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?

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    $\begingroup$ Perhaps you could provide some equations. It could make it easier to follow your thoughts. $\endgroup$ Feb 23, 2021 at 9:59
  • $\begingroup$ @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. $\endgroup$
    – brunni
    Feb 23, 2021 at 10:06
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    $\begingroup$ 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. $\endgroup$ Feb 23, 2021 at 10:08
  • $\begingroup$ @Jakob I've added my general equation for the probabilities to the question $\endgroup$
    – brunni
    Feb 23, 2021 at 10:16
  • $\begingroup$ 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... $\endgroup$
    – Roger V.
    Feb 23, 2021 at 11:02

1 Answer 1


Certain systems - like electron spin - can be described by a two-state system plus changes of basis. The change of basis corresponds to measurement of spin along a different direction in space and can look like an unbalanced superposition from the point of view of the new basis.

All possible states of a two-state system can be represented as points on the Bloch sphere and opposite points on that sphere always form a basis https://en.wikipedia.org/wiki/Bloch_sphere


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