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The angles $\phi$ and $\theta$ never contain a non-zero imaginary part (they are always purely real numbers). The general solution to Eq. 1 is: $$ \tag{1} |\Phi(t)\rangle = e^{-\textrm{i}H t} | \Phi(0)\rangle. $$ This can be written as: $$ \tag{2} \begin{bmatrix} a(t) \\ b(t) \end{bmatrix} = U \begin{bmatrix} a(0) \\ b(0) \end{bmatrix} . $$ where the ...


6

Higher-order integrators are used, but usually the way they perform the calculation is not through directly calculating higher order derivatives, but essentially through multiple force calculations. The Wikipedia page on symplectic integrators gives some information on this, including 3rd order and 4th order examples. I've personally used the 4th order ...


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