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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?

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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 $\hat H\vert\psi^n_i\rangle=E_i \vert\psi^n_i\rangle$.

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The assumption is an example of the Born Rule, which was formulated by Max Born in 1926. More generally the rule states that if you make a measurement of an observable property of a particle associated with some operator O, say, then the result must be one of the eigenvalues of O, and the wave function of the particle immediately after the measurement must be the corresponding eigenfunction of O. The probability of obtaining a particular eigenvalue is related to the overlap between the associated eigenfunction and the initial wave function of the particle before the measurement. Intuitively it is a very intuitive result, in my opinion.

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