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Quantum Mechanics
• Erwin Schrödinger
developed a mathematical
treatment into which both
the wave and particle nature
of matter could be
incorporated.
• It is known as quantum
mechanics.
The Quantum Mechanical Model
• Energy is quantized - It comes in chunks.
• A quantum is the amount of energy needed to move from
one energy level to another.
• Since the energy of an atom is never “in between” there
must be a quantum leap in energy.
• In 1926, Erwin Schrodinger derived an equation that
described the energy and position of the electrons in an
atom.
Schrodinger’s Wave Equation

d

V 
8  m dx
h
2
2
2
2
 E
Equation for the probability of a
single electron being found along a
single axis (x-axis)
Erwin Schrodinger
Quantum Mechanics
• The wave equation is designated with
a lower case Greek psi ().
• The square of the wave equation, 2,
gives a probability density map of
where an electron has a certain
statistical likelihood of being at any
given instant in time.
Quantum Numbers
• Solving the wave equation gives a set of wave
functions, or orbitals, and their corresponding
energies.
• Each orbital describes a spatial distribution of
electron density.
• An orbital is described by a set of three quantum
numbers.
Principal Quantum Number, n (Shell)
• The principal quantum number, n, describes the energy level
on which the orbital resides.
• The values of n are integers ≥ 0.
distance of e- from the nucleus
n=1
n = 1, 2, 3, 4, ….
n=2
n=3
Azimuthal Quantum Number, l
• This quantum number defines the shape of the orbital.
• Allowed values of l are integers ranging from 0 to n − 1.
• We use letter designations to communicate the different
values of l and, therefore, the shapes and types of orbitals.
for a given value of n, l = 0, 1, 2, 3, … n-1
n = 1, l = 0
n = 2, l = 0 or 1
n = 3, l = 0, 1, or 2
Value of l
0
1
2
3
Type of orbital
s
p
d
f
l=1
l=0
l=2
p orbital (principal)
s orbital (Sharp)
d orbital (diffuse)
l=3
f orbital (fundamental)
Magnetic Quantum Number, ml
• Describes the three-dimensional orientation of the orbital.
• Values are integers ranging from -l to l:
−l ≤ ml ≤ l.
• Therefore, on any given energy level, there can be up to 1s
orbital, 3p orbitals, 5d orbitals, 7f orbitals, etc.
if l = 1 (p orbital), ml = -1, 0, or 1
if l = 2 (d orbital), ml = -2, -1, 0, 1, or 2
orientation of the orbital in space
Magnetic Quantum Number, ml
• Orbitals with the same value of n form a shell.
• Different orbital types within a shell are subshells.
Quantum Numbers and Atomic Orbitals
An atomic orbital is specified by three quantum numbers.
n the principal quantum number - a positive integer
l the angular momentum quantum number - an integer from 0
to n-1
ml the magnetic moment quantum number - an integer from -l
to +l
Table 7.2 The Hierarchy of Quantum Numbers for Atomic Orbitals
Name, Symbol
(Property)
Allowed Values
Quantum Numbers
Principal, n
(size, energy)
Positive integer
(1, 2, 3, ...)
1
Angular
momentum, l
(shape)
0 to n-1
0
0
0
0
Magnetic, ml
(orientation)
-l,…,0,…,+l
2
3
1
0
1
2
0
-1 0 +1
-1 0 +1
-2
-1
0
+1 +2
Sample Problem
Determining Quantum Numbers for an Energy Level
What values of the angular momentum (l) and magnetic (ml)
PROBLEM: quantum numbers are allowed for a principal quantum number
(n) of 3? How many orbitals are allowed for n = 3?
PLAN: Follow the rules for allowable quantum numbers found in the text.
l values can be integers from 0 to n-1; ml can be integers from -l through 0 to + l.
SOLUTION:
For n = 3, l = 0, 1, 2
For l = 0 ml = 0
For l = 1 ml = -1, 0, or +1
For l = 2 ml = -2, -1, 0, +1, or +2
There are 9 ml values and therefore 9 orbitals with n = 3.0
Determining Sublevel Names and Orbital Quantum Numbers
Sample Problem
PROBLEM: Give the name, magnetic quantum numbers, and number of orbitals
for each sublevel with the following quantum numbers:
(a) n = 3, l = 2
(b) n = 2, l = 0
(c) n = 5, l = 1 (d) n = 4, l = 3
PLAN: Combine the n value and l designation to name the sublevel.
Knowing l, we can find ml and the number of orbitals.
SOLUTION:
n
l
sublevel name possible ml values # of orbitals
(a)
3
2
3d
-2, -1, 0, 1, 2
5
(b)
2
0
2s
0
1
(c)
5
1
5p
-1, 0, 1
3
(d)
4
3
4f
-3, -2, -1, 0, 1, 2, 3
7