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6.2: Notes

Last updated

Oct 10, 2020
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- 6.1: Time Dependence
- 6.3: Example - Oscillation in a fully mixing two state system

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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.

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