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The Old Quantum Theory

1.1 Introduction

The Old Quantum Theory refers to the early developments, largely during the years 1900 to 1925, which became precursors to modern quantum mechanics. While it lasted, the OQT produced many significant advances in theoretical chemistry, including an understanding of the periodic structure of the elements.

Physics around 1900 was regarded by many scientists as a completed theory, needing only some refinements “to a few decimal places.” The foundations of classical physics, as this is now known, are based on Isaac Newton’s laws of mechanics, James Clerk Maxwell’s theory of electromagnetism, and statistical mechanics as developed principally by Ludwig Boltzmann, J. Willard Gibbs, and J. C. Maxwell. From our modern viewpoint, we can identify three failures of classical physics: its inadequacy to explain blackbody radiation, the photoelectric effect, and the origin of line spectra. (Sometimes, the heat capacity of crystals at low temperatures is cited.) As it turns out, these seemingly minor flaws were ultimately responsible for the demolition of the entire foundation of classical physics.

1.2 Blackbody Radiation

It is a matter of common experience that a hot object can emit radiation. A piece of metal stuck into a flame can become “red hot.” Josiah Wedgwood, the famous pottery designer, invented a pyrometer (ca 1782) based on his observation that different materials become red hot at the same temperature. The radiation given off by material bodies when they are heated is called blackbody radiation, a blackbody being an idealized perfect absorber and emitter of all possible wavelengths of radiation. Figure 1.1 shows experimental wavelength distributions of thermal radiation at several temperatures. The maximum in the distribution, which determines the predominant color, increases with temperature in accordance with Wien’s displacement law

Figure 1.1 Intensity distributions of blackbody radiation at three different temperatures. The total radiation intensity varies as (Stefan-Boltzmann law) so the total radiation at 2000 K is actually times that at 1000 K. The visible region of wavelengths is shown.

where the wavelength is expressed in nanometers (nm). Integration of a distribution for a given temperature over all wavelengths gives the total radiation energy density per unit volume, known as the Stefan-Boltzmann law:

At room temperature (300 K), the maximum occurs around 9700 nm, in the infrared region. In Figure 1.1, the approximate values of are 2900 nm at 1000 K, 1450 nm at 2000 K, and 500 nm at 5772 K, the approximate surface temperature of the Sun. The Sun’s is near the middle of the visible range (380-750 nm) and is perceived by our eyes as white light.

The origin of blackbody radiation was a major challenge to nineteenth century physics. Lord Rayleigh proposed that the electromagnetic field could be represented by a collection of oscillators of all possible frequencies. We need first to calculate the number of oscillators per unit volume for each wavelength . The reciprocal of the wavelength, , is known as the wavenumber, equal to the number of wave oscillations per unit length. The wavenumber actually represents the magnitude of the wavevector , which also gives the direction in which a wave is propagating. Now, all the vectors of constant magnitude in a 3-dimensional space can be considered to sweep out a spherical shell of radius and infinitesimal thickness . The volume (in -space) of this shell is equal to and can be identified with the number of modes of oscillation per unit volume (in real space). Expressed in terms of , the number of modes per unit volume thereby equals . Sir James Jeans recognized that this must be multiplied by 2 to take account of the two possible polarizations of each mode of the electromagnetic field.

Rayleigh assumed that every oscillator contributed equally to the radiation, in accordance with the equipartition principle. Assuming equipartition of energy, each oscillator has the energy , where here is Boltzmann’s constant J K−1. We obtain thereby the energy per unit volume per unit wavelength range:

which is known as the Rayleigh-Jeans law. This agrees fairly well with experiments at lower frequencies (higher wavelengths), in the infrared region and beyond. But the formula implies that increases without limit as . Indeed, if ultraviolet rays and higher frequencies were really produced in increasing numbers, we might get roasted like marshmallows by sitting in front of a fireplace! Fortunately, this doesn’t happen. A theory with such disagreements with observation, which classical physics could not reconcile, is said to suffer from an “ultraviolet catastrophe.”

Max Planck in 1900 derived the correct form of the blackbody radiation law by introducing a bold postulate. He proposed that energies involved in absorption and emission of electromagnetic radiation did not belong to a continuum, as implied by classical theory, but were actually made up of discrete bundles, which he called “quanta.” On this basis, Planck is traditionally regarded as the father of quantum theory. A quantum associated with radiation of frequency is proposed to carry an energy

where the proportionality factor J sec is known as Planck’s constant. Using the relation between frequency and wavelength

where m/sec, the speed of light, we can alternatively express Planck’s formula in terms of wavelength:

Our development of the quantum theory of atoms and molecules will make extensive use of Planck’s iconic formula.

Planck realized that the fatal flaw was equipartition, which is based on the assumption that the possible energies of each oscillator belong to a continuum ( ). If, instead, the energies of an oscillator of wavelength come in discrete bundles , then the possible energies are given by

By the Boltzmann distribution in statistical mechanics, the average energy of an oscillator at temperature is given by

More explicitly,

Mathematica can evaluate this:

Therefore

This implies that the higher-energy modes are less populated than what is implied by the equipartition principle. Substituting this value, rather than , into the Rayleigh-Jeans formula (1.1), we obtain the Planck distribution law

Note that, for large values of and/or , the average energy (1.8) is approximated by and the Planck formula reduces to the Rayleigh-Jeans approximation. The Planck distribution law accurately accounts for the experimental data on thermal radiation shown in Figure 1.1. Remarkably, measurements by the Cosmic Microwave Background Explorer satellite (COBE) give a perfect fit for a blackbody distribution at temperature 2.73K, as shown in Figure 1.2. The cosmic microwave background radiation, which was discovered by Penzias and Wilson in 1965, is a relic of the Big Bang 13.8 billion years ago.

Figure 1.2 Cosmic Microwave Background. Adapted from G. F. Smoot and D. Scott.

From the Planck distribution law one can calculate the wavelength at which is a maximum at a given . This is somewhat tricky since it involves a transcendental equation. We first find the derivative of and find the value of for which this is equal to 0:

The result agrees with the Wien displacement law:

By integration of Eq (1.26) over all wavelengths , we obtain the total radiation energy density per unit volume:

Therefore

in accord with the Stefan-Boltzmann law.

1.3 The Photoelectric Effect

A common device in modern technology is the photocell or “electric eye,” which runs a variety of useful gadgets, including automatic door openers. The principle involved in these devices is the photoelectric effect, which was first observed by Heinrich Hertz in the same laboratory in which he discovered electromagnetic waves. Visible or ultraviolet radiation impinging on clean metal surfaces can cause electrons to be ejected from the metal; see Figure 1.3. Such an effect is not, in itself, inconsistent with classical theory since electromagnetic waves are allowed to carry energy and momentum. But the detailed behavior as a function of radiation frequency and intensity can not be explained classically.

Figure 1.3 Photoelectric effect.

The energy required to eject an electron from a metal is determined by its work function . For example, sodium has eV. The...