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The Quantum (Wave) Mechanics Model
• In 1924, a French physicist named Louis de Broglie suggested that, like
light, electrons could act as both particles and waves.
• De Broglie's hypothesis was soon confirmed in experiments that showed
electron beams could be diffracted or bent as they passed through a slit much
like light could.
• The waves produced by an electron confined in its orbit about the nucleus
sets up a standing wave of specific wavelength, energy and frequency (i.e.,
Bohr's energy levels) much like a guitar string sets up a standing wave when
plucked.
• De Broglie's vision of Bohr's atom
Quantum (Wave) Mechanics
Quantum mechanics, or wave mechanics, is the
treatment of atomic structure through the wavelike
properties of the electron
Erwin Schrödinger developed an
equation to describe the hydrogen
atom
A wave function is a solution to the
Schrödinger equation and represents
an energy state of the atom
Wave Mechanics = Probability
Wave mechanics provides a probability of where an
electron will be in certain regions of an atom
This region of space where there’s a high probability
of finding an electron is called an orbital
Wave mechanics led to the idea of a “cloud of
electron density” rather than a discrete location
Quantum Numbers and
Atomic Orbitals
A wave function with a given set of these three
quantum numbers is called an atomic orbital
In quantum mechanics the atomic orbitals require
three integer quantum numbers to completely
describe the energy and the shape of the 3-D space
occupied by the electron (n, l, and ml)
Principal Quantum Number
(n)
• Is independent of the other two quantum numbers
• Can only be a positive integer
• indicates the size of an orbital (distance from
the nucleus) and its electron energy
• n can be 1, 2, 3, 4, …
•These orbitals are the volume
around the atom that the
electrons occupy 90-95% of the time.
1
2
3
Orbital Angular Momentum Quantum
Number (l)
(aka Azimuthal quantum number)
• Determines the shape of the orbital: s, p, d, f , which corresponds to values l
values of: 0, 1, 2, 3
• Possible values of l: 0 to n
– 1; e.g. if n = 2, l can only be 0 or 1
• Each of these orbitals is in a different region of space and has a different shape
•All the ‘l’ quantum values represent different sublevels or subshells
•When n = 1, there is only one “l” value meaning there is only one sublevel in the
first energy level; when n= 2; there are two values for ‘l’ indicating two sublevels in
the second energy level
Summary
n
l
subshell name
1
0
s ( sharp)
2
1
p (principal)
3
2
d (diffuse)
4
3
f (fundamental)
Magnetic Quantum Number
(ml)
Determines the orientation in space of the orbital;
so named because in a magnetic field, these
different orientations have different energies
Possible values: –l to +l;
e.g., if l = 2,
ml can be –2, –1, 0, 1, 2
The magnetic quantum number, ml, defines the
number of orbitals in a sublevel. E.g. in the l = 0
sublevel, there is only one ml value, therefore there
is only orbital in this sublevel; when l=1; there are 3
possible ml values (-1, 0, +1) 3 orbitals in this
sublevel
Quantum Numbers Summary
Taken together the three quantum numbers specific
the orbital the electron occupies. Namely:
the energy of the orbital, the shape of the orbital, and
the orientation of the orbital
.
• writing 3 quantum numbers to indicate
every possible orbital an electron can
occupy is cumbersome; instead do we do
the following:
– retain the numeric value of the principal
quantum number and use a letter to indicate
the azimuthal quantum number:
l = 0  s; l = 1 p; l = 2  d; l = 3  d
- When combined, they indicate an a
specific orbital e.g. 1s orbital; 2s orbital; 2p
orbital
Radial Distributions
Electrons are most likely to reside nearest the
nucleus because of electrostatic attraction
Probability of finding an electron
decreases as distance (radius) from the
nucleus increases
Electron Probabilities
and the 1s Orbital
The 1s orbital looks very much like a fuzzy ball,
that is, the orbital has spherical symmetry (the
probability of finding an electron is the same in
direction)
The electrons are more concentrated near the center
Electron Probabilities
and the 2s Orbital
The 2s orbital has two regions of high electron
probability, both being spherical
The region near the nucleus is separated from the
outer region by a spherical node - a spherical shell
in which the electron probability is zero
EOS
The Three p Orbitals
-There are three p orbital; each orbital is cylindrically
symmetrical with respect to rotation around one of the
3 axes, x, y, or z
Each ‘p’ orbital has two lobes of high probability
density separated by a node (region of zero
probability)
The Five d Orbitals
• The first d orbitals
appear
in the n=3 shell.
• The five d orbitals
have two different
shapes
( 4 are clover and 1 is …
complex)
• There are 5 d orbitals
per n level
They have an l=2
value, therefore
ml= -2, -1, 0, +1 , +2
The f orbitals
• The first f orbital
appears in the
the n=4 shell.
• There are seven
f orbitals per n
levels
• They have an
l=3 value
ml= -3,-2,-1,0,
+1, +2, +3
Electron Spin (ms)
The electron spin quantum number explains some
of the finer features of atomic emission spectra
The spin refers to a magnetic field induced by the
moving electric charge of the electron as it spins
Only possible values
= –1/2 to +1/2
EOS
Pauli’s Exclusion Principle
“ No two electrons in an
atom can have the same
set of 4 quantum numbers”