jostpuur
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If the Hamilton's operator H(t) depends on the time parameter, what is the definition for the time evolution of the wave function \Psi(t)? Is the equation
<br /> i\hbar\partial_t\Psi(t) = H(t)\Psi(t)\quad\quad\quad (1)<br />
or the equation
<br /> \Psi(t) = \exp\Big(-\frac{it}{\hbar}H(t)}\Big)\Psi(0)\quad\quad\quad (2)<br />
These are not equivalent, because if the wave function satisfies the equation (2), then it also satisfies
<br /> i\hbar\partial_t\Psi(t) = H(t)\Psi(t) + t\big(\partial_tH(t)\big)\Psi(t)<br />
Because these alternatives are not equivalent now, I don't which one to believe in.
<br /> i\hbar\partial_t\Psi(t) = H(t)\Psi(t)\quad\quad\quad (1)<br />
or the equation
<br /> \Psi(t) = \exp\Big(-\frac{it}{\hbar}H(t)}\Big)\Psi(0)\quad\quad\quad (2)<br />
These are not equivalent, because if the wave function satisfies the equation (2), then it also satisfies
<br /> i\hbar\partial_t\Psi(t) = H(t)\Psi(t) + t\big(\partial_tH(t)\big)\Psi(t)<br />
Because these alternatives are not equivalent now, I don't which one to believe in.