Below we outline the formulation of the Dirac equation incorporating momentum and energy as linear terms in the Hamiltonian. We find that it leads to the prediction of particles with spin 1/2.

We found in our discussion on the subject of relativistic quantum mechanics in Chapter 31 that, as a consequence of the quadratic nature of the energy–momentum relation, negative energy solutions must exist and that the probability density is not positive definite. To solve these problems, Dirac set out to write a quantum-mechanical equation that was linear in energy and momentum based on the following Hamiltonian for a free particle,

with *m*_{0} the rest mass, p the momentum of the particle and the unknown *α* and ...

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