Heisenberg equations of motion for the position, momentum, and spin of an electron are derived using the Heisenberg equation of motion formula, accounting for the particular Hamiltonian given. The exact equations depend on the commutation relations between the observables and the Hamiltonian and could include additional time-dependent terms if any observables exhibit explicit time dependence.
To derive the Heisenberg equations of motion for the given Hamiltonian H = p^2/2m + aS · p + ñ · S, where p is the momentum operator, S is the spin angular momentum operator, and ñ is an arbitrary unit vector, we apply the time evolution operator in the Heisenberg picture. The Heisenberg equation of motion for an observable O is given by
dO/dt = i/ħ [H, O] + (∂O/∂t)H,
where [H, O] denotes the commutator of H and O. For the position operator x, using [p^2/2m, x] = ip/ħ and no explicit time-dependence, the equation becomes dx/dt = p/m, which is the velocity of the particle. For the momentum operator p, since it does not commute with S, we get the equation dp/dt = -i/ħ [aS · p + ñ · S, p]. For the spin components S, the equation is dS/dt = -i/ħ [aS · p + ñ · S, S]. Compatible observables will commute with the Hamiltonian, but without knowing the explicit form of a and ñ, it's difficult to provide a complete list.