Final answer:
The non-relativistic solution to Dirac's equation is represented by the time-independent Schrödinger equation, while for ultra-relativistic particles, the wavelength is found using the relativistic momentum equation, which closely resembles that of a massless particle.
Step-by-step explanation:
Dirac's Equation for Non-Relativistic and Ultra-Relativistic Particles
The non-relativistic limit of Dirac's equation when the mass energy is significantly larger than the kinetic energy can be approximated by the Schrödinger equation. For a particle in one dimension, this would be Y = (E - V)y, where ‘Y’ is the time-dependent wave function, and ‘y’ is the time-independent wave function.
For ultra-relativistic particles where the mass can be ignored, the momentum is given by p = E/c, where ‘E’ is the total energy. The wavelength λ for such a particle is nearly identical to that of a massless particle, like a photon, given the relationship λ = h/p.
In summary, the solution forms depend on the energy regime in which the particle operates. For non-relativistic scenarios, Schrödinger’s equation is used, while for ultra-relativistic cases, simplifications using relativistic momentum are made for calculating de Broglie wavelengths.