Two-state quantum systems in a superposition of energy eigenstates oscillate with time between the possible states. If both energy eigenstates contribute with the same amplitude, the oscillation is between definite state $ \uparrow $ and definite state $ \downarrow $:
$$ A_1^2 + A_2^2 = 1 $$ $$ \psi(\uparrow,t) = A_1 \cdot \frac{1}{\sqrt{2}} \cdot e^{ -i \cdot E_1 \cdot t / \hbar } + A_2 \cdot \frac{1}{\sqrt{2}} \cdot e^{ -i \cdot E_2 \cdot t / \hbar } $$ $$ \psi(\downarrow,t) = A_1 \cdot \frac{1}{\sqrt{2}} \cdot e^{ -i \cdot E_1 \cdot t / \hbar } - A_2 \cdot \frac{1}{\sqrt{2}} \cdot e^{ -i \cdot E_2 \cdot t / \hbar } $$ $$ |\psi(\uparrow,t)|^2 = \frac{1}{2} + A_1 \cdot A_2 \cdot \cos (t \cdot ( E_2 - E_1 ) / \hbar ) $$ $$ |\psi(\downarrow,t)|^2 = \frac{1}{2} - A_1 \cdot A_2 \cdot \cos (t \cdot ( E_2 - E_1 ) / \hbar ) $$
$ A_1 $ is the contribution of the eigenstate with energy $ E_1 $ and $ A_2 $ is the contribution of the eigenstate with energy $ E_2 $. $ A_1 $ and $ A_2 $ are assumed to be real in this calculation.
There seem to be solutions with unequal amplitudes $ A_1 $ and $ A_2 $ of the energy eigenstates where the oscillation of probabilities is reduced in amplitude and so never reaches the definite states. It seems that after measuring the state in such a system, the amplitudes of energy eigenstates need to be equal and the expectation value for the energy changed.
Yet I cannot find any mention of such solutions in theory or in nature. Have I made a mistake or are they prevented by some principle?
Edit: Maybe measuring the spin of an electron in a magnetic field along an arbitrary axis is the same as switching to an arbitrary basis? That might look like an unequal energy superposition in the new basis?
