Discussion: Energy quantization
Normally we perceive light as continuous flow, just as normally we perceive the table
as continuous matter. But if we look closely at the table with an appropriate kind of
microscope, we see that it is made of atoms. Similarly, if we look at very lowintensity light with a sufficiently sensitive detector, we find that the light arrives in
packets, which we call “photons” and which are particle-like. Moreover, red light
consists of photons each of which has energy of about 1.8 eV (1 eV = 1.610-19 J).
Blue light consists of photons each of which has energy of about 3.1 eV.
Ponderable: Solar collector engineering
WID 1132818 437y
solar_radiation_map.pdf
Solar insolation at Earth’s orbital radius is 1400 W/m2.
Area of that sphere is 4pi(1.5e11)2 = 2.83 e23 m2.
So the Sun puts out 1400 W/m2 * 2.83 e23 m2 = 4e26 W= 4e26 J/s.
Divide by c2 to get 4.4e9 kg converted to energy every second.
Average value much less at Earth’s surface. Depends on climate and latitude. Find
what it is (probably at NREL, use annual average flat plate collector aimed south,
tilted at latitude.) Get about 5.25 kWh/m2/day. Fiddle with units to find J/s per square
meter = 5250 Wh/m2/24 h = 220 W/m2.
See map (solar_radiation_map.pdf).
Then find number of photons per second per square meter.
Average visible photon is about 2.4 eV (1.6e-19 J/eV) = 3.8 e-19 J per photon. So,
220 J/s per m2 divided by 3.8 e-19 J per photon gives 5.8 e 20 photons per second per
square meter.
If photovoltaic cell is between 11% and 12% efficient, 220 W/m2 turns into 25 W/m2
of useable output.
Average daily usage (http://www.washingtonco-op.com/pages/understand.htm) is
about 18 kWh per day = 18000 Wh/24 h = 750 W power demand on average. This
would require 750/25 = 30 m2, or collector 5.5 m = 18 feet on edge (basically the
roof). And what about at night?
Chapter 8
1
Tangible: Sorry, I have a code…
real_prize_inside.pdf (TPT article describing the activity)
UPC_Data.jpg (or UPC_Handout.pdf)
UPC_Tangible.doc
Give them UPC_Data.jpg (or UPC_Handout.psd) and have them
figure out as much as they can about UPC codes from it:
 All the barcodes are black and white.
 Every barcode begins and ends with two longer bars.
 Two sets of five digits separated by another set of longer bars.
 Different codes for identical numbers to the left and right of the longer middle
bars.
 Each code starts with zero (see article for discussion of this hypothesis).
 The first five digits are the same for the same manufacturer.
 There was a separate digit at the end (most notice the digit, but no one discerns the
underlying checksum algorithm).
 Codes for manufacturer seem duplicated
Give them UPC_Tangible.doc on paper and have abc groups decode their own UPC,
identifying numbers and manufacturer if possible.
Chapter 8
2
Compare engineering (solar panel problem) to science—finding the underlying
patterns and theorizing.
Relate to identifying elements by particular color patterns. Recall helium discovered
on Sun before found on Earth. But why do hot elements give off particular color
patterns.
VPython Demo: Bohr model and energy levels
Bohr_levels.py
The energy (K+U) of an atom is quantized. As an overview, show the program
07_Bohr_levels.py, illustrating the simplified Bohr model of the hydrogen atom,
proposed by the Danish physicist Niels Bohr in 1913. On the left is shown an electron
in a circular orbit, on the right the energy-level diagram for the hydrogen atom, and
below is a graph of kinetic energy, potential energy, and K+U for the circular orbit.
Click in the orbit window to cause a transition to a larger-radius orbit, with
accompanying jump in the energy-level diagram, and a rise in K+U in the graph.
Continue clicking to go to higher and higher energies, corresponding to absorbing
energy from (for example) a bombarding electron. Then click to observe transitions to
lower-energy states, which would be accompanied by the emission of a photon whose
energy is the difference in the energies of the two levels.
Chapter 8
3
-0.85 eV
-1.5 eV
-0.28 eV
-0.54 eV-0.38 eV
-3.4 eV
Horizontal is meaningless
-13.6 eV
From solving Schrodinger Equation (one of two simple systems):
Chapter 8
4
EN = K + U e =
-13.6 eV
, N = 1, 2,...
N2
After the full development of quantum mechanics, it became clear that the Bohr model
for the hydrogen atom is too simple to capture many important features of hydrogen.
For example, instead of Bohr's circular orbits, quantum mechanics predicts that the
electron exists as a probability cloud around the proton. Yet quantum mechanics
predicts that in higher-energy states the electron is on average farther from the proton,
just as predicted by the Bohr model. And Bohr's proposed “jumps” between energy
levels, with photon emission, does capture the main points about energy quantization.
