PHYS 172: Modern Mechanics
Spring 2011
EXAM 2 is next week
Time: 8:00-10:00 pm Wed March 9th
Place: Elliott Hall
Material: lectures 9-14 (may require previous material as well)
Problems: multiple choice, 14 questions (70 points)
write-up part, hand graded (30 points)
Equation sheet: provided
Practice exam: will be posted
Note: no lecture on March 10
Lecture 15 – Energy Quantization
Quantization
•Classical Physics:
quantities are continuous.
• Quantum Physics:
Some quantities are limited
to a discrete set of values.
Example: charge, Q = N.e
Read 8.1 – 8.8
Energy Levels are quantized for many
microscopic systems:
Energy of a photon
Photon energy and wavelength: E photon = hν light =
hc
λlight
Planck’s constant: h = 6.6 10-34 J.s
Planck: light is emitted by quanta (1900)
Einstein: light consists of quanta! (1905)
Visible light Electromagnetic spectrum
E = 3.1 eV
ν = 7.5×1014
Wavelength 400
E = 1.8 eV
ν = 4.2×1014
450
500
550
Higher energy
600
photons
650
Lower energy
700
750
nm
Experimental fact: Atoms have QUANTIZED energy levels.
Light from these lamps comes only in a few narrow colors, which means the
photons in those lines have a fixed, well-defined energy.
The photon energy is the difference between two quantized energy levels.
If atomic energy levels were not quantized, the light would be white (all energies).
Hydrogen atom: electron energy
Ue = −
1 e2
4πε 0 r
Energy levels of H atom:
EN = K + U e = −
13.6 eV
N2
Emission spectra
Hydrogen atom:
Energy of emitted photon:
EN = K + U e = −
hν = E4 − E1
13.6 eV
N2
Line spectrum – light is emitted at fixed frequencies
Absorption spectra
Hydrogen atom:
Energy of absorbed photon:
EN = K + U e = −
hν = E2 − E1
13.6 eV
N2
Line spectrum – absorption at fixed frequencies
Different atoms – different energies
Atomic spectra – signature of element
Example: He was discovered on Sun first
Clicker Question
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Effect of temperature
Hydrogen atom:
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
“ground state”
Population of levels for visible light transition
at room temperature: E ~ 2 eV, T=300K
~ exp ( − E / kT ) = 10−33
Sun, 6000K: ~0.02
Effect of temperature
Hydrogen atom:
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
“ground state”
Clicker question:
At very high temperature, the number of
particles in state E2 can be larger than
number of particles in the ground state E1
A) TRUE
B) FALSE
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.
First laser:
1960, Theodore Maiman (Hughes Laboratory, California)
Laser
Ruby: aluminum oxide crystal (sapphire) where some Al were
replaced by Cr
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
Quantum harmonic oscillator:
Classical harmonic oscillator:
U = kss2
U = kss2
ω0 =
E2 = 2 ω0 + E0
E1 = ω0 + E0
E0 = 12 ω0
ks
m
equidistant spacing
2
E = 12 mv 2 + 12 kx 2 = 12 kAmax
Any value of A is allowed
→ any E is possible.
ground state
ω0 = k s / m
≡
h
= 1.05 × 10−34 J ⋅ s
2π
Energy levels:
EN = N ω0 + 12 ω0
Quantized vibrational energy levels
U = ksx2
U = ksx2
EN = N ω0 + E0
ω0 = 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)