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Partial differential equation which describes the time evolution of the wavefunction of a quantum system. It is one of the first and most fundamental equations of quantum mechanics.

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Time evolution unclear

You seem to be making the assumption $e^{i(H_1 \otimes H_2)t} = e^{iH_1t}\otimes e^{iH_2t}.$ This is not true in general. Write down the spectral decompositions of the operators and then exponentiate …
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