Light: oscillating electric and magnetic fields electromagnetic (EM) radiation - travelling wave
Characterize a wave by its wavelength, l, or frequency, n
c=ln
Black-body radiation
Planck: vibrating atoms in a heated object give rise to the
emitted EM radiation; these vibrations are quantized
Energy of vibration = h n
Photoelectric effect
Einstein: light consists of massless particles called photons
Ephoton = h n = h c / l
E of a mole of photons = No h n
Calculate the energy of each photon of blue light of
frequency 6.40 x 1014 Hz.
What is the energy of a mole of photons of this frequency?
E per photon = h n = (6.626 x 10-34 J s) (6.40 x 1014 s-1) =
4.20 x 10-19 J
E for a mole of photons = 4.20 x 10-19 J x 6.023 x 1023/mole
= 2.53 x 105 J/mole or 253 kJ/mol
Atomic line spectra
(nm)
For the H atom
n = ( 1 2 - 1 2 ) 3.29 x 1015 s-1
n1
n2
n1 = 3, 4, …
n2 = n1 + 1, n1 + 2, ...
The Bohr Model of the Atom: explains spectra of oneelectron atoms such as H, He+, Li2+. Also accounts for the
stability of the atom.
Classical physics prediction
Bohr theory
1) Quantization of angular momentum: For electrons, only
those orbits, or energy state that have certain values of
angular momentum are allowed
Allowed orbits: angular momentum = n h / 2 p (n = 1, 2, 3, ..)
2) As long as an electron stays in an allowed orbit it does not
absorb or emit energy
3) Emission or absorption occurs only during transitions
between allowed orbits. The emission and absorption are
observed in spectra.
From the Bohr model
nh
angular momentum = me vr 
2p
Radius of allowed orbitals
n2
rn 
ao
Z
Z: positive charge on the nucleus
ao : Bohr radius
= 5.29177 x 10-11 m
ao 
Total energy of electron in a stable orbital
En  
Z2
2
n 8p
h2
2
me a2o
En  B
1
n2
h2
4p 2me e2
Electrons can move from one allowed orbit to another,
changing n is the energy absorbed or released equals the
energy difference between allowed orbits of two different n
values
c
1
1
E  h  h  B[ 2  2 ]
l
n2 n1
E  h  Z (
2
1
n22

1
n12
)Ry
An electron in an atom can exist only in a series of discrete
levels
When an electron makes a transition, its energy changes
from one of these levels to another.
The difference in energy E = Eupper - Elower is carried away as
a photon
The frequency of the photon emitted, n = E/h
Energy from light of frequency n can also be absorbed by
the H atoms if
h n = E = Eupper - Elower
In this case, an electron in a lower energy level is excited to
an upper energy level.
If h n  Eupper - Elower the electron cannot undergo a
transition.
The frequency of an individual spectral line is related to the
energy difference between the two levels:
h n = E = Eupper - Elower
animation
The observation of discrete
spectral lines suggest that an
electron in an atom can have only
certain energies/
Electron transitions between
energy levels result in emission
or absorbption of photons in
accord with the Bohr frequency
condition.
Lyman series: n1 = 1
Balmer series: n1 = 2
Paschen series: n1 = 3
Brackett series: n1 = 4
Pfund series: n1 = 5
Problems with Bohr’s theory
Could not be used to determine energies of atoms with more
than one electron.
Unable to explain fine structure observed in H atom spectra
Cannot be used to understand bonding in molecules, nor can
it be used to calculate energies of even the simplest
molecules.
Bohr’s model based on classical mechanics, used a
quantization restriction on a classical model.
Wave-Particle Duality
Light is a traveling wave.
Wave properties demonstrated through interference and
diffraction
Interference of light
Diffraction of light
Einstein’s experiments are explained by the “particle”
nature of light - photons
Light behaves as both a wave and a particle
Wave-Particle Duality of Matter
Louis de Broglie: suggested that all particles should have
wave like properties.
