Q13. Bohr’s Triumph: Explaining The Rydberg Constant

In this Essay I present the essence of Bohr’s 1913 work on the hydrogen atom. It builds on the four postulates summarized in Essay Q12 and differs from the simplified rewriting of history (circular orbits and quantized angular momentum) that appears in many textbooks. (I will do a little rewriting of history myself by referring to photons, which Bohr had not yet accepted. It is just easier to say “photon” than “bundle of energy emitted in a quantum jump,” and it doesn’t alter the essence of Bohr’s reasoning.)

Bohr knew of the Balmer series of spectral lines (partly in the visible, partly in the infrared) and the Paschen series (all in the infrared) emitted by hydrogen. The frequencies of these lines (and others discovered later) are expressed by

In this formula n = 2 (Balmer) or 3 (Paschen)—or another small integer for series discovered later)—and m is an integer larger than n. The other symbols in the formula are f, the frequency of the spectral line; c, the speed of light; and ℛ, the so-called Rydberg constant, named after the Swedish physicist Johannes Rydberg who first drew attention to simple regularities in various spectral series.

I must emphasize that when Bohr undertook to construct a quantum theory of the hydrogen atom, the Rydberg constant was an empirical constant, known to high accuracy from measurements of the frequencies (or wavelengths) of spectral lines, but with no known relation to other constants of physics. To give away the end of the story (see the next-to last equation in this Essay), Bohr succeeded in expressing ℛ in terms of the mass m of the electron, the charge e of the electron, Planck’s constant h, the speed of light c, and the electric force constant k.¹

Multiplication of both sides of this equation by Planck’s constant h gives the energy radiated in the quantum jump, hf, which Bohr assumed to be equal to the energy difference, ΔE, between two stationary states:

A single set of energy values to account for these energy differences is

The integer n identifies a particular stationary state. The first (and lowest) is characterized by n = 1, the second by n = 2, and so on. The negative sign in this equation reflects the fact that the electron is bound to the positive nucleus, like the Moon to the Earth, or the Earth to the Sun. It is often convenient to use the binding energy W, a positive quantity,

As shown in the figure below, the Balmer series is produced by quantum jumps ending at the second stationary state, and the Paschen series by quantum jumps ending at the third stationary state. Without any evidence for the lowest energy state (n = 1), Bohr confidently predicted its existence. Before long, the Lyman series, produced by quantum jumps ending at the first stationary state, was identified, vindicating the prediction.

So far I have used the ideas of stationary states, quantum jumps, and energy conservation. The last equation above, for the binding energies of the stationary states, is still an empirical formula, chosen to fit the facts of the spectrum (in the framework of these ideas). Bohr’s great achievement, expressing The Rydberg constant in terms of other fundamental constants, required the use of his fourth essential idea, the correspondence principle.

Classically, an electron far from the nucleus radiates continuously at a frequency equal to its own frequency of revolution about the nucleus. It spirals inward, radiating at ever higher frequencies. Quantum-mechanically, the electron moves in one stationary state, then jumps to a lower state, then to a still lower state, and so on, cascading toward the nucleus while emitting a series of photons that contribute to discrete spectral lines. According to the correspondence principle, these two seemingly very different descriptions of the atom should merge into one when the stationary states are fractionally close together in energy, and the successively emitted photons are fractionally close together in frequency. Then the granularity of the quantum description gives way to the continuity of the classical description. It is evident from the energy-level diagram above that this classical limit could be approached in the hydrogen atom only where the energy levels cluster together, at high values of n.

Classically, an electron in a planetary orbit about a fixed proton rotates with a frequency f_e given by

where W is the electron’s binding energy, e is the magnitude of the electron charge (and also the magnitude of the proton charge), m is the mass of the electron, and k is the Coulomb force constant. (I provide a derivation of this expression for circular orbits in an appendix to this Essay. It is also valid for elliptical orbits.) In order for a quantum description of a cascading electron to correspond to a classical description of a spiraling electron, the cascading electron must pass successively through every stationary state. In a transition from state n to state n – 1, the electron emits a photon whose frequency is given by

Here the subscript r designates radiation. This equation can be rewritten

For very large n, the factor 2n – 1 in the numerator can be replaced by 2n, and the factor (n – 1)² in the denominator replaced by n², to yield

According to the correspondence principle the radiated frequency f_r should equal the electron frequency f_e when n is very large. If Equation (1) above for W_n is substituted into Equation (2) for f_e, the correspondence condition f_r = f_e can be written

An important thing to note about this equation is that the factor n³ on the left cancels against an equal power of n on the right. This confirms that the correspondence principle holds generally for all large values of n, not just for a particular transition. It is in fact possible to prove by a somewhat more general argument that the requirement of the correspondence principle can be met only if the binding energy varies in proportion to 1/n² at large n.

Notice next that this equation can be solved for ℛ in terms of other fundamental constants. The result is

This splendid unification of electron constants (e and m), quantum constant (ħ), and spectral radiation constants (c and ℛ) remains valid today. The Rydberg constant, now known to 1 part in a trillion,² helps, through this equation, to determine an accurate value for Planck’s constant. With the accuracy of constants available to him, Bohr was able to verify the correctness of his equation for ℛ to within 6%.If we substitute this equation for ℛ into the earlier equation for the binding energies W_n of the electron in the hydrogen atom, we get an alternative expression for the binding energies (now omitting the subscript n):

This is expressed in terms of Planck’s constant (recall ħ = h/2π) and two basic properties of the electron, m and e. Note: Wherever k appears in this Essay or in the Appendix that follows, it can be replaced by 1/(4πε₀).

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Appendix

Derivation of expression for the classical orbital frequency of an electron in a circular orbit around a proton

Start by equating the centripetal force on the electron to the electrical force acting on it.

Solve this for the orbital radius r to get

The orbital frequency of the electron is its speed divided by the circumference of its orbit,

Into this equation substitute the expression above for r to getTaking advantage of the fact that the electron’s kinetic energy is K = 1/2mv², rewrite this equation for orbital frequency:

But you want to express this frequency in terms of the binding energy W rather than the kinetic energy K. Fortunately, they are equal! To show that this is the case, go back to the first equation in this Appendix, and multiply both sides by r. Then the left side becomes twice the kinetic energy K and the right side becomes the magnitude of the potential energy U (which is the negative of the potential energy):

2K = – U.

Then the total energy can be written

E = K + U = – K.

Since the binding energy is the negative of the total energy,

W = K

and the formula for the orbital frequency is

This is the equation that we set out to derive.

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