this equation given in (2.12). (Note that this method of evaluating a determinant is not valid, in general, for matrices involving submatrices that do not commute. In the present case, however, the submatrices We can obtain the solutions (wave functions) of the Dirac equation, in this case, by directly solving the set of four coupled equations in (2.9). Alternatively, we can introduce two component wave functions
where
We note that for the positive energy solutions
the set of coupled equations takes the form
Writing out explicitly, (2.17) leads to
The two component function
Let us note here parenthetically that the first relation in (2.18) also leads to the same relation (as it should), namely,
where we have used the property of the Pauli matrices, namely,
Choosing the two independent solutions for û as
we obtain respectively
and
This determines the two positive energy solutions of the Dirac equation (remember that the energy eigenvalues are doubly degenerate). (The question of which components can be chosen independently follows from an examination of the dynamical equations. Thus, for example, from the second of the two two-component Dirac equations in (2.18), we note that
Similarly, for the negative energy solutions we write
and the set of equations (2.9) becomes
We can solve these as
Choosing the independent solutions as
we obtain respectively
and
and these determine the two negative energy solutions of the Dirac equation.
The independent two component wave functions in (2.21) and (2.27) are reminiscent of the spin up and spin down states of a two component spinor. Thus, from the fact that we can write
the positive and the negative energy solutions have the explicit forms
The notation is suggestive and implies that the wave function corresponds to that of a spin
From the structure
