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Lecture 1
Discretization of energies
(c) So Hirata, Department of Chemistry, University of Illinois at Urbana-Champaign. This material has
been developed and made available online by work supported jointly by University of Illinois, the
National Science Foundation under Grant CHE-1118616 (CAREER), and the Camille & Henry Dreyfus
Foundation, Inc. through the Camille Dreyfus Teacher-Scholar program. Any opinions, findings, and
conclusions or recommendations expressed in this material are those of the author(s) and do not
necessarily reflect the views of the sponsoring agencies.
Discretization of Energies

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Energies are discrete (quantized) and not
continuous.
This quantization principle cannot be derived
– rather it should be accepted as physical
reality.
We will survey historical developments in
physics that led to the important discovery
that energies are quantized. The details of
each of the experiments or theories are not
so important – the conclusion is important.
Quantization of Energy


Classical mechanics: Any value of energy is
allowed. Energy is continuous.
Quantum mechanics: Not all values of
energy are allowed. Energy is discrete
(quantized).
Black-body radiation



A heated metal emits
light.
As the temperature
becomes higher, the
color of the emitted light
shifts from red to white
to blue.
How can physics
explain this effect?
Light

Wavelength (λ) and frequency (ν) of light are
inversely proportional: c = νλ (c is the speed
of light).
Black-body radiation


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What is “temperature”? – translation, rotation,
vibrations, etc. of atoms in molecules and
solids.
Light of frequency v can also be viewed as an
oscillator having temperature.
Equipartition principle: Heat flows from
high to low temperature area; each oscillator
has the same thermal energy kT at
equilibrium.
Black-body radiation

Experimentally, increasing
the temperature increases
the intensity and decreases
the maximum of
wavelengths of light.
Black-body radiation

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
Classical explanation –
Rayleigh-Jeans law – has the
limitation.
energy distribution
~ kT / λ4 ~ kTv2
Ultraviolet
Rayleigh-Jeans ~ kTv2
catastrophe.
c = vλ
Experimental
short wavelength
large frequency
v
Black-body radiation


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Planck could explain the
experimental energy
density by postulating that
the energy of each
electromagnetic oscillator
is limited to discrete
values (quantized).
E = nhν (n = 0,1,2,…).
h is Planck’s constant.
0
Density of lights with frequency v
kT
Black-body radiation
ν
∞
Each
electromagnetic
oscillator of a
frequency v is
given an equal
share of energy
kT
v2
v
hν hν hν hν hν hν hν hν hν hν
0
Density of lights with frequency v
kT
Black-body radiation
hν
hν
hν
ν
hν
hν
v2
v2 × hv/(ehv/kT−1)
hv/(ehv/kT−1) Bose-Einstein statistics
v
∞
Electromagnetic
oscillators with
smaller
frequencies are
unaffected,
leading to
Rayleigh-Jeans
results
Continuous vs. quantized
Each oscillator has a price tag of hv and
kT may not be enough to buy one hv if v is high
kT
kT
kT
kT
Higher frequencies
Planck’s constant h

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E = nhν (n = 0,1,2,…)
h = 6.63 x 10–34 J s. (Joule is the units of
energy and is equal to Nm or Newton x
meter). The frequency has the units s–1.
Because h is small, classical limit works well
in so many cases.
In the limit h → 0, E becomes continuous and
any arbitrary value of E is allowed. This is the
classical limit.
Heat capacities

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
Heat capacity is the amount of energy one
needs to heat up a substance by 1 K. The
greater the heat capacity, the more thermally
responsive the substance is.
“Heat” is a macroscopic concept of a flow of
energies of a collection of oscillators.
Dulong-Petit’s law of heat capacity: molar
heat capacity of monatomic solids is the
same.
Heat capacities

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There are NA (Avogadro’s) number of atoms
in a mole of a monatomic solid. Each can act
as a three-way oscillator (that oscillates in x,
y, and z directions independently).
According to the equipartition principle,
each of the three degrees of freedom of an
oscillator is entitled to kT energy.
E = 3NAkT → C = dE/dT = 3NAk = 3R
R is the gas constant.
Heat capacities
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Dulong-Petit’s law
holds for large T.
For small T, it does
not.
The deviation at
low T has been
explained by
Einstein
Heat capacities

For smaller T, the thermal energy kT
ceases to be able to afford the smallest
allowed quantum hν (ν is the frequency
of oscillation).
kT
hv
kT
hv
hv
Small T
kT
hv
hv
hv
hv
hv
hv
hv
hv
…
hv
Large T
Heat capacities
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Einstein assumed that
there was only one
available frequency of
oscillation v.
When Debye used a
more realistic distribution
of frequencies, he
obtained a better
agreement between
theory and experiment.
Atomic & molecular spectra


Colors of matter originate
from the light emitted or
absorbed by constituent
atoms and molecules.
The frequencies of light
emitted or absorbed are
discrete.
Atomic & molecular spectra


This has been explained
by atoms and molecules
having discrete energies
(E1, E2, …).
When light is emitted or
absorbed, an atom or
molecule shifts in energy,
so hv = En – Em.
Summary



Energies of stable atoms, molecules,
electromagnetic radiation, and vibrations of
atoms, etc. are discrete (quantized). They
are not continuous.
Macroscopically observed phenomena, such
as red color of hot metals, heat capacity of
solids at a low temperature, and colors of
matter are all due to quantum effects.
Quantized nature of energy cannot be
derived. We should simply accept this.