The Hydrogen Spectrum

One of the greatest accomplishments of atomic physics is Neils Bohr’s model for the structure of the hydrogen atom, and his prediction of the hydrogen spectrum. While Bohr gets the credit for deriving the formula for the wavelengths, λ, of light emitted by hydrogen—one of the early triumphs of quantum mechanics—it was first discovered empirically from the spectroscopic analysis of Johannes Rydberg. In the 4th edition of Intermediate Physics for Medicine and Biology, Russ Hobbie and I introduce Rydberg’s formula in homework Problem 4 of Chapter 14.

Problem 4 (a) Starting with Eq. 14.7, derive a formula for the hydrogen atom spectrum in the form

where n and m are integers. R is called the Rydberg constant. Find an expression for R in terms of fundamental constants. 

(b) Verify that the wavelengths of the spectral lines a-d at the top of Fig. 14.3 are consistent with the energy transitions shown at the bottom of the figure.

Our Fig. 14.3 is in black and white. It is often useful to see the visible hydrogen spectrum (just four lines, b-e in Fig 14.3) in color, so you can appreciate better the position of the emission lines in the spectrum.

The hydrogen lines in the visible part of the spectrum are often referred to as the Balmer series, in honor of physicist Johann Balmer who discovered this part of the spectrum before Rydberg. Additional Balmer series lines exist in the near ultraviolet part of the spectrum (the thick band of lines just to the left of line e at the top of Fig. 14.3). All the Balmer series lines can be reproduced using the equation in Problem 4 with n = 2.

An entire series of spectral lines exists in the extreme ultraviolet, called the Lyman series, shown at the top of Fig. 14.3 as the line labeled a and the lines to its left. These lines are generated by the formula in Problem 4 using n = 1. The new homework problem below will help the student better understand the hydrogen spectrum.

Section 14.2

Problem 4 ½ The Lyman series, part of the spectrum of hydrogen, is shown at the top of Fig. 14.3 as the line labeled a and the band of lines to the left of that line. Create a figure like Fig. 14.3, but which shows a detailed view of the Lyman series. Let the wavelength scale at the top of your figure range from 0 to 150 nm, as opposed to 0-2 μm in Fig. 14.3. Also include an energy level drawing like at the bottom of Fig. 14.3, in which you indicate which transitions correspond to which lines in the Lyman spectrum. Be sure to indicate the shortest possible wavelength in the Lyman spectrum, show what transition that wavelength corresponds to, and determine how this wavelength is related to the Rydberg constant.

Many spectral lines can be found in the infrared, known as the Paschen series (n = 3), the Brackett series (n = 4) and the Pfund series (n = 5). The Paschen series is shown as lines f, g, h, and i in Fig. 14.3, plus the several unlabeled lines to their left. The Paschen, Brackett, and Pfund series overlap, making the hydrogen infrared spectrum more complicated than its visible and ultraviolet spectra. In fact, the short-wavelength lines of the Brackett series would appear at the top of Fig. 14.3 if all spectral lines were shown.

Rydberg’s formula, given in Problem 4, nicely summarizes the entire hydrogen spectrum. Johannes Rydberg is a Sweedish physicist (I am 3/8th Sweedish myself). His entry in Asimov’s Biographical Encyclopedia of Science and Technology reads

“RYDBERG, Johannes Robert (rid’bar-yeh) Swedish physicist Born: Halmstad, November 8, 1854. Died: Lund, Malmohus, December 28, 1919.

Rydberg studied at the University of Lund and received his Ph.D. in mathematics in 1879, and then jointed the faculty, reaching professorial status in 1897.

He was primarily interested in spectroscopy and labored to make sense of the various spectral lines produced by the different elements when incandescent (as Balmer did for hydrogen in 1885). Rydberg worked out a relationship before he learned of Balmer’s equation, and when that was called to his attention, he was able to demonstrate that Balmer’s equation was a special case of the more general relationship he himself had worked out.

Even Rydberg’s equation was purely empirical. He did not manage to work out the reason why the equation existed. That had to await Bohr’s application of quantum notions to atomic structure. Rydberg did, however, suspect the existence or regularities in the list of elements that were simpler and more regular than the atomic weights and this notion was borne out magnificently by Moseley’s elucidation of atomic numbers.”

Yesterday was the 158th anniversary of Rydberg’s birth.