**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.

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,
...

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...

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.

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.

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