Momentum kinematic operator

For the electro-nuclear model, it is the charge the only homogeneous element between electron and nuclear states. The electronic part corresponds to fermion states, each one represented by a 2-spinor and a space part. Thus, it has always been natural to use the Coulomb Hamiltonian Hc(q,Q) as an entity to work with. The operator includes the electronic kinetic energy (Ke) and all electrostatic interaction operators (Vee + VeN + Vnn)- In fact this is a key operator for describing molecular physics events [1-3]. Let us consider the electronic space problem first exact solutions exist for this problem the wavefunctions are defined as /(q) do not mix up these functions with the previous electro-nuclear wavefunctions. At this level. He and S (total electronic spin operator) commute the spin operator appears in the kinematic operator V and H commute with the total angular momentum J=L+S in the I-ffame L is the total orbital angular momentum, the system is referred to a unique origin. 

K is a kinematic operator that damps the high momentum part of the wave function while keeps the low momentum part intact. Its appearance makes the DK-SOC operator bounded from below and variationally stable and a much better SOC operator than the BP-SOC. The effect of the K operator is illustrated by Fig. 8.7, where the 6p orbitals of Tl, At, and Rn are taken as examples. While the radial distribution function of the original Tl 6p orbital (Tl 6p) and the one after the operator of K (Tl K6p) are highly similar in Fig. 8.7a, the squares of these functions... 

In the scattering operator Tr(z), all kinematical insertions contained in Hr are summed up in the external lines (see (Antonelli et al, 2001) for details). We calculate the matrix element of the scattering operator Tr(z) between the ir d states at 0(a). After removing the CM momentum, the spin-nonflip part of this matrix element on energy shell is equal to... 

The presence of this property in the system allows one to compose with the dynamic viscosity ri a path linking the concentration of momentum to the local pressure P (shear stress), which constitutes a definition for the operator ambiguously named kinematic viscosity... 

By noting that 7t = p — A the vector product of the kinematic momentum operator with itself can be simplified to... 

The basic formula to take Fermi motion into account can be derived along lines very similar to those followed in the appendix to Chapter 16 [see eqn (16.9.19)]. The difference, of course, is that now V represents the momentum distribution of a nucleon in the nucleus. The following convolution formula emerges either using the techniques of the operator product expansion, or more simply, by considering the kinematics of Fig. 17.11 which shows a nucleus of atomic number A in a reference frame in which it is moving very fast with momentum P along OZ. A nucleon i inside the nucleus has -component of momentum pz = zP/A and a parton. 

The form invariance of the Schrodinger equation will then lead to gauge invariant expectation values of the Hamiltonian. However, this will not be the case for an arbitrary operator. In particular, it turns out that expectation values of the canonical momentum operator, given in Eq. (2.45), are not gauge invariant, whereas expectation values of the mechanical or kinematical momentum operator, given in Eq. (2.97), are gauge invariant [see Exercise 2.15]... 

The mechanical or kinematical momentum operator is therefore sometimes also called the gauge invariant momentum operator. 

Exercise 2.15 Prove equation (2.118) for a one-electron system, i.e. with the kinematical momentum operator in Eq. (2.57) and with a one-electron transformation operator... 

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