The Hydrogen Atom

In 1885 a Swiss secondary school teacher named Johann Jacob Balmer published a short note (entitled “Note on the Spectral Lines of Hydrogen”, Annalen der Physik und Chemie25, 80-5) in which he described an empirical formula for the four most prominent wavelengths of light emitted by hydrogen gas. These wavelengths had been measured with great precision by Vogel and Huggins, giving the four values 6562.10, 4860.74, 4340.10, and 4101.20 Angstroms (10-10 m).  Balmer's note does not make clear whether he was also aware of the measured series limit, l¥ = 3645.6 A, or whether he deduced this himself.  In any case, one can find by numerical experimentation that the four characteristic wavelengths are closely proportional to the following products of small primes

Three of these are divisible by 23, three are divisible by 5, three are divisible by 7, and two are divisible by 33.  Thus we can easily express these numbers as simple fractional multiples of 840 = 23×3×5×7, which corresponds to the series limit l¥ = 3645.6 A.  It may have been just this kind of numerical experimentation that led Balmer to recognize that the four prominent wavelengths are given very closely by (9/5)l¥, (16/12)l¥, (25/21)l¥, and (36/32)l¥.  He also noticed that the numerators of the coefficients are consecutive squares, and each denominator is 4 less than the numerator. He speculated that the pattern would continue up to the series limit, which is indeed the case. In terms of the wave number k(=1/l), Balmer's formula can be written as

where R = n2/l¥ with n = 2.  The parameter RH is now called Rydberg's constant for hydrogen, and the best empirical value is 10967757.6 m-1.  As Balmer also speculated, if we take different values of n we get different series of spectral lines. The series with n = 1, 2, 3, 4, and 5 are now known as the Lyman, Balmer, Paschen, Brackett, and Pfund series, respectively, which characteristic the spectral lines of the hydrogen atom.  (The Balmer series was observed first because its frequencies are in the visible and near ultra-violet range.)  This is an outstanding example of a successful empirical fit (like Bode's Law in astronomy) for a class of physical phenomena that was not based on any underlying physical model or theory, i.e., no reason was known for why wavelengths of light emitted from a hydrogen atom should exhibit this pattern.

In classical terms a hydrogen atom consists of a proton and an electron bound together by their mutual electrical attraction.  To keep them from collapsing together, we might imagine that the electron is revolving in “orbit” around the proton, similar to a planet revolving around the Sun, with the centrifugual force balancing the electrical attraction.  However, this model is not satisfactory, because the electron would be continuously accelerating, and according to classical theory an accelerating charge radiates energy in the form of electro-magnetic waves. As a result, the orbiting electron would very quickly radiate away all of its kinetic energy and spiral into the proton. Thus the existence of stable atoms was inexplicable in the context of classical physics, as was the characteristic set of discrete energy levels of atoms.

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