Spinor wave function

Section VI shows the power of the modulus-phase formalism and is included in this chapter partly for methodological purposes. In this formalism, the equations of continuity and the Hamilton-Jacobi equations can be naturally derived in both the nonrelativistic and the relativistic (Dirac) theories of the electron. It is shown that in the four-component (spinor) theory of electrons, the two exha components in the spinor wave function will have only a minor effect on the topological phase, provided certain conditions are met (nearly nonrelativistic velocities and external fields that are not excessively large). 

This part of our chapter has shown that the use of the two variables, moduli and phases, leads in a direct way to the derivation of the continuity and Hamilton-Jacobi equations for both scalar and spinor wave functions. For the latter case, we show that the differential equations for each spinor component are (in the nearly nomelativistic limit) approximately decoupled. Because of this decoupling (mutual independence) it appears that the reciprocal relations between phases and moduli derived in Section III hold to a good approximation for each spinor component separately, too. For velocities and electromagnetic field strengths that ate nomrally below the relativistic scale, the Berry phase obtained from the Schrddinger equation (for scalar fields) will not be altered by consideration of the Dirac equation. 

We see that a Lorentz transformation A not only affects the argument of a spinor-wave function, but also acts directly on the spinor components via the matrix M. An exception are the pure translations, which are of the form... 

In case of four-component spinor-wave functions the procedure described above can be done, in principle, for each of the four components separately. It is, however, useful, to choose a basis set that is better adapted to the solution of the Dirac equation in the presence of spherical symmetry. 

In this appendix, we describe the change of metric that occurs when a 4-spinor wave function is modified by operations on the small component. The unmodified spinor is expressed as... 

Similarly, for wave functions, hereafter called spinors, we define operations isomorphic to, r and complex conjugation. Thus if u is a column spinor we define... 

Abstract Hilbert space, 426 Accuracy of computed root, 78 Acharga, R., 498,539,560 Additive Gaussian noise channel, 242 Adjoint spinor transformation under Lorentz transformation, 533 Admissible wave function, 552 Aitkin s method, 79 Akhiezer, A., 723 Algebra, Clifford, 520 Algebraic problem, 52 linear, 53... 

This two-component wave function is known as a spinor. 

As we can see, the FW two-component wave function is not the large component of the Dirac spinor, but it is related to it by an expression involving X. Consider a similarity transformation based on U parameterized as... 

The inclusion of relativistic effects is essential in quantum chemical studies of molecules containing heavy elements. A full relativistic calculation, i.e. based upon Quantum Electro Dynamics, is only feasible for the smallest systems. In the SCF approximation it involves the solution of the Dirac Fock equation. Due to the four component complex wave functions and the large number of basis functions needed to describe the small component Dirac spinors, these computations are much more demanding than the corresponding non-relativistic ones. This limits Dirac Fock calculations, which can be performed using e.g. the MOLFDIR package [1], to small molecular systems, UFe being a typical example, see e.g. [2]. 

In the inner core region, the pseudospinors are smoothed, so that the electronic density with the pseudo-wave function is not correct. When operators describing properties of interest are heavily concentrated near or on nuclei, their mean values are strongly affected by the wave function in the inner region. The four-component molecular spinors must, therefore, be restored in the heavy-atom cores. 

Then one can extract a free heavy particle spinor from the wave function in (1.19)... 

Finally, the eight-component wave function ip p, En) (four ordinary electron spinor indices, and two extra indices corresponding to the two-component... 

EDE in the external Coulomb field in Fig. 1.6. The eigenfunctions of this equation may be found exactly in the form of the Dirac-Coulomb wave functions (see, e.g, [10]). For practical purposes it is often sufficient to approximate these exact wave functions by the product of the Schrodinger-Coulomb wave functions with the reduced mass and the free electron spinors which depend on the electron mass and not on the reduced mass. These functions are very convenient for calculation of the high order corrections, and while below we will often skip some steps in the derivation of one or another high order contribution from the EDE, we advise the reader to keep in mind that almost all calculations below are done with these unperturbed wave functions. 

In another physics application, the Dirac equation for states of an electron in relativistic space-time requires wave functions taking values in the complex vector space C" = (ci, C2, c, C4) ci, C2, C3, C4 e C. These wave functions are called Dirac spinors. 

Relativity becomes important for elements heavier than the first row transition elements. Most methods applicable on molecules are derived from the Dirac equation. The Dirac equation itself is difficult to use, since it involves a description of the wave function as a four component spinor. The Dirac equation can be approximately brought to a two-component form using e.g. the Foldy-Wouthuysen (FW) transformational,12]. Unfortunately the FW transformation, as originally proposed, is both quite complicated and also divergent in the expansion in the momentum (for large momenta), and it can thus only be carried out approximately (usually to low orders). The resulting equations are not variationally stable, and they are used only in first order perturbation theory. 

Thus, the non-relativistic wave function (1.14) of an electron is a two-component spinor (tensor having half-integer rank) whereas its relativistic counterpart is already, due to the presence of large (/) and small (g) components, a four-component spinor. The choice of / in the form (1 + l — l ) is conditioned by the requirement of a standard phase system for the wave functions (see Introduction, Eq. (4)). 

The energy spectrum of atoms and ions with j j coupling can be found using the relativistic Hamiltonian of iV-electron atoms (2.1)-(2.7). Its irreducible tensorial form is presented in Chapter 19. The relativistic one-electron wave functions are four-component spinors (2.15). They are the eigenfunctions of the total angular momentum operator for the electron and are used to determine one-electron and two-electron matrix elements of relativistic interaction operators. These matrix elements, in the representation of occupation numbers, are the parameters that enter into the expansions of the operators corresponding to physical quantities (see general expressions (13.22) and (13.23)). 

T.2.C Non-Hermitean Hamiltonians Taking into Account the Indirect Damping T.2.D Kets, Spinors, Density Operators, and Wave Functions... 

Representation theory of molecular point groups tells us how a rotation or a reflection of a molecule can be represented as an orthogonal transformation in 3D coordinate space. We can therefore easily determine the irreducible representation for the spatial part of the wave function. By contrast, a spin eigenfunction is not a function of the spatial coordinates. If we want to study the transformation properties of the spinors... 

We are not going to review here the transformation properties of spatial wave functions under the symmetry operations of molecular point groups. To prepare the discussion of the transformation properties of spinors, we shall put some effort, however, in discussing the symmetry operations of 0(3)+, the group of proper rotations in 3D coordinate space (i.e., orthogonal transformations with determinant + 1). Reflections and improper rotations (orthogonal transformations with determinant -1) will be dealt with later. 

We consider a ladder of N = 2M spins 1/2. The wave function of this system is described by the Nth-rank spinor... 

We partition the system into pairs of spins located on rungs of the ladder. The wave function can then be written as the product of M second-rank spinors... 

We now choose a Hamiltonian H for which the wave function (55) is an exact ground-state wave function. To do so, we consider the part of the system (cell) consisting of two nearest neighbor spin pairs. In the wave function (55) the factor corresponding to the two spin pairs is a second-rank spinor ... 

Thus, the singlet ground-state wave function of the model (46) can be also written in a spinor form (55). 

Figure 8 Graphical correspondence of the model wave function. The indices of the site spinors depend on the site index (not shown in the figure).

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