Does an electron change its energy when the spin projection is changed?

In presence of a uniform magnetic field, the spin of a particle mostly enters the Hamiltonian, and therefore the energy, via a Zeeman term $ hat{H}_{Zeeman}=-muvec{B}cdothat{vec{S}}$, characterized by the scalar product of the magnetic field vector $vec{B}$ and the spin operator $hat{vec{S}}$.

Given some spin state $|chirangle$ consisting of a superposition of spin-up and spin-down along $vec{B}$, the Zeeman energy contribution is given by the expectation value of the corresponding term in the Hamiltonian, i.e. by $langle hat{H}_{Zeeman} rangle=langle chi | hat{H}_{Zeeman} | chi rangle $. However, notice that it makes sense to consider such a contribution for an ensemble of electrons in most cases, interpreting the expectation value as a mean value.