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PHYS 172: Modern Mechanics
Lecture 14 – Energy Quantization
Summer 2012
Read 8.1-8.8
Quantization
•Classical Physics:
quantities are continuous.
• Quantum Physics:
Some quantities are limited
to a discrete set of values.
Example: charge, Q = N.e
Quantum means quantized
Answers come in whole numbers
Example: The number of unopened Coke cans
in your refrigerator is quantized.
Quantum Waves are Quantized
There are discrete vibrational modes (normal modes)
1D: One Dimension
Violin string, jumprope
2D: Two Dimensions
Modes of a drumhead, coffee sloshing in your mug
http://demonstrations.wolfram.com/NormalModesOfACircularDrumHead/
3D: Three Dimensions
Electron Waves around Atomic Nuclei!
http://www.daugerresearch.com/orbitals/index.shtml
Higher Frequency = Higher Energy
Photons
Photons come in discrete particles, or packets of energy.
One PHOTON = One packet of light
And yet it's still a wave:
= Wavelength (crest to crest)
wavelength [m]
frequency [1/s]
speed of light [m/s]
Number of wavelengths
which go by per second
Photons
Photons come in discrete particles, or packets of energy.
One PHOTON = One packet of light
Photon energy and wavelength: E photon = hn light
hc
=
llight
Visible light Electromagnetic spectrum
E = 3.1 eV
ν = 7.5x1014 s-1
Wavelength 400
450
E = 1.8 eV
ν = 4.2x1014 s-1
500
550
600
650
700
750
nm
Atoms and Light
Absorb a Photon
3S
Absorb a Photon
2S
1S
Adding a photon
increases the energy
of the atom
Atoms and Light
Release a Photon
3S
2S
Releasing a photon
decreases the energy
of the atom
Release a Photon
1S
Atoms and Light
QUANTUM MECHANICS
says each ELEMENT (type of atom)
can only have specific, QUANTIZED energies.
Each atomic transition has a
CHARACTERISTIC COLOR
Photon Energy = Frequency = Color
The Sun
Dark lines correspond to specific atomic transitions,
such as “1s to 2s in Hydrogen”,
or “1s to 2p in Helium”.
Hydrogen atom: electron energy
3S
2S
1S
Emission spectra
Hydrogen atom:
Energy of emitted photon:
3S
EN = K + U e = -
hn = E 4 - E1
13.6 eV
N2
2S
Line spectrum – light is emitted at fixed frequencies
1S
Absorption spectra
Hydrogen atom:
Energy of absorbed photon:
EN = K + U e = 3S
13.6 eV
N2
hn = E 2 - E1
Line spectrum – absorption at fixed frequencies
2S
Different atoms – different energies
Atomic spectra – signature of element
Example: He was discovered on Sun first
1S
Clicker Question
Suppose that these are the
quantized energy levels (K+U) for
an atom. Initially the atom is in its
ground state (symbolized by a dot).
An electron with kinetic energy
6 eV collides with the atom and
excites it. What is the remaining
kinetic energy of the electron?
A) 9 eV
B) 6 eV
C) 5 eV
D) 3 eV
E) 2 eV
Effect of temperature
Boltzmann constant: k=1.4×10-23J/K
Population of level:
~ exp (- E / kT )
Temperature, K
Energy of the level above
the “ground state”, EN – E1
~ exp (- E / kT ) = 10
-33
Population of levels for visible light transition (E = 2eV):
At Room Temperature, 300K:
On the Sun, 6000K:
Energy conversion: light and matter
Absorption:
• photon is absorbed
• electron jumps to higher
level
Spontaneous emission:
• photon is emitted
• electron jumps to lower level
Stimulated emission:
• external photon causes
electron jump to lower level
• a photon is emitted
• the original photon is not
absorbed!
Makes laser work!
Laser
L ight
A mplification by
S timulated
E mission of
R adiation
Laser media
Requirement:
inverted population, more atoms must
be in excited state E’ than in state E.
Quantizing two interacting atoms
E
Spring (harmonic oscillator)
r
U for two atoms
If atoms don’t move too far from equilibrium, U looks like Uspring.
Thus, energy levels should correspond to a quantized spring . . .
Quantizing two interacting atoms
Classical harmonic oscillator:
Quantum harmonic oscillator:
U = (1/2)kss2
U = (1/2)kss2
w0 =
E2 = 2 w0 + E0
E1 = w0 + E0
E0 = 12 w 0
ks
m
ground state
equidistant spacing
2
E = 12 mv 2 + 12 kx 2 = 12 kAmax
Any value of A is allowed
and any E is possible.
w0 = k s / m
h
=
h
= 1.05 X10 34 J s
2
Energy levels:
EN = N w 0 + 12 w 0
Yes, Tiny Harmonic Oscillators are Quantized
Quantum harmonic oscillator:
U = (1/2)kss2
w0 =
E2 = 2 w0 + E0
E1 = w0 + E0
E0 = 12 w 0
ks
m
ground state
equidistant spacing
w0 = k s / m
=
h
= 1.05 ｴ10 34 J s
2p
Energy levels:
EN = N w 0 + 12 w 0
WEB DEMO:
http://web.ift.uib.no/AMOS/MOV/HO/
Quantized vibrational energy levels
U = (1/2)ksx2
U = (1/2)ksx2
EN = N w 0 + E0
w0 = k s / m
Larger resonance frequency –
larger level separation
Anharmonic oscillator:
Not an equidistant spacing of levels
Home study:
Rotational energy levels (8.5,page 338)
Nuclear & Hadronic energy levels (8.6)
Comparison of energy level spacing (8.7)
Laser
Ruby: aluminum oxide crystal (sapphire) where some Al were
replaced by Cr