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Physics LibreTexts

6.3: Example - Oscillation in a fully mixing two state system

( \newcommand{\kernel}{\mathrm{null}\,}\)

Consider the expectation value of a quantity S in a system which has two non-degenerate energy eigenstates |1 and |2, and where the Hermitian operator ˆS is defined by ˆS|1=|2, ˆS|2=|1.

The general state can be written:

|ϕ=c1 exp(iE1t/)|1+c2 exp(iE2t/)|2

if we assume real c1, c2 it follows that the expectation value ˆS will be:

ˆS=ϕ|ˆS|ϕ=[c1eiE1t/1|+c2eiE2t/2|][c1eiE1t/|2+c2eiE2t/|1]=c1c2[eiω21t+eiω21t]=2c1c2cos(ω21t)

Thus the expectation value of ˆS oscillates in time at frequency ω21=(E2E1)/. This arises because ˆS is not compatible with the hamiltonian, and hence does not define a constant of the motion.


This page titled 6.3: Example - Oscillation in a fully mixing two state system 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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