Orthogonal spin quantization

The Hartree-Fock model leads to an effective one-electron Hamiltonian, called the Fockian F. The second quantized representation of the Fockian has that same form as any other one-electron operator. In the basis of orthogonalized spin-orbitals one can write ... 

The discussion in the previous sections dealt with the second quantization formalism for orthonormal spinorbitals. In this section, we will generalize the formalism to treat cases where the set of spin orbitals is non-orthogonal. Consider a set of n spin orbitals with the general metric... 

The simplest procedure is to take the origin of a global I-frame so that P = 0 and linear momentum conservation forces kj = —k2. At the antipodes, kd = k2 so that the common I-frame is restricted now. The particle model in this frame becomes strongly correlated. If spin quantum state for I-frame, one corresponds to the linear superposition (a /S)[CiC2]i and the other I-frame system should display the state (a P)[c, — Ci]2, namely, an orthogonal quantum state. The quantization of three axes is fixed. Spin and space are correlated in this manner. Now, the label states (a P)[c2 — cji and (a )[C C2]2 present another set of possibilities. This is because quantum states concern possibilities. All of them must be incorporated in a base state set. At this point, classical and quantum-physical descriptions differ radically. The former case handles objects that are characterized by properties, whereas the latter handle objects that are characterized by quantum states sustained by specific materiality. 

Circular polarization is equivalent to a superposition of linear polarizations in two perpendicular directions, say, horizontal (H) and vertical (V). Whereas two quantized states of spin-j particles differ in orientation by 180°, for photons the two orthogonal states of polarization differ by 90°. Formulas derived for spin- j pailidcs can generally be applied to photons with B/2 replaced by B. It is... 

By using four detectors in a standard Mott detector, a single gold foil can be used for spin analysis along two orthogonal directions. The polarimeter depicted schematically in Figure 3.2.2.32 combines two Mott detectors, one oriented at 90° with respect to the other, allowing for complete three-dimensional spin polarimetry, that is, with quantization axes =x, y, and z [39]. 

Our discussion so far has been concerned with the development of the second-quantization formalism for an orthonormal basis. Occasionally, however, we shall find it mote convenient to work with spin orbitals that are not orthogonal. We therefore extend the formalism of second quantization to deal with such spin orbitals, drawing heavily on the development in the preceding sections. 

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