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Consider that we have a system with a time dependent hamiltonian $H(t)$. We consider that the hamiltonian changes from $H(0)$ to $H(t)$, ($t$ large), and we assume that the change occurs slowly. So, the adiabact theorem states that, if the system is initially in the $n$th eigenstate of $H(0)$, in the instant $t$ it will be in the $n$th eigenstate of $H(t)$.

A condition to it be true is that the spectrum is non-degenerate and discrete, so there is no ambiguity in the ordering of the eigenstates. My question is: Suppose our system is simply a spin 1/2 particle subject to a potential that does not depend on the spin of the particle; the eigenstates of its hamiltonian will be degenerate with respect to the spin (right?). In this case the adiabatic theorem would be appliable? To me, it look likes we can use the theorem, because I think the change in hamiltonian can't change the spin of the particle.

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This system can be decomposed into a tensor product of two uncoupled subsystems, one which does not depend on spin and one which depends only on spin. The adiabatic theorem can be applied to the former subsystem.

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