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# Small mass compared to height

When the mass is small or even zero, the coefficient matrix of the Dirac equation is nearly a scalar multiple of the unit antisymmetric matrix . Accordingly solutions take the form seen in Equations (13) and (13). For the potential the results for the interval are

 = = = (17)

and accordingly

 = (18)

half of whose trace is , independently of the strength of the potential, and even of its form so long as it is periodic. This is evident in Figure 6 and even in the contour plot of Figure 7, whose apparent structure is more likely due to the precision of the computation than to the effects of mass, height, and energy.

Mass and energy may be separated in the Dirac equation by writing the coefficient matrix as M = A + B where and . Factoring the solution into and solving

 = (19)

to get the result already shown in Equation 18 (where the solution was called ), there remains the solution of the equation for ,
 = (20)

When m0 is small, it is approximately solved by the so-called first Born approximation
 = (21) = (22)

the second form being due to the anticommutation of the quaternion matrices. Taking into account that is a combination of and while the integral is a mixture of and , changes in the dispersion relation due to a small mass will be of at least second order.

Next: Mass comparable to height Up: Periodic Potentials in One Previous: Dirac equation