If the Hamilton's operator [itex]H(t)[/itex] depends on the time parameter, what is the definition for the time evolution of the wave function [itex]\Psi(t)[/itex]? Is the equation(adsbygoogle = window.adsbygoogle || []).push({});

[tex]

i\hbar\partial_t\Psi(t) = H(t)\Psi(t)\quad\quad\quad (1)

[/tex]

or the equation

[tex]

\Psi(t) = \exp\Big(-\frac{it}{\hbar}H(t)}\Big)\Psi(0)\quad\quad\quad (2)

[/tex]

These are not equivalent, because if the wave function satisfies the equation (2), then it also satisfies

[tex]

i\hbar\partial_t\Psi(t) = H(t)\Psi(t) + t\big(\partial_tH(t)\big)\Psi(t)

[/tex]

Because these alternatives are not equivalent now, I don't which one to believe in.

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# What if Hamiltonian is not constant in time?

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