By R. Boudet (auth.)
When one techniques the examine of the quantal relativistic thought of the electron, one should be shocked by means of the space which lies among the body of the experiments, i.e. the genuine geometry of the gap and time, and the abstraction of the complicated matrices and spinors formalism hired within the presentation of the idea. This booklet makes use of a conception of the electron, brought by means of David Hestenes, within which the mathematical language is equal to the single of the geometry of the distance and time. one of these language not just permits one to discover back the well-known effects about the one-electron atoms conception yet additionally leads simply to the answer of difficulties thought of for a very long time with no solution.
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Extra resources for Relativistic Transitions in the Hydrogenic Atoms
1) Let vector n may be expressed in the frame (I, J, K): n = cos θˆ K + sin θˆ U, U = cos ϕˆ I + sin ϕˆ J. 3) ⊥ ⊥ e⊥ 1 = cos θ0 I, e2 = 0, e3 = sin θ0 I, u⊥ = cos θ0 cos ϕ I + sin ϕ J, v⊥ = − cos θ0 sin ϕ I + cos ϕ J. 4) that the component of T⊥ 1 (±k) upon I and the (±k) upon J are null for the three cases = m1 − m2 = 0, 1, component of T⊥ 2 or −1. 11) 1. 5) ˆ e±ikr cos θ (a(r, θ) cos ϕ cos θ0 + c(r, θ) sin θ0 )dτ ]I. 6) 2. 7) ˆ e±ikr cos θ (( b(r, θ) sin2 ϕ + a(r, θ) cos2 ϕ) cos θ0 + c(r, θ) cos ϕ sin θ0 )dτ ] I.
We denote ψ1 the wave function of a state S1/2 and ψ2 the wave function of a state P 1/2 and P 3/2. (a) Transitions P 1/2–S1/2. 5) and Case 1 in Sect. 3 with l = 0 ∞ U 1 = 0, U2 = 2 0 1 g1 f2 + g2 f1 r2 dr e3 . 1) (b) Transitions P 3/2–S1/2. 5) and Case 2 in Sect. 3 with l = 0 √ 4 2 ∞ g2 f1 r2 dr e3 . 3) 48 7 Case of the Transitions P1/2-S1/2 and P3/2-S1/2 be the Schr¨ odinger wave functions corresponding to two states of energy E1 and E2 . 4) i where αa = /mc, which gives j 2 (r) = αa(φ1 ∇φ2 − φ2 ∇φ1 ).
50) 2 Then the energy E = E0 = mc is given by E(n, κ) = mc2 [1 + α2 Z 2 (n − |κ| + (κ2 − α2 Z 2 )2 ]−1/2 . 51) The ﬁrst terms of the Taylor development of the formula giving the energy E(n, κ) of le level n, a state corresponding to the number κ, are E(n, κ) mc2 1 − α2 Z 2 − α4 Z 4 2n2 1 3 − 2n3 |κ| 8n4 . 52) An useful formula is the following . Because mcα/ = 1/a, E(n, κ) mc2 1 c − α Z2 + α2 Z 2 a 2n2 1 3 − 4 3 2n |κ| 8n . 53) Labels Given to the State According to Their Levels of Energy. E depends on Z and on the couple of numbers (κ, n).