Spin transformations

We now consider some aspects of the theory of electronic spin angular momentum. What follows here is a relatively brief and simple exposition we will return to a comprehensive description of the details of electron spin theory in chapter 3. For a single electron, the spin vector. S is set equal to (1 /2) r and in the representation where Sz (or a y ) is diagonal, the components of the vector a may be represented by 2 x 2 

There are therefore two eigenfunctions fM= i/2 and from the matrix representation of S2 and Sy each fM must be a two-component function, 

These functions are called spinors and any other two-component function (spinor) can be written as a linear combination of them, 

Since (a n)2 = 1 it follows that A. = 1. A rotation from the first representation (which we now identify as space-fixed) to the second (instantaneous molecule-fixed) can be represented by a rotational matrix, i.e., 

We are now in a position to investigate the effects of 3/30, 9/90 and 9/9y on the electron spin functions. When the electron spins are quantised in the molecule-fixed axis system, we see that each component of the 2 -rank spinor is an implicit function of p, 0 and x through its dependence on the transformation matrix (2.99). The total spinor f(S) may be expressed as a product of one-electron spinors, 

---

The consequence of these relations is that every proper 2n rotation on S + — in the present instance the Poincare sphere—corresponds to precisely two unitary spin rotations. As every rotation on the Poincare sphere corresponds to a polarization/rotation modulation, then every proper 2n polarization/rotation modulation corresponds to precisely two unitary spin rotations. The vector K in Fig. lb corresponds to two vectorial components one is the negative of the other. As every unitary spin transformation corresponds to a unique proper rotation of S +, then any static (unipolarized, e.g., linearly, circularly or ellipti-cally polarized, as opposed to polarization-modulated) representation on S + (Poincare sphere) corresponds to a trisphere representation (Fig. 3a). Therefore... 

A /l = 7, where 7 is the identity matrix. Thus, a spin transformation is defined uniquely up to sign by its effect on a static instantaneous snapshot representation on the S+ (Poincare) sphere ... 

It seems that a better suited approach should make use of a series of actual measurements on an evolving material quantum system. In fact, two-level atoms, equivalent to spin systems, show a formal analogy with light polarisation. Whereas the configuration space of polarisation transforms according to symmetry group SO(3), the symmetry of spin transformation is SU(2), which is a double covering of SO(3), and locally isomorphic with the latter one [17]. Thus, similar visualisations of the dynamics of both systems apply. 

The term involving p is independent of spatial coordinates. The effect of the spin transformation on is... 

The spin transformation here is the same as in the case considered in the previous section with the operator working on the four-component spinor expressed in our standard basis. Expanding a in terms of a yields... 

THE DIRAC EQUATION SOLUTIONS AND PROPERTIES Expanding the spin transformation in separate components, we get... 

We refer only to diagonal matrices and the diagonal elements of the matrices (applications of the gates considered here to a diagonal density matrix do not produce off-diagonal elements). For a many-spin system, RESET of the reset spin, transforms the diagonal of any diagonal density matrix, p, as follows ... 

---