Transformation to two components

The second possibility to reduce the four-component Dirac spinor to two-component Pauli form is to decouple the Dirac equation, i.e., to transform the Dirac Hamiltonian to block-diagonal form by a suitably chosen unitary transformation U, 

Before deriving the explicit form of the matrix U in terms of the operator X it should be mentioned that the spectrum of the Dirac operator Hd is invariant under arbitrary similarity transformations, i.e., non-singular (invertible) transformations U, whether they are unitary or not. But only unitary transformations conserve the normalisation of the Dirac spinor and leave scalar products and matrix elements invariant. Therefore a restriction to unitary transformations is inevitable as soon as one is interested in properties of the wavefunction. Furthermore, the problem experiences a great technical simplification by the choice of a unitary transformation, since the inverse transformation U can in general hardly be accomplished if U was not unitary. 

Employing the most general ansatz for the unitary matrix U, 

Similarly, the requirement — Q for positronic (class II) solutions yields together with X = —y the relation 

Finally, the form of U and U22 is determined by the unitarity condition UU = 1, which is only satisfied if 

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In this notation the presence of two upper and two lower components of the four-component Dirac spinor fa is emphasized. For solutions with positive energy and weak potentials, the latter is suppressed by a factor 1 /c2 with respect to the former, and therefore commonly dubbed the small component fa, as opposed to the large component fa. While a Hamiltonian for a many-electron system like an atom or a molecule requires an electron interaction term (in the simplest form we add the Coulomb interaction and obtain the Dirac-Coulomb-Breit Hamiltonian see Chapter 2), we focus here on the one-electron operator and discuss how it may be transformed to two components in order to integrate out the degrees of freedom of the charge-conjugated particle, which we do not want to consider explicitly. 

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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