A Prelude to Quantum Field Theory by John Donoghue and Lorenzo Sorbo

Author:John Donoghue and Lorenzo Sorbo
Language: eng
Format: epub
Publisher: Princeton University Press
Published: 2021-11-29T00:00:00+00:00

To get a probability we square it:

The product of two delta functions can be interpreted by treating one of them as

where we have enforced from the other delta function, and where T is the entire duration of the measurement. This yields

which is Fermi’s Golden Rule in its most basic form. For time-independent interaction terms, such as those that we will use, we can just send . While the math has been quick and dirty, the result is correct. It even makes intuitive sense, with the amplitude-squared factor and energy conservation.

Equation (6.6) is fine if we want to study the transition from some initial state to an isolated final state. But what if there are many final states, possibly a continuum, that are accessible to the system? In that case, we have to sum on all those final states:

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