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It is on the red edge of colors that you can see by eye. What happens if the box (molecule) is smaller, say 0.7 nm instead of 0.8 nm? The energy of the light that is absorbed becomes higher and, therefore, the wavelength of light that is absorbed becomes shorter as the box becomes smaller. The energy absorbed goes inversely as L2 (L2 is in the denominator), which means as the box gets smaller, the energy levels get further apart, and the energy difference increases as the square of the box length. So for a 0.7 nm box, the absorbed wavelength is λ = 540 nm, which is green light. If the box is even smaller, say 0.6 nm, then λ = 397 nm, which is very blue, on the blue edge of the light that can be seen with the unaided eye.

These simple results are also generally true for molecules, although there are a lot of details that come into play. However, for a sequence of molecules that have essentially the same type of structure (types of atoms, etc.) the bigger the molecule, the further to the red it will absorb. The results presented for a particle in a box show very qualitatively why things have different colors. Small molecules absorb light in the ultraviolet part of the spectrum. We can’t see ultraviolet light, so absorption by small molecules does not result in a color. We see colors by the light that bounces off an object. The colors that are absorbed don’t bounce off. Large molecules absorb in the visible part of the spectrum, and these molecular absorptions give objects color.

Cherries are red and blueberries are blue because they have different molecules in them that absorb different colors of light strongly. These molecules have quantized electronic transitions. They can only absorb light from their ground electronic states to their excited states at wavelengths that are determined by their quantized energy levels. In the particle in a box, the transition energies of an electron are determined solely by the length of the box and the mass of the electron. For molecules, the quantized transition energies, and therefore wavelengths and colors, are determined both by the sizes of the molecules and by the details of the molecular structures, that is, the shape of the molecule, the types of atoms that make up the molecule, and how the atoms are arranged. Dyes are molecules that have specific strong absorptions in the visible. Dyes are used to give our clothes different colors. Brightly colored plants, green leaves and red roses, contain a wide variety of molecules with different sizes and shapes that absorb distinct colors of light strongly. The sizes and shapes of these molecular absorbers give the plants their brilliant colors. If the molecules absorb green and red strongly, then blue will bounce off of an object, and it will look blue. If blue and green are absorbed strongly, mainly red will bounce off, and an object will look red. The colors that are absorbed are determined by the quantized energy levels of the molecules in the object.

We observe color constantly in everyday life. Color is one of many phenomena that we encounter that is inherently quantum mechanical. There are many others. For example, when you turn on an electric stove, the element gets hot. Why electricity moving through metal produces heat (Chapter 19) is another everyday quantum phenomenon. Why is carbon dioxide a greenhouse gas (Chapter 17)? What is a trans fat (Chapter 16)? It is necessary to go into the quantum mechanics of molecular structure to understand such systems. In the following chapters, the quantum description of atoms and molecules will be developed and applied to a number of common issues and problems. The necessary machinery for understanding atoms and molecules is developed in Chapters 9 through 14. These chapters supply a great deal of interesting information about how atoms and molecules behave and provide the bridge between the general ideas of quantum theory that we have just developed and understanding many phenomena that surround us.

9

The Hydrogen Atom: The History

IN CHAPTER 8, WE DISCUSSED the particle in a box