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Physics LibreTexts

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:

i˙c1=V11c1+V12c2eiω12ti˙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/)|c1|21c2it0V21eiω21tdt

Including the constant of integration for c1(0)=1.


This page titled 6.1: Time Dependence is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by Graeme Ackland via source content that was edited to the style and standards of the LibreTexts platform.

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