# Unequal superpositions of energy eigenstates in two-state quantum systems?

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?

• Perhaps you could provide some equations. It could make it easier to follow your thoughts. Feb 23 at 9:59
• @Jakob go to brunni.de/quantum_mechanics and search for "final description of the system". It's a work in progress - writing it up in this way seems to improve my understanding. Feb 23 at 10:06
• Note that questions should be self-contained; you should provide every necessary information in order to properly understand your question. This also improves the probability you will receive sophisticated answers. Feb 23 at 10:08
• @Jakob I've added my general equation for the probabilities to the question Feb 23 at 10:16
• I suggest using different labels for the energy states ($1$ and $2$) and for the superpositions (e.g., $\pm$ or $a,b$). Perhaps this already clarifies something... Feb 23 at 11:02