 Solution

ID: 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/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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