Quantum Dynamics Of Atomic Electrons

In this Thesis we study the Quantum Dynamics of Atomic Electrons. Using the mathematicalrnderivation of the system of two coupled di erential equations for the radial wavernfunction, we have calculated the Sommerfeld expression for energy eigenvalues for electronsrnbound to nuclei. Then we have calculated the wave function at the diving point.rnApplying the Sommerfeld- ne structure formula, we have calculated the binding energiesrnfor 1s1=2. And we have classi ed the bound states of the electron according to the Diracrnequation for Z = 1 (hydrogen atom). We found that the energy levels descend progressivelyrnwith increasing Z. The energy of the lowest state (1s; k = 1) becomes negativernwhen the nuclear charge Z > 150 and 1s level nally reaches the values E1s = À€€m0c2 at arncritical charge Z1srncr ' 173. Hence the Sommerfeld energies of the states with k = ð€€€1(ns1=2)rnand k = +1(np1=2) break o with a vertical tangent at Z = 1 and with the nite nuclearrnradius taken into account all levels reach the edge of the lower continuum E = ð€€€m0c2 atrna corresponding critical charge Zcr.rnUsing perturbation potential V 0 we have determined the Fano's formalism for therndescription of resonances. And using Fano's formalism we have calculated the nal holernprobability in the bound state and the spectrum of the emitted positrons depending on therndiving duration. Then, we found that the modi ed continuum wave E(r) thus displaysrnexactly the same asymptotic behavior as E(r) but it is shifted by an angle E. Thernprobability of nding a hole in state 0 after a time T thus decreases exponentially asrndetermined by the decay width ð€€€0 = ð€€€Er . An oscillating function with maximum atrnE = Er having a width decreasing with the inverse of T. The peak height increasesrnquadratically and it consistent linearly in T until saturation is reached at T > 1=ð€€€0

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