How can I calculate the probability that the measured energy is equal to some given energy? I find it hard to solve question b of the following example. 
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$\begingroup$ Possible duplicate of Calculating the probability of a given energy $\endgroup$– sammy gerbilCommented Mar 23, 2018 at 22:06
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According to the Born rule, the probability of measuring an energy $E_i$ (meaning the system is in state $\psi_i$) is given by $|\psi_i \cdot \psi(x,t)|^2$. Since energy eigenstates are orthogonal, $\psi_i \cdot \psi_j = \delta_{ij}$. Since you want the probability that the system measures $E_2$ for the energy, you want the probability that the wavefunction is $\psi_2$, so you can calculate $\psi_2 \cdot \psi(x,t)$ and take the magnitude squared.
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$\begingroup$ After conducting research, I found online that the probability of measuring an energy $E_i$ is given by $\mid{c_i}\mid^2$. Is it equivalent to $|\psi_i \cdot \psi(x,t)|^2$? $\endgroup$– AndrewCommented Mar 25, 2018 at 18:22
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$\begingroup$ If you read my answer then you should be able to answer that yourself. If there is something unclear about my answer then say so. $\endgroup$ Commented Mar 25, 2018 at 18:26
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$\begingroup$ Ok, I showed they are equivalent using orthogonality and basic algebra, but it wasn't trivial for me. $\endgroup$– AndrewCommented Mar 26, 2018 at 7:20