# The justification for the probability of definite energy states in quantum mechanics

In quantum mechanics, if the energy of a system is measured at some $$t$$ the probability of obtaining the energy eigenvalue $$E_i$$ is: $$\left| \int_{-\infty} ^{\infty} {\psi_i^* (x)\Psi(x,t)} dx^2 \right|^2$$

where $$\Psi(x,t)$$ is the complete wavefunction of the system and $$\psi_i(x)$$ is the energy eigenfunction of corresponding to $$E_i$$.

What I would like to know is where does this assumption come from? Is this a fundamental postulate of quantum theory or can this be derived/justified from a more foundational principle?

It is a postulate: the probability of outcome $$E_i$$ is given by the inner product of the eigenfunction associated with $$E_i$$ with the initial state. In Dirac notation $$\vert \langle \psi_i\vert \Psi(t)\vert^2$$.
If more than one eigenfunction is associated with the eigenvalue $$E_i$$ then one must sum over these eigenfunctions: $$\sum_{n} \vert \langle \psi^n_i\vert \Psi(t)\vert^2$$ where the sum is over all $$\vert\psi^n_i\rangle$$ that satisfy $$\hat H\vert\psi^n_i\rangle=E_i \vert\psi^n_i\rangle$$.