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## Main Question or Discussion Point

I'm working my way through some QM problems for self-study and this one has stumped me. Given the Hamiltonian as [itex]H(t) = f(t)H^0[/itex] where [itex]f(t)[/itex] is a real function and [itex]H^0[/itex] is Hermitian with a complete set of eigenstates [itex]H^0|E_n^0> = E_n^0|E_n^0>[/itex]. Time evolution is given by the Schrodinger equation [itex]i \hbar \frac{d}{dt}|\phi (t)> = H(t)|\phi (t)>[/itex]. I am supposed to write a solution to the Schrodinger equation as a linear combination of the eigenstates of [itex]H^0[/itex]. Now clearly

[itex]|\phi (t)> = \sum\limits_{n=1}^N c_n (t)|E_n^0>[/itex]. But where do I go from there. The second part is to convert the Schrodinger equation into a first order diff eq and solve for the [itex]c_n (t)[/itex]. Any help is appreciated. Thanks.

[itex]|\phi (t)> = \sum\limits_{n=1}^N c_n (t)|E_n^0>[/itex]. But where do I go from there. The second part is to convert the Schrodinger equation into a first order diff eq and solve for the [itex]c_n (t)[/itex]. Any help is appreciated. Thanks.