Symbol t

The passage around this formula

…to the swap operator between the two, and the Schrödinger equation solves in closed form. Starting from |$\psi(0)$$\rangle$ = |e,g$\rangle$ , $|\psi(t)\rangle = \cos(gt)\,|e,g\rangle - i\sin(gt)\,|g,e\rangle$. the standard two-level exchange solution, exact for all t under this Hamiltonian. From it, $\langle$ $H_S$ $\rangle(t)$ = ($\hbar$$\omega$/2)[$\cos^2(gt)$ - $\sin^2(gt)$] = ($\hbar$$\omega$/2)$\cos(2gt)$ , while $\langle$ $H_f$ $\rangle$ is exactly time-independent — a general fact for any state evolving under its own generator, not special to this one — and equal here to…

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