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# Matrix differential equations

All in all, the Dirac harmonic oscillator is a good place to study resonances, especially if much smaller masses are considered than the ones which characterize the actual experience of physicists. Of course, that makes a comparison with experiment correspondingly difficult.

The one-dimensional Dirac equation for a particle of rest mass m0 has a matrix form

 = (1)

The substitutions

lead to a convenient solution matrix,
 = (2)

but only for a constant energy difference E - V, and for real k. Formally the result holds for immaginary k but in that case, a classically forbidden region, it is preferable to redefine both k and to make them real and then use hyperbolic functions.

There are other approximations which are useful; for example when the mass is small enough to be neglected, the coefficient matrix is antisymmetric. Setting

 = (3)

the solution becomes

 = (4)

which is a pure rotation in the phase plane.

There is a standard procedure for correcting an approximate coefficient matrix in a system of differential equations. Suppose that the full coefficient M is split into a sum

 M = A + B, (5)

wherein A is the convenient approximation and B is a correction, not necessarily small. All kinds of splittings are possible: symmetric and antisymmetric parts, averages and deviations, large and small components, to mention three.

Supposing that the auxiliary equation

 = (6)

has already been solved subject to the initial condition , and that we intend to write , we find that must solve the differential equation
 = (7)

subject to the same initial condition as .

Next: Numerical integration with graphical Up: Resonance in the Dirac Previous: The Dirac equation