Finite-Size Corrections to Fermi’s Golden Rule

The transition process in quantum mechanics has been studied with the probability ΓT proportional to the time interval T between the initial and final states, where Γ is the rate. Recently it was found that the constant term P(d) is added as P = ΓT +P(d). P(d) has been ignored for a macroscopic T, but that is not ignoable in various systems. We find P(d) in such processes that Γ ≈ 0 and show that it has wave-like properties and leads unusual quantum phenomena. P(d) is