## The Four Quantum Numbers

James Richard Fromm

The locations described by the solutions to the Schroedinger equation are not circular orbits, as they are in the Bohr model of the atom, but orbitals which may have a more complex geometry. Orbital geometry is taken up in later sections; for the moment we restrict ourselves to the permitted orbital energy levels of electrons within atoms.

Solutions to the Schroedinger equation describe these energy levels in terms of four quantum numbers. Each electron in an atom is described by these quantum numbers. The quantum numbers are all integers, which is not surprising if electrons have wave properties and the levels they occupy must correspond to an integral number of wavelengths. The four quantum numbers are called the principal or shell quantum number n, the momentum or subshell quantum number l, the magnetic quantum number m, and the spin quantum number s.

The integer values which one quantum number can have depends upon the values of the other quantum numbers describing the same electron. The permitted or allowed values of the subshell quantum number l depends upon the value of the principal quantum number n with which it is associated. The allowed values of the magnetic quantum number m depend upon the value of the subshell quantum number l, and thus also depend upon the principal quantum number n with which it is associated.

#### Principal Quantum Number

The principal quantum number, generally symbolized by n, denotes the major shell in which the electron is located. Usually this is simply called the shell. It corresponds to the orbit in the Bohr model of hydrogen. The values it can take are those of any integer greater than zero, which is expressed as a series: 1, 2, 3, 4, ... or K, L, M, N, ...

#### Subshell Quantum Number

The subshell quantum number, generally symbolized by l, denotes the subshell in which the electron is located. The values it can take are those of any integer from l = 0 up to and including l = n - 1. When n = 1, l = 0; when n = 2, l = 0 and 1; when n = 3, l = 0, 1, and 2; and so forth. The values of subshell quantum numbers may be expressed as the series 0, 1, 2, 3, 4 ..., or s, p, d, f, g...

#### Magnetic Quantum Number

The magnetic quantum number, generally symbolized by m, denotes the energy levels available within a subshell. It has the value of any positive or negative integer whose absolute magnitude does not exceed the value of l. This interdependence is summarized in the Table below.

##### Table: Interrelationships of the Quantum Numbers
 Orbital Values Number of Values s l=0, m=0 (1) p l=1, m=-1,0,+1 (3) d l=2, m=-2,-1,0,+1,+2 (5) f l=3, m = -3,-2,-1,0,+1,+2,+3 (7) g l=4, m = -4,-3,...,+3,+4 (9)

The different values of the magnetic quantum number do not correspond to different values for the energy of an electron in the absence of an external magnetic field. Experimentally, electrons described by different principal or subshell quantum numbers are found to have different energies regardless of the presence of an external magnetic field, while those with different magnetic quantum numbers and the same principal and subshell quantum numbers show differences only in the presence of such a field.

#### Spin Quantum Number

The spin quantum number, generally symbolized by s, denotes the direction of the electron spin. It can take either of two values, traditionally given as +1/2 or -1/2, sometimes as upspin and downspin. Any other two-valued assignment, such as male-female, would have done as well.

Many atoms have an odd number of electrons or an arrangement of electrons in which the number of positive and negative spins are not the same. These atoms or electrons are said to have unpaired spins. The presence of unpaired spins in the electronic structure of atoms can be detected in various ways, and chemists can make use of it to study the structures of molecules which contain these atoms.

Copyright 1997 James R. Fromm