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Given that a particle is in a state defined by the wavefunction: $$\Psi (x,t) = \psi_0(x)e^{-iE_0t/\hbar}+\psi_1(x)e^{-iE_1t/\hbar}$$ where $\psi_0(x)$ and $\psi_1(x)$ are the energy eigenfunctions of the two different energy levels, is it possible to predict with certainty the outcome of an measurement for the energy of a particle in this state?

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The wave function you've provided is a linear superposition of two distinct energy eigenfunctions, $\psi_1(x)$ and $\psi_2(x)$, that are assumed to have distinct energy eigenvalues, $E_1$ and $E_2$ respectively. It is not possible to predict with absolute certainty the outcome of a measurement of the energy observable.

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The short answer is No. Because there is no definite energy state of this particle. It is in a superposition of two energy states $E_0$ and $E_1$.So, all we know is that any measurement will result in the collapse of wavefunction in one of the energy eigenstates. But where will it collapse is completely random. All we know is that there is equal probability for both the states.

But once you've made one measurement, the wavefunction changes to one of the eigenstates (say of $E_0$)

$\psi_f$ = $\psi_0 e^{-iw_0t}$

or the $E_1$ eigenstate. Thus the result of the second measurement will be same as the first. This is ceratin.

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