The interpretation of quantum numbers is key in understanding why chemists make such extensive use of the concept.

n, the principal quantum number, is most important in determining the energy of the electron. The lower n, the more energy will be needed to remove the electron from an atom or ion. n tells the number of nodal surfaces in the orbital. A nodal surface, a surface of zero probability, reduces the influence of the positive nucleus on the negative electron. The more nodal surfaces, the less tightly bound the electron.

The always is one nodal surface at infinite distance from the nucleus. So, the lowest value n can have is 1.

l, the azimuthal quantum number or orbital quantum number, is next most important in determining the electronic energy. l tells the number of nodal surfaces that pass through the nucleus. An orbital may have none of these -- no nodal surfaces passing through the nucleus. It may have several of them. However, there always is one nodal surface at infinite distance -- the ultimate container for the electron. So, the smallest value l can have is 0, but the largest value is n-1.

Because it determines the number of nodal surfaces passing through the nucleus, the orbital quantum number is a key factor in determining the so-called shape of the orbital. In fact: all orbitals with l = 0 are called s orbitals, all those with l = 1 are called p orbitals, all those with l = 2 are called d orbitals, and all those with l = 3 are called f orbitals,

The orbitals usually have the quantum number n & l written separated by a comma, so, 3,0 represents a 3s orbital, while 3,2 represents a 3d orbital.

m, the magnetic quantum number, leads to small differences in energy. For a given pair of values of n and l, several orbitals may exist that differ from one another in orientation. m distinguishes these orbitals. The number of orbitals that can share an l value is 2l + 1. The smallest value of m is -l, the largest is l, and the values change one unit at a time, including 0, going from -l to l.

There are 5 orbitals with quantum numbers n=3 and l=2; these are the five 3d orbitals.

3,2,-2

3,2,-1

3,2,0

3,2,1

3,2,2

Finally, the electron has an intrinsic property called electron spin. This property is represented by s, the spin quantum number. s can have a value of either -1/2 or +1/2. Changing from one value to another involves a unit change in the number.

Four quantum numbers determine an electron's energy. Three of these -- n, l, and m -- come from the orbital. The fourth, s, comes from the electron spin.

Wolfgang Pauli [local] set fourth an important principle: in an atom or molecule, no two electrons can have one and the same set of four quantum numbers. This is called the Pauli Exclusion Principle. To say that two electrons had the same quantum numbers would be to say that they had exactly the same wavefunctions. In a sense this is the ultimate transition from particulate thinking to quantum thinking.

A Manthey image for one of the 3 3p orbitals. There is one nodal surface at infinite distance. There is a (flat) plane passing through the nucleus. There also is a sphere that separates the small "lobes" inside from the larger "lobes" outside.

Table summary of quantum numbers. [local]