By Professor Hartmut M. Pilkuhn (auth.)
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In the following, we consider a particularly simple rotation, namely one by an angle a about the z-axis. 272) The resulting transformation matrix DAa) is diagonal, (Dz(a))mn = bmne-ima, Dz(a) = e- iiza . 275) One says that the Hermitian lz = -i8q, "generates" rotations about the z-axis. 276) The simplest unitary matrix is a multiple of the unit matrix, Uo = e- iao . 265), it results in the multiplication of all expansion coefficients by eiao . /J by e- iao . For definiteness, one sets Qo == O. 276), namely trH = 0, for the following reason: Remembering det(AB)=(det A)(det B), 1 = uut implies det(UUt) = IdetUI 2 = 1 or detU = e- iau (the case U = e- iao would have Qu = Qodim(U), where dim(U) is the number of diagonal elements of U).
One may define a "KG potential" for use in the KG equation (1. 76) or (1. 77): VKG = V(l - V/2E). 136) Even without the approximation V / E ~ V / mc 2 , ßz is strictly independent of n. The expression for E N to lowest nonvanishing order in has, of course, ßz = 0, E N = -a1mc2 /2n 2 , in agreement with the Bohr-Sommerfeld formula. 128). And as a given l-value contains already 2l + 1 magnetic sublevels, the total degeneracy of the nonrelativistic energy levels of the spinless hydrogen atom is a1 gspinless(n) = E~r/(2l + 1) = n 2 .
2) = 1 - 2z/b + z2 /[b(b + 1)]. 179) below. 127) The integers n r and n are the "radial" and "principal" quantum numbers, respectively. 128) n=nr+l+l. 124) is given in Appendix A. 127), K, must be eliminated. 127) becomes a z E/(m 2c4 - E 2)1/2 = nß. Resolving this expression for E, one finds E = mc 2 ( 1+ ~~ -1/2 ) , 2 azE ( hcK, = - - = mc nß Note the absence of odd powers of az. 37). As ßz == ß is also small, one expands a1/n~ = a1/(n - ß)2 = a1/n 2(1 + 2ß/n + 3ß2/n2 + 4ß3/n3). 133) Expansion of the square root in powers of ßz = a1/(2l a1 gives: + 1)[1 + a1/(2l + 1)2 + 2ai/(2l + 1)4].
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