QUANTUM NUMBERS OF ELECTRONS - EXTENSION
The Quantum theory makes the following postulates:
1. Atoms and molecules can only exist in certain states, characterised by definite
amounts of energy. When an atom or molecule changes its state, it must
absorb or emit an amount of energy just sufficient to bring it to another state.
2. When atoms or molecules absorb or emit light in the process of changing their
energies, the wavelength  of the light is related to the magnitude of the
energy change Eby the equation
E = hc/ where h = Planck’s constant, c is the speed of light.
3. The allowed energy states of atoms and molecules can be described by sets
of numbers called quantum numbers.
In 1926 Schrödinger developed and equation, now known as the Schrödinger Wave
Equation. It is extremely complex and for a one-particle system takes the form:
 2   2   2  8 2 m
( E  V )  0
+
+ 2 +
h2
x 2  y 2
z
Where m is the mass of the particle, E and V are its total and potential energies,
respectively, h is Planck’s constant and the first three terms are partial derivatives
with respect to x, y and z, of , the wave function to be associated with the particle.
Schrödinger found that the electron in the H atom could be described by three
quantum numbers, which are now called n,  and m. There are three such numbers
because the electron requires three coordinates to determine its motion. The
quantum numbers of the electron establish its state and fix both its energy and wave
function.
First Quantum Number, n. The first or principal quantum number is given the
symbol n and it determines the energy of the electron. As the quantum number
increases the energy of the electron increases. The principal quantum number is
always integral and takes the values
n = 1, 2, 3, 4, 5, …. but not 0
When electrons in an atom have the same value for n they are said to be in the
same energy level or shell.
Second Quantum Number, . Each energy level of electrons in an atom has one or
more sublevels, or subshells. The sublevels are denoted by the second quantum
number, . The general geometric shape of the electron cloud associated with the
electron is determined by . The quantum number  is related to the quantum
number n: for a given value of n, is limited to the values:
 = 0, 1, 2, 3, ….. (n -1)
An electron with the n = 1, will have to have  = 0, which means that in the first
energy level there will be only one sublevel, in which  = 0. When n = 2 (in the
second energy level,  will be either 0 or 1, so there are 2 sublevels etc.
Third Quantum Number, m. Each sublevel contains one or more orbitals, each
designated by the third quantum number, m. This quantum number is associated
with the orientation of the electron cloud with respect to a given direction, unusually
one which is imposed on the atom by a strong magnetic field. For a given value of ,
m can have any integral value between  and - .
That is:
m = , -1, ….o, -1, …-
When  = 0 then m must also equal 0. Since there is one orbital for each different
value of m, there is only one orbital in the  = 0 sublevel. If  = 1, mtakes the values
1, 0, -1: thus in an  = 1 an electron may be in any one of three orbitals. In general,
within a given sublevel there will be 2 + 1 orbitals, with essentially the same energy.
Fourth Quantum Number, ms. In order to completely describe an electron in an
atom, we must have a fourth quantum number, ms, called the spin quantum number,
which is associated with the spin of the electron about its own axis. The quantum
number ms is not related to the vales of the other quantum numbers and only has
two possible values, +½ and -½, depending on the direction of rotation of the
electron about its axis. Two electrons in the same orbital, having the ms values of +½
and -½ are said to be paired. Both of these electrons would have the same energy.
The Pauli exclusion principle states that no two electrons can have the same set of
four quantum numbers. This means that if the first three quantum numbers are the
same the electrons must have different values for ms. This is why we can state as a
general rule that no more than two electrons can fit into any atomic orbital.
Level n
Sublevel 
Orbital m
Spin ms  = +½
=-½
Allowed sets of Quantum Numbers for Electrons in Atoms.
1
2
3
0
0
1
0
1
0
0
1
0
-1
0
1
0
-1
2
1
 
   
  
 
2
0

-1

-2

Questions:
If the n quantum number of an electron equals 2, what values could its  quantum
number have? Its m quantum number? And its ms quantum number?