Measuring Spectrum of Boxes A & B: Same QM System?

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zetafunction
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let us suppose we have two boxes labelled by A and B

i can not open none of them (the boxes) but i can use an spectrometer to measure the absortion and emission spectrum

i chek that the eigenvalues of both systems are equal , then my question is A and B boxes contain the same QM system??

the idea is, is a system completely defined by knowing its spectrum ??
 

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I don't think so... if you're talking about the absorption/emission spectra resulting from transitions between states, the spectrum tells you only the energy eigenvalues (or rather, the differences between them). I think you could, in principle, have different systems with the same energy eigenvalues but different corresponding eigenstates (wavefunctions).

If you know the energy eigenvalues and the eigenstates, then I think that does characterize everything you need to know about the system. (Good luck figuring out the eigenstates though ;-)