Wavelength associated with a particle of mass m and speed v
h
l=mv
h
l= p
where p = m v, the linear momentum of the particle
For a golf ball of mass 1.62 ounces (0.0459 kg), propelled at
an average speed of 150. Mi/hr (67.1 m/s), the deBroglie
wavelength is 2.15 x 10-34 m; too small to be measured
Wavelength of an electron (me = 9.11 x 10-31 kg) moving at a
speed of 3.00 x 107 m/s is 2.42 x 10-11 m; corresponds to the
X-ray region of the electromagnetic spectrum
Electron diffraction pattern (Davisson, Germer, Thompson,
1927)
The Uncertainty Principle
In classical mechanics a particle has a definite path, or
trajectory, on which location and linear momentum are
specified at each instant.
However, if a particle behaves as a wave, its precise location
cannot be specified.
Heisenberg’s uncertainty principle: impossible to fix both the
position of an electron in an atom and its energy with any
degree of certainty if the electron is described by a wave
Heisenberg uncertainty principle: if the location of a particle
is known to within an uncertainty of x, then the linear
momentum parallel to the x axis can be known only to
within p, where
x p ≥ 1 h
2
h=
h
2p
Max Born: If the energy of an electron in an atom is known
with a small uncertainty, then there must be a large
uncertainty in its position. Can only assess the probability of
finding an electron with a given energy within a given region
of space
Wave Mechanics
E. Schrödinger: Electron in an atom could be described by
equations of wave motion
Standing waves: vibrations
set up by plucking a string
stretched taut between two
fixed pegs
Ends are fixed; l of allowed
oscillations satisfy
nl =L
2
n = 1, 2, 3,….
node
third harmonic
n2 = 2 n 1
n3 = 3n1
second harmonic
fundamental or first harmonic
Standing waves - example of quantization; only certain
discrete states allowed
Electrons are wave-like and exist in stable standing waves,
called stationary states, about the nucleus.
Wave mechanics depicts the electron as a “wave-packet”
Stationary states with the
circumference being divisible into
an integral number of wavelengths
From wave mechanics can show
that allowed stationary states
satisfy the condition:
me v r = n h
2p
Wavefunctions and Energy Levels
Since particles have wavelike properties cannot expect them
to behave like point-like objects moving along precise
trajectories.
Erwin Schrödinger: Replace the precise trajectory of particles
by a wavefunction (y), a mathematical function that varies
with position
Max Born: physical interpretation of wavefunctions.
Probability of finding a particle in a region is proportional
to y2.
y2 is the probability density. To calculate the probability that
a particle is in a small region in space multiply y2 by the
volume of the region.
Probability = y2 (x,y,z) dx dy dz
Schrödinger Equation
The Schrödinger equation describes the motion of a particle
of mass m moving in a region where the potential energy is
described by V(x).
2 y
d
-h
2 m dx2 + V(x) y = E y
(1-dimension)
Only certain wave functions are allowed for the electron in
an atom
The solutions to the equation defines the wavefunctions
and energies of the allowed states
An outcome of Schrödinger’s equation is that the particle can
only possess certain values of energy, i.e. energy of a
particle is quantized.
For example, one of the simplest example is that of a particle
confined between two rigid walls a distance of L apart “particle in a box”
Only certain wavelengths can exist in the box, just as a
stretched guitar string can support only certain
wavelengths.
yn (x) = (2 / L)1/2 sin (n p x / L)
n = 1, 2, …..
n is called the quantum number
Energy of the particle is
quantized, restricted to discrete
values, called energy level.
En = n2 h2 / (8 m L2)
Also, a particle in the container
cannot have zero-energy - zeropoint energy
At the lowest level, n = 1, E1 > 0
The probability density for a particle at a location is
proportional to the square of the wavefunction at that point
To find the energy levels of an electron in the H atom, solve
the Schrödinger equation.
In the H atom the potential that the electron feels is the
electrostatic interaction between it and the positive
nucleus
V(r ) = - e2 / (4 p eo r)
r: distance between the electron and the nucleus.
Solution for allowed energy levels is:
En = - h R
n2
n = 1, 2,...
R = (me e4) / (8 h3 eo2) = 3.29 x 1015 Hz
Showed that quantum mechanics did indeed explain behavior
of the electron
Note that the energy of an electron bound in a H atom is
always lower than that of a free electron (as indicated by
the negative sign)
n: principle quantum number.
Labels the energy levels
When n = 1 => ground state of
the H atom. Electron in its
lowest energy
n > 1 : excited states; energy
increases as n increases
E = 0 when n = ∞ , electron has
left the atom - ionization