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[QUE/QM-09003]

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\(\newcommand{\dd}[2][]{\frac{d#1}{d#2}}\)
Solve the  differential equation
\begin{equation}\label{EQ21} \dd{t} U(t,t_0) = \lambda \hat{F}(t) U(t,t_0) \end{equation}
for \(U(t,t_0)\) subject to the condition \(U(t,t_0)=\hat{I}\) for the following cases. Here \(\hat{F}\) is an operator (or a matrix).
[A] Assume the operator \(\hat{F}\) to be independent of time. Expand \(U(t,t_0)\) in a power series in \(\lambda\) and prove    \[\begin{equation} U(t,t_0) = \exp(\lambda  (t-t_0)\hat{F}).\end{equation}\] [B] When \(\hat{F}\) depends on time but is such that the commutator \(\big[\hat{F}(t), \hat{F}(t')\big]\) vanishes for all \(t,t'\) show that \begin{equation}   U(t,t_0) = \exp\big(\lambda \int_{t_0}^t \hat{F}(\tau)\, d\tau\big).\end{equation}

NOTE: The case when \(\hat{F}(t)\) and \(\hat{F}(t^\prime)\) do not commute for \(t \ne t^\prime\), requires more work  and requires use of Time Ordered Exponentials in the solution.

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