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Additional info for An Introduction to the Electron Theory of Solids. Metallurgy Division
11) can be further simplified in some circumstances. 11 becomes -JLlAu 2m ôx2 + Vuifi = £ # = /# — . 14) Dividing by u\j/9 h2 1 d2\l/ .. 1 du rA ... 21 +v = E = ih . 15) is a function of time only, and it therefore follows that the total energy E is a function of time only. 15) depends only on position. It follows therefore that in this case the total energy of the system E is independent of both position and time. 13) is valid for potential fields in which energy is conserved. Such potentials are called "conservative fields".
Exact agreement is obtained if the transitions are restricted to those in which the angular quantum number changes by unity: Afc« ± 1 . This is called a selection rule. 43) 27 ATOMIC SPECTRA The Third Degree of Freedom—the Zeeman Effect So far, only two quantum numbers have been used to describe the motion, and this is a necessary corollary of the restriction introduced earlier to the effect that the possible orbits are planar. The third degree of freedom is related to the angle between the normal to the orbit plane and some external reference axis.
The energy of the orbits taking into account the relativistic mass change is given by the formula where as before ri is the radial quantum number, fi0 is the rest reduced mass, and a is the "fine structure constant" a = — = 7-2973 x 10" 3 . 30) may be derived: Enk 2 . c /ML = R- H£ +, RH* —ii— In I _ - 3\ - 1 +, ,terms of order a44 . 42) This means that the degeneracy of the orbits of the same major semi-axis but different eccentricity is removed by the relativistic mass change, with a resultant splitting in the spectral lines.