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.

 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.

 Classical mechanics: Any value of energy is allowed. Energy is continuous.
 Quantum mechanics: Not all values of energy are allowed. Energy is discrete
(quantized).

 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?

 Wavelength ( λ ) and frequency ( ν ) of light are inversely proportional: c = νλ ( c is the speed of light).


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.

 Experimentally, increasing the temperature increases the intensity and decreases the maximum of wavelengths of light.

 Classical explanation –
Rayleigh-Jeans law – has the limitation.
 energy distribution
~ kT / λ 4 ~ kTv 2
 Ultraviolet catastrophe.
Rayleigh-Jeans ~ kTv 2 c = vλ
Experimental short wavelength large frequency v




Planck could explain the experimental energy density by postulating that the energy of each electromagnetic oscillator is limited to discrete values (quantized) .
E = nh ν ( n = 0,1,2,…).
h is Planck ’ s constant.

0 v 2
ν ∞
Each electromagnetic oscillator of a frequency v is given an equal share of energy kT v

h ν
0 h ν hν hν hν hν hν hν hν hν h ν h ν h ν h ν h ν
ν ∞ v 2 v 2 × hv /( e hv / kT −1) hv /( e hv / kT −1) Bose-Einstein statistics
Electromagnetic oscillators with smaller frequencies are unaffected, leading to
Rayleigh-Jeans results v

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



E = nh ν ( 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 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.

 There are N
A
(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 = 3 N
A kT → C = dE / dT = 3 N
A k = 3 R
R is the gas constant.

Heat capacities
 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 h ν ( ν is the frequency of oscillation).
kT hv hv kT hv kT hv hv hv hv hv hv hv hv hv
…
Small T Large T

Heat capacities
 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
( E
1
, E
2
, …) .
When light is emitted or absorbed, an atom or molecule shifts in energy, so hv = E n
– E m
.

 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.