If I define a state ket in the traditional way, Say:(adsbygoogle = window.adsbygoogle || []).push({});

$$|\Psi \rangle =\sum _{ i }^{ }{ a_{ i }|\varphi _{ i }\rangle \quad } $$

Where $$a_i$$ is the probability amplitude.

How does:

$$\hat {H } |\Psi \rangle =E|\Psi \rangle $$ if the states of $$\Psi$$ could possibly represent states that have different energy levels, and one would not be able to factor the energy eigenvalue out of the summation. I know I am missing something big here. Could someone point it out?

Edit: Does $$ \hat {H} $$ collapse psi to one state phi, and so render only energy eigenvalue of the one phi that remains?

Thanks,

Chris Maness

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# State vectors and Eigenvalues?

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