The Magnetic Moments of Hydrogen-like Atoms

Abstract

DR. BREIT'S letter in NATURE of Oct. 27 seems to imply that the magnetic moment of a hydrogen-like atom has so far been calculated only for radial quantum number zero. I therefore venture to give the general result, expecting, however, that it has already been calculated by others. The calculation is easily performed by expressing Darwin's functions in terms of Laguerre's polynomials of non-integral rank. It is convenient to write j = k + 1 when it is positive and - k when it is negative, and to write J = √(j² - γ²), N = J + p n = √(N² + γ²), where p is the radial quantum number and γ = 2πZe²/hc. We find that the magnetic moment is j(2l + 1)(2Nj + n)(2j - 1)(2j + 1)n); l being the equatorial quantum number. This is the expression of spacial quantisation in Dirac's system.