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Basic Quantum Mechanics

Overview

Optical processes such as absorption, emission, and their combination in a process called laser-induced fluorescence have been proven useful for studying engines. Several examples of their utility are provided elsewhere on this webpage. In many cases, proper application of optical techniques requires some understanding of quantum mechanics.

Details

The fundamental issue here is that both the optical part of the problem (photons) and the gas-phase part of the problem (molecules) are composed of small particles. For example, a typical gas molecule such as the hydroxyl radical (OH) can be modeled as a sphere with a diameter of 0.3 nm. At such a small scale, some features of physics are peculiar and unfamiliar. Consider rotation of the O-H molecule as suggested below. One might expect that OH can rotate about the arrow shown with any rotational speed. In fact, this is not the case. Nature only allows certain speeds (including, in this case 0, 1×1013, 1.7×1013, and 2.5×1013 revolutions / second; or 0, 600 trillion, 1 quadrillion, and 1.5 quadrillion rpm). Although these speeds are higher than the speeds of macroscopic objects, the associated energy stored in the rotation of the molecules is relatively small (0, 7.5×10-22, 2.3×10-21, and 4.5×10-21 Joules), because of the small size and mass of the rotating molecule.

molecules1

Note that the OH molecule is simply forbidden to spin at any speed between 0 and 600 trillion rpm; this behavior is inconsistent with our experiences with spinning objects of ordinary size. In brief, the reason for this quantization of rotational speeds rests in the fact that objects have both wave-like and particle-like properties. The wave-like property requires that only certain resonant “rotational waves” can exist. To illustrate: imagine that the O and the H atoms are represented by enormous waves on a planet that contains no land, only ocean. The waves circulate about the planet (analogous with the rotation of the O and the H) but can only do so if an integer number of waves fit around the planet (otherwise there would have to be a jump in the ocean level at some longitude). In this way, only certain resonant rotational speeds are allowed.

A consequence of this quantization of rotational speeds is that the OH molecule interacts with light at only certain specific colors. For example, on average there are ~2 million OH molecules per cm3 in the air above Madison, WI. About 22% of them are spinning at 600 trillion rpm (see the figure below); these can emit optical energy, then spin at a slower speed to accommodate the loss of energy. However, the only possible slower speed represents complete lack of rotation. So these molecules can completely stop their rotation by emitting photons, but the emitted photons must have a particular energy value (7.5×10-22 J, see above) which means these photons have a particular color (far infrared: 265 µm). Thus absorption, emission, and fluorescence spectra, particularly of small molecules, are characterized by emission at discrete colors, as shown in the absorption and emission tutorials.