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26 The hydrogen atom Fig. 2.1Polar plots of the squared modulus of the angular wavefunctions for the hydrogen a ...
2.1 The Schr ̈odinger equation 27 can be cast in a convenient form by the substitutionP(r)=rR(r): − ^2 2 me d^2 P dr^2 + { ^2 ...
28 The hydrogen atom Table 2.2 Radial hydrogenic wavefunctionsRn,lin terms of the variableρ = Zr/(na 0 ), which gives a scaling ...
2.2 Transitions 29 equation. The dependence on the principal quantum numbernalso seems to follow from eqn 2.21 but this is coinc ...
30 The hydrogen atom D 12 = ∫∞ 0 Rn 2 ,l 2 (r)rRn 1 ,l 1 (r)r^2 dr. (2.28) The angular integral is Iang= ∫ 2 π 0 ∫π 0 Yl∗ 2 ,m 2 ...
2.2 Transitions 31 whereAπdepends on the component of the electric field along thez- axis and the component in thexy-plane is wr ...
32 The hydrogen atom when the polarized light interacts with an atom that has a well-defined orientation, e.g. an atom in an ext ...
2.2 Transitions 33 selection rules. The parity transformation is an inversion through the origin given byr→−r. This is equivalen ...
34 The hydrogen atom Fig. 2.2Allowed transitions between the configurations of hydrogen obey the selection rule ∆l=±1. The confi ...
2.3 Fine structure 35 2.3.1 Spin of the electron In addition to the evidence provided by observations of the fine structure itse ...
36 The hydrogen atom Fig. 2.3The representation of (a) spin- up and (b) spin-down states as vectors precessing around thez-axis. ...
2.3 Fine structure 37 has a magnitude close to one Bohr magneton (μB =e/ 2 me). The interaction of the electron’s magnetic mome ...
38 The hydrogen atom where the spin–orbit constantβis (from eqns 2.51 and 2.23) β= ^2 2 m^2 ec^2 e^2 4 π 0 1 (na 0 )^3 l ( l+^ ...
2.3 Fine structure 39 as shown in Fig. 2.5(b). For both configurations, it is easy to see that the spin–orbit interaction does n ...
40 The hydrogen atom exact relativistic solution of the Dirac equation and the non-relativistic energy levels, three relativisti ...
2.3 Fine structure 41 Lamb shift Spin−orbit Fig. 2.7The fine structure of the n =2andn = 3 shells of hydro- gen and the allowed ...
42 The hydrogen atom transitions between the levels with differentjare as follows: 2P 3 / 2 −3S 1 / 2 , 2P 3 / 2 −3D 3 / 2 , 2P ...
Exercises for Chapter 2 43 (2.3)Radial wavefunctions Verify eqn 2.23 forn=2,l= 1 by calculating the radial integral (forZ=1). (2 ...
44 The hydrogen atom (a) Show that Θ(θ) satisfies the equation 1 Θ(θ) ∂Θ(θ) ∂θ =mmax cosθ sinθ . (b) Find the solution of the eq ...
Helium 3 3.1 The ground state of helium 45 3.2 Excited states of helium 46 3.3 Evaluation of the integrals in helium 53 Further ...
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