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6.2: Notes

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    • The ‘Matrix element’ \(V_{21}\) determines whether there is a transition from an initial state 1 to a final state 2 even if \(\hat{V}\) is independent of time. It also determines the rate of the transition.
    • If the states \(|1 \rangle\) and \(|2 \rangle\) are eigenstates of the perturbation \(\hat{V}\) then \(V_{21} = V_{12} = 0\) and no transition occurs.
    • Over a long period of time, the system will oscillate between the two states.
    • Perturbation theory, in essence, ignores the third-order possibility of ending up in state 2 via \(|1\rangle \rightarrow |2\rangle \rightarrow |1\rangle \rightarrow |2\rangle\)
    • The mathematics is the same as for two coupled pendula, where the energy moves back and forth between the two bobs.
    • The states can represent anything, and oscillation will occur whenever there are off diagonal terms in the matrix.
    • Examples: (see Feynman III Ch.9-11) Nitrogen atom in ammonia, electron in H\(^+_2\), pion exchange, benzene, electron spins, photon polarization, neutrino oscillations, neutral kaons.

    This page titled 6.2: Notes 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; a detailed edit history is available upon request.