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Can someone give me an Eli5 description of what the matrix element is, particularly in regards to Fermi's Golden Rule?

Fermi's golden rule describes the likelihood of a transition per unit time. Does the matrix element describe the the likelihood of a transition itself without the time element?

Is the matrix element just a variable/operator that facilitates the description of the finite spatial extent of a scattering body?

Or does the matrix element just describe the number of final states available for a given interaction/transition?

The matrix element wiki wasn't very helpful.

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The matrix element could be interpreted as a term that gives information about whether a transition from an initial to final state is possible and if so the strength of that transition.

Consider for example, a specific case in solid state physics where an electric field is the perturbation. This perturbation allows for transitions of electrons from one a state in one band (i) to a state in another band (f).In the dipole approximation, the material field interaction is given by the product of the electric field amplitude and the dipole moment between the final and initial states. Here the dipole moment $ <f|\vec{r} |i> $ is simply the matrix element and its value tells us whether the transition is allowed and its strength.

The matrix element does not have anything to do with the density of states in the final state, there is explicitly another term in the Fermi Golden Rule to account for this.

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  • $\begingroup$ so the matrix element is essentially a description of the likelihood of a transition? $\endgroup$ – mcsquared Jul 29 '18 at 15:39
  • $\begingroup$ Yes, it is one of the factors that determine the likelihood of the transition, the other factor is of course the density of states of the final state, which also appears in the Golden Rule $\endgroup$ – Hari Jul 30 '18 at 12:43

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