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Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 4 / 34B / a / Solution

In the symmetric gauge,

\partial_{i} A_{j} + \partial_{j} A_{i} = 0, \partial_{i} A_{j} - \partial_{j} A_{i} = ε_{ijk} B_{k} .

Together with

[p_{i}, f (x)] = - i ℏ \partial_{i} f

, this gives

[p_{i} + e A_{i}, p_{j} - e A_{j}] = - e [p_{i}, A_{j}] + e [A_{i}, p_{j}] = i e ℏ (\partial_{i} A_{j} + \partial_{j} A_{i}) = 0,

whereas

[p_{i} + e A_{i}, p_{j} + e A_{j}] = e [p_{i}, A_{j}] + e [A_{i}, p_{j}] = i e ℏ (\partial_{j} A_{i} - \partial_{i} A_{j}) = - i e ℏ ε_{ijk} B_{k} .

Thus

[p_{i} + e A_{i}, p_{j} - e A_{j}] = 0, [p_{i} + e A_{i}, p_{j} + e A_{j}] = - i e ℏ ε_{ijk} B_{k} .

These are the commutators of the kinetic momentum and magnetic pseudomomentum.

Solved by gpt-5.6-sol high.

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