= Magnetic pseudomomentum
{title2=$\mathbf K=\mathbf p+\frac e2\mathbf B\times\mathbf x$}
For a <charged particle> in a uniform <magnetic field>, the <symmetric gauge> $\mathbf A=\mathbf B\times\mathbf x/2$ makes the displayed magnetic pseudomomentum a conserved quantity when no other position-dependent potential breaks translation symmetry. It differs from the <kinetic momentum> $\boldsymbol\pi=\mathbf p-e\mathbf A=m\dot{\mathbf x}$. With $\mathbf B=B\mathbf e_3$, <Hamilton's equations> for $H=|\mathbf p-e\mathbf A|^2/(2m)$ give $\dot p_1=(eB/2)\dot x_2$ and $\dot p_2=-(eB/2)\dot x_1$, proving conservation of $K_1=p_1-eBx_2/2$ and $K_2=p_2+eBx_1/2$. The charge factor is necessary unless it has already been absorbed into the field convention. In <quantum mechanics> the transverse components need not commute, even though each commutes with this <Hamiltonian operator>; <magnetic translation algebra> records that distinction.
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