6.1: Time Dependence
( \newcommand{\kernel}{\mathrm{null}\,}\)
The exact expression for the time dependence of a system with N states required a set of N simultaneous differential equations. One case where we can solve this problem exactly is when we have a small number of states. Consider a system which requires only two basis states. Say we prepare it in initial state |1⟩ and we want to know how long it will take to go to the other state |2⟩. From section 5, we have two coupled equations in the time dependent c1 and c2:
iℏ˙c1=V11c1+V12c2eiω12tiℏ˙c2=V22c2+V21c1eiω21t
where c1(0)=1 and c2(0)=0.
If the change is slow, we can use first order time-dependent perturbation theory. We thus replace the cn(t) by cn(0), and integrate whence:
c1≈ exp(iV11t/ℏ)|c1|2≈1c2≈−iℏ∫t0V21eiω21tdt
Including the constant of integration for c1(0)=1.