## Quantum Number

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# Quantum Number

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A quantum number is a number that specifies the particular state of motion of an atom or molecule. Consequently, it describes the energies of electrons in atoms because of that motion of an elementary particle or system. Each quantum number describes the value (as integers or half integers) of a conserved quantity in the quantum system.

By 1900, several phenomena were recognized that could not be explained by accepted scientific theories. One such phenomenon was the behavior of light itself. In 1900, however, German physicist Maxwell Planck (18581947) developed a new theory that successfully described the nature of light. Part of this theory required that light having a certain frequency also had to have a certain specific energy. One way to state this is that the energy of a certain frequency of light was quantized. Light was considered as acting as a particle of energy, later called a photon.

Some of the unexplainable phenomena were related to atoms and molecules and, between 1925 and 1927, German physicist Werner Heisenberg (19011976) and Austrian physicist Erwin Schrodinger (18871961) considered that subatomic particles like electrons can act as waves (just like light waves can act as particles) and simultaneously developed quantum mechanics. They used different ways to describe their theories mathematically, and today most scientists use Schrodingers way. Since Schrodinger used wave equations to describe the behavior of electrons in atoms and molecules, quantum mechanics is sometimes also referred to as wave mechanics.

Schrodingers wave mechanics assumed that the motions of electrons, which are the basis of almost all chemistry, can also be described mathematically as waves, and so the idea of the wave function was established. A wave function is an equation that describes the motion of an electron. An electron whose motion can be described by a particular wave function is said to be in a particular state.

One of the more unusual (but useful) parts of Schrodingers wave functions is that an electron having a particular state has a certain, specific quantity of energy. That is, wave mechanics predicts that the energy of electrons is quantized. In almost all of the wave functions, a whole number (i.e., either 1, 2, 3, 4,...) is part of the wave equation. This whole number is a quantum number and, for electrons in atoms, it is called the principle quantum number. The value of the energy associated with that wave function depends on the quantum number. Therefore, the quantum number ultimately predicts what value of energy an electron in a state will have. Other quantum numbers are related to other properties of an electron. In particular, the value of the angular momentum of an electron (that is, the momentum that the electron has as it circles about the nucleus in an atom) is also quantized, and it is related to a whole-number quantum number called the angular momentum quantum number. There is also a magnetic quantum number for electrons in atoms, which is related to how much an electron in an atom interacts with a magnetic field. The amounts of such interactions are also quantized, that is, they can have only certain values and no others.

Molecules have other types of motions that are associated with certain values of energy. For example, the atoms in molecules vibrate back and forth. Molecules in the gas phase can also rotate. For each of these kinds of motions, quantum mechanics predicts that the motions can be expressed using a wave function. Quantum mechanics further predicts that each wave function will have a certain quantized value of energy, and that this energy can be expressed by a quantum number. Hence, vibrational and rotational motions also have quantum numbers associated with them. These quantum numbers are also whole numbers.

Quantum mechanics predicts a previously unknown property of subatomic particles that is called spin. All electrons, for example, have spin. So do protons and neutrons. However, quantum mechanics predicts that the quantum number associated with spin does not necessarily have to be a whole number; it can also be a half-integer number. For electrons, the quantum number for spin is ½. Since it can spin in either one of two directionsthat is, an electron can behave as if it is spinning either clockwise or counterclockwiseelectrons are labeled as having spin quantum numbers of either +½ or -½. The curious thing about the spin quantum number is that it cannot have any value other than ½ for an electron. Other subatomic particles have their own characteristic spin quantum numbers. Including spin, electrons in atoms can be assigned four separate quantum numbers: a principle quantum number, an angular momentum quantum number, a magnetic quantum number, and a spin quantum number. Stating the values of these four numbers expresses the complete energy state of an electron in an atom.

The principal quantum number (n) can have values of n = 1, 2, 3,.... The angular momentum quantum number (l) has values of l = 0, 1, ..., n-1. The magnetic quantum number (ml) has values ml = -l, -l + 1, ..., 0, ..., l - 1, l. The spin quantum number (ms) has values ms = -½ or +½.

### KEY TERMS

Wave function A useful mathematical construct commonly employed in quantum mechanics to represent both a particles wavelike characteristics and its uncertainty in location.

## Resources

### BOOKS

Duck, Ian. 100 Years of Plancks Quantum. Singapore and River Edge, NJ: World Scientific, 2001.

Huang, Fannie, ed. Quantum Physics: An Anthology of Current Thought. New York: Rosen Publishing Group, 2006.

Lewin, Roger. Making Waves: Irving Darkik and his Superwave Principle. Emmaus, PA: Rodale, 2005.

Mehra, Jagdish. The Golden Age of Theoretical Physics. Singapore and River Edge, NJ: World Scientific, 2000.

David W. Ball