State Vectors of a Sum of Hamiltonians

harrydent
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I'm reading from my textbook "Foundations of Quantum Physics," and came across a statement that sounds trivial but I don't know how to prove. The passage can be found at http://books.google.com/books?id=3rxOHtn85qcC&pg=PA289#v=onepage&q=&f=false".

Because the particles do not interact, the Hamiltonian is the sum of individual Hamiltonians, one for each particle... The eigenkets are products of the individual eigenkets and the total energy is the sum of the energy eigenvalues of the particles.

Does anyone have any information on the proof of this theorem?
 
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Maybe someone could just explain it in plain English rather than a proof?
 

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