According to Quantum Mechanics, these orbitals are
mathematically defined and are described by a uniquely different
math
function for each electron in the atom called an "eigen function" and a
differential equation generated by the following equation: H(eigen
function) = Energy ( eigen function)
The H in the above equation stands for a mathematical operator called
the Hamiltonian. N can have a value of any positive integer beginning with the value of 1.
There is a rule that defines the maximum number of
electrons that can
have a value of N as part of the physical description of the atomic
orbital. This is called the 2N squared rule. The maximum number of electrons that can be assigned an N value specified is 2N2.
For example for an N value of 1 you could have a maximum of 2(1)2 or 2
electrons whose probability regions are partially described by an N
value of 1. The maximum number of electrons in an atom whose orbitals could be partially described by an N value of 2 would be 2(2)2 or 8. As the N value increases the probability region
radiates out further from the nucleus and the energy state associated
with that probability region increases.
The second quantum number in the set is called the subsidiary number and is symbollically represented by L. L can have values beginning with zero and increasing in value one integer until the integer N-1 is reached. The maximum number of electrons in an atom that can
have an L value of zero while having the same N value in their
description is 2. With an L value = 1 the shape will be double lobed approximating two obloid spheroids tangeant to one another. The maximum number of electrons that can have an L value of 1 while having the same N value is 6. The L value of 2 would indicate a shape that is quadra lobed somewhat like a four leaf clover.
The maximum number of electrons that could have an L value of 2 while
having the same N value in the partial description of the orbital is 10.
For an L value of 3 the shape would be octa lobed and the maximum
number of electrons that can have an L value of 3 while having the same
N value is 14.
The third quantum number in the set is called the magnetic quantum number and is symbolized as M. M values begin at -L values and proceed with every integer until L+1 is reached. So for example if L=3 then M= -3,-2,-1,0,+1,+2,+3 or seven values.
There can be a maximum of two electrons that can have the same value of
M while having the same value of L and N in their partial description.
The magnetic number sometimes referred to as "orbital" number
physically represents the direction in space that the orbital is
pointing in. For example for an L=0 value the M value can only be zero.
One value suggests that the orbital must have only one direction in
space that it is oriented toward. A sphere has been described as a
solid body that extends equally in all directions. A sphere is
uni-directional therefore the reason for only one M value. If L=1 then
M can have values of -1, 0, +1. Three values of M
suggest that the orbitals thus described which are double lobed in
shape (L=1) would have three distinctly different directions in three
dimensional space that they could extend into. An electron particle spinning on an imaginary axis
just as a planet in our solar system rotates around an axis could spin
either clockwise or counter-clockwise, two ways, hence two values for
the Spin number. Once those parameters have been calculated which
would represent theoretical parameters based upon the model, then one
can compare those predicted values with experimental values derived
from an experimental setting. If the
experimental values which we assume to be reliable do not match and
therefore support the mathematical results of the model, then one has
to conclude that the model is somehow defective and must be modified to
account for the experimental difference. Such a mathematical component is not necessary, but certainly improves the model''s predictive powers.
With the introduction of the Principles of Quantum Mechanics which
proposed the quantum
mechanical model for atomic systems, the model was
purely mathematical consisting of rather complex wave equations for
each electron in an atomic system.