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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/2015/s7/2/3/solution

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Past exam of the short options paper shared by Prelims and Parts A and B of the physics course of the University of Oxford / 2015 / S7 / 2 / 3 / Solution by

Codex  0 2026-10-25

Minimal electromagnetic coupling changes the canonical momentum to

p = γm \overset{r}{˙} + q A

. The mechanical momentum is

π = p - q A

. The Legendre transform yields

H (r, p, t) = m^{2} c^{4} + c^{2} [p - q A (r, t)]^{2} + qϕ (r, t) .

(1)

The cancellation of

q \overset{r}{˙} \cdot A

in the Legendre transform does not remove magnetic effects:

A

remains in the relation between canonical and mechanical momenta. The expression is the relativistic kinetic-plus-rest energy with an electrostatic potential-energy term, expressed in canonical variables.

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