Old Atomic Quantum Theories

A black body is a material that absorbs perfectly at all wavelengths. The intensity of electromagnetic radiation emitted from a black body at some temperature is found to increase with frequency to some maximum (nm) and then decrease to zero at some greater frequency. The peak and maximum frequency of the envelope of emitted radiation is found to increase with temperature as shown in Figure 1.4. In the classical treatment, the black body radiator is described as a sea of electrons moving as harmonic oscillators in the walls of the object. This model predicted the intensity of the emitted radiation should increase quadratically with frequency. This agreed with experiment at low frequencies, but failed at higher frequencies.

Figure 1.4. Black body radiator spectra distribution at 500K and 1500K. Dashed line at left shows the Rayleigh-Jeans Law of classical physics.

In 1900, Max Planck solved the problem of the black body radiator [3] by imposing conditions of quantization on the energies of the electron harmonic oscillators. Taking adjacent energy levels to be separated (DE) by an amount proportional to frequency (n), he proposed:

(1)

The constant h later came to be known as Planck's constant, and now has the accepted value 6.626x10-34 J-sec.

An explanation for the spectral discreteness and stability of the hydrogen atom was provided by Bohr in 1913 [4]. In solving for the orbital motion of the electron in an orbit of radius r, Bohr imposed the condition that the angular momentum (l) of the electron must equal some integral multiple of h:

(2)

where p is the linear momentum of the electron. He also postulated that electromagnetic radiation is emitted when an electron discontinuously changes its motion from one energy level to another, the frequency of the emitted radiation being as predicted by equation (1). As the difference in energies is equal to the energy of the emitted photon, the above relation is the same as Einstein's postulate that the energy of a photon is equal to its frequency times Planck's constant. On the basis of observation, Bohr also postulated that the feature of classical theory that predicted the emission of electromagnetic radiation by the orbiting electron was not valid for an atomic electron. With his restriction on the orbital angular momentum of the electron and suspension of an aspect of classical electromagnetic theory, the stability and energy levels of the hydrogen atom electrons were accurately predicted. This represented the first quantized model of the atom. The electron energy levels 'n' of hydrogen are shown in Figure 1.5. Electromagnetic emission corresponds to transitions of electrons between electron energy levels as shown.

Figure 1.5. Electron energy levels of the hydrogen atom. Also shown are observed transitions between states resulting in emitted electromagnetic radiation. The 'n' quantum numbers represent orbitals predicted by Bohr's model while the 'l' quantum numbers indicate additional energy levels explained by Sommerfeld's model.

In nature, the energy levels of hydrogen predicted by the Bohr model are actually split into separate closely spaced energy levels. This 'fine structure' is seen in the energy levels of all atoms, but the Bohr model did not directly predict this. In 1916, Wilson and Sommerfeld proposed a quantum condition for physical systems that are periodic functions of time, called the Wilson-Sommerfeld rule. Planck's quantization of energy levels of the harmonic oscillator in the black body radiator and the quantization of angular momentum in the Bohr model were shown to be special cases of the Wilson-Sommerfeld rule.

Bohr's model described the orbit of the electron as circular. The Sommerfeld model [5] described the orbit of the electron as a generic ellipse. Using the Wilson-Sommerfeld rule, describing the electron orbit in polar co-ordinates lead to two integer quantum numbers n and nq, and therefore to sets of allowed elliptical orbits with each Bohr circular orbit.

(3)

The set of spherical orbits associated with each circular orbit had the same total energy as the Bohr orbit unless account was made of relativity. In elliptical orbits, electrons pass closer to the nucleus than in circular orbits. Accordingly, electrons would travel faster as they approached the nucleus. By applying special relativity to the set of allowed elliptical orbits, Sommerfeld obtained the same fine structure of energy levels observed in hydrogen.

As indicated in Figure 1.5, not all transitions between energy levels resulting in the emission of a photon take place. The old quantum theory of atoms does not explain this. Though selection rules for transitions were established and the Correspondence Principle, described by Bohr in 1923, ensured consistency between model predictions and transition selection rules in the quantum and classical limits, the old quantum theory of the atom was an ad-hoc mixture of classical and quantum principles. The classical limit was defined as the point where quantum numbers describing the state of a system get very large.

The angular momentum condition of the Bohr atom was invented for the case of the hydrogen atom, and the model only worked reasonably well for one-electron atoms. The alkali elements (Li, Na, K, Rb, Cs) are similar to a one-electron atom, so limited success was found with these elements. In addition, the model says nothing about the rates at which transitions will occur. Finally, the Wilson-Sommerfeld rule limits the old quantum theory to periodic systems. There are many significant physical systems that are not periodic.

In 1924, de Broglie's doctoral thesis suggested the wave-particle duality of radiation (light waves and photons) was equally applicable to matter (electrons, etc.). As with radiation, the energy E of a particle is related to the frequency n of the wave associated with its motion and the momentum p is related to the wavelength l of the wave:

(4)

Substituting the second part of equation (4) into equation (2) results in:

(5)

As 2pr is the circumference of the orbit, the Bohr condition can be seen to be the same as requiring a standing wave around the atom. This new interpretation of matter led to the Schrödinger equation, quantum mechanics, and the modern quantum theory.

While the Bohr model has been replaced by quantum mechanics, its simplicity has ensured it is still a useful tool in the description of atomic physics. The persistence of such Bohr model concepts as 'orbital' in quantum physics is tribute to this.

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