Past exam of the short options paper shared by Prelims and Parts A and B of the physics course of the University of Oxford / 2018 / S18 / 3 / a / i

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- Solution i

Solution

For a free particle the Dirac matrices are constant and the momenta commute. Squaring the Hamiltonian cancels all mixed terms by their anticommutators:

H_{D}^{2} = c^{2} p^{2} + m^{2} c^{4} .

Applying another time derivative to the Dirac equation therefore gives

(\frac{1}{c ^{2}} \partial_{t}^{2} - \nabla^{2} + \frac{m ^{2} c ^{2}}{ℏ ^{2}}) ψ_{α} = 0

for every component. This is the Klein-Gordon equation. Its componentwise satisfaction is a necessary consequence of the first-order Dirac equation, not a converse selecting all spinor solutions of the squared equation.

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