. (2) particle density operator incorporates an additional contribution in the first term due to the movement (kinetic energy) of electrons. The most important question is for what k is the kinetic term small compared with (0p2nk term Answer is the inequality... 

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