Schrodinger spinor form

When shown that (a p)2 = p2, the spinor form of Schrodinger s equation reduces to the normal complex form with E and p in operator notation ... 

The proof takes different forms in different representations. Here we assume that quantum states are column vectors (or spinors ) iji, with n elements, and that the scalar product has the form ft ip. If ip were a Schrodinger function, J ftipdr would take the place of this matrix product, and in Dirac s theory of the electron, it would be replaced by J fttpdr, iji being a four-component spinor. But the work goes through as below with only formal changes. Use of the bra-ket notation (Chapter 8) would cover all these cases, but it obscures some of the detail we wish to exhibit here. 

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. 

According to general properties of Pauli matrices (a p)2 = p2 hence (9) is recognized as Schrodinger s equation, with E and p in operator form. On defining the electronic wave functions as spinors both Dirac s and Schrodinger s equations are therefore obtained as the differential equation describing respectively non-relativistic and relativistic motion of an electron with spin, which appears naturally. 

Abstract. This chapter concerns a presentation of the Darwin solutions of the Dirac equation, in the Hestenes form of this equation, for the central potential problem. The passage from this presentation to that of complex spinor is entirely explicited. The nonrelativistic Pauli and Schrodinger theories are deduced as approximations of the Dirac theory. 

We have demonstrated that by replacing it with a-it we can indeed introduce spin into the nonrelativistic Schrodinger equation. In this form, spin appears explicitly in the wave function through the Pauli spinors (or products of these for many-electron systems), and its interaction with magnetic fields appears naturally in the Hamiltonian and need not be grafted on ad hoc when required. However, apart from the fact that it yields a convenient form of the Schrodinger equation, it is not immediately evident why the operator a it should be used. And we still have the problem that the free-electron part of the Hamiltonian is not Lorentz invariant. So we must look for an alternative route to a relativistic quantum theory for the electron, one which preferably also accounts for spin. Our experiences from the derivations in this section show us that this route may lead to multicomponent wave functions. 

Before taking the limit c oo, this equation must be rearranged for two reasons first, because we need to change it to a form where c occurs in some form of denominator—this will provide us with terms that vanish and hopefully other terms that remain finite—and, second, because the nonrelativistic wave function is a scalar function, whereas the Dirac wave function is a four-component vector function. If we use the two-component nonrelativistic Schrodinger equation that we derived in section 4.2, we can write the nonrelativistic wave function in terms of spin-orbitals, which can be transformed to two-component spinors. Then it is only necessary to reduce the Dirac equation from four-component to two-component form. 

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