An important drawing convention: Often we only care about the discrete energy levels
and just draw horizontal lines without drawing the potential energy curve U. In
this case the position along the x-axis is essentially meaningless. For example, we can
draw the energy-level scheme for atomic hydrogen as a set of horizontal lines of equal
length, disregarding the additional information about maximum separation shown
when we display the U curve.
Discussion: Excitation and emission
Describe the process of exciting an atom to a higher (bound) state by hitting it with
an energetic electron.
Initial state
Final
state
e–
Electron with
Ki
Ki
Ki
Ki
e–
Ground state
atom
Electron with
Kf
N=4
N=3
N=4
N=3
N=2
N=2
N=1
N=1
Energy of this electron is
not absorbed
Chapter 8
Excited
atom
“excited” state
“ground” state
5
Kf electron
Ki
emitted photon
If really high incoming energy, you ionize the atom.
DEsystem = E f , system - Ei, system = Wexternal + Q = 0
(m
electron
) (
System is the whole universe
)
c 2 + K f , electron + E f , atom - melectron c 2 + K i, electron + Ei, atom = 0
E f , atom - Ei, atom = K i, electron - K f , electron ³ 0
If Ki, electron < Egap, Ki, electron = Kf, electron
DEatom = -DK electron
VPython Demo: Absorption and emission
absorb_emit.py
Show and discuss the program 07_absorb_emit.py, which shows an atom’s electron
levels as it absorbs energy and emits light.
Explain that we're considering a collection of many atoms in a gas, each of which has
only 5 bound states, indicated by horizontal bars. We start with all of the atoms in the
lowest "ground" state. (We ignore the overall motion and kinetic energy of the atoms
in the gas.) Introduce convention of a dot on a level meaning that is the state of an
Chapter 8
6
atom, or several dots on a level means that many atoms in a collection are in that state.
Click to shoot an electron into the gas and you'll see an atom absorb some of the
electron's kinetic energy and be raised to an excited state. Each time you click you
may see another such absorption event, or instead you may see one of the excited
atoms drop to a lower state with the emission of a photon whose color is determined
by its energy, red for low energy, blue or violet for high energy.
The photon energy is the difference in the upper and lower atom energies. At the
bottom of the window you see a "spectrum" as would be seen with a prism or
"diffraction grating" that spreads out the colors of the rainbow. Continue clicking and
observing all the kinds of absorption and emission events that are possible. When all
possible photon emissions have been seen, and the spectrum at the bottom of the
window is complete with 10 different "spectral lines", the electron beam is
automatically turned off in the program, and further clicks lead solely to photon
emissions, until finally all the atoms are in their ground states.
The program starts with Eexcite = 4, which means that the electron beam has enough
energy to reach the N = 4 level, where N=0 is the ground state. It is useful to run the
program again with, say, Eexcite = 2, and see that some high-energy photon emissions
no longer occur, because the electron beam doesn't have enough energy to excite the
higher energy states of the atoms.
Ponderable: Activity - CSI identification of elements
WID 1132805 rtu47
Have them predict energies and hence colors from atomic hydrogen.
N Energy
1 -13.60
2
-3.40
3
-1.51
4
-0.85
5
-0.54
6
-0.38
7
-0.28
8
-0.21
9
-0.17
10
-0.14
11
-0.11
to 1
10.20
12.09
12.75
13.06
13.22
13.32
13.39
13.43
13.46
13.49
to 2
1.89
2.55
2.86
3.02
3.12
3.19
3.23
3.26
3.29
to 3
0.66
0.97
1.13
1.23
1.30
1.34
1.38
1.40
to 4
0.31
0.47
0.57
0.64
0.68
0.71
0.74
Talk about Balmer series, etc.
Show animation at http://www.bigs.de/en/shop/anim/termsch01.swf
Chapter 8
7
Chapter 8
8
Demo: Spectra for Hydrogen, Helium, Neon, fluorescent lights
Give them diffraction gratings and set up spectral tubes to see which is hydrogen. Can
also show them nitrogen, helium, neon. Practical hint: students should hold the grating
close up against the eye, then look to the side for the colored vertical lines.
Relate spectra to the UPC barcodes from “Tangible: Sorry, I have a code…”
Also show a glowing filament as an example of an apparently continuous spectrum,
shifting toward the red as you lower the temperature, toward the blue (white) as you
raise the temperature. Then show gases (nice to show molecular nitrogen if available,
since its band spectrum is quite different from the simple line spectrum of the atomic
gases.) Talk about stars like red Betelgeuse and blue Rigel in Orion. Can see photo in
Wikipedia:
Review hydrogen spectra using the flash animation found at:
http://www.bigs.de/en/shop/anim/termsch01.swf
Chapter 8
9
Comment on simulation error since Balmer series shows excitations from N = 2.
Ponderable: Activity - Let’s be Franck (Hertz)
WID 1132810 (15 minutes)
Franck_Hertz.py
How do atoms get excited in the first place? We can try shooting electrons at the
atoms. Show them the program 07_franckhertz.py, which is a simplified form of the
Franck-Hertz experiment of 1914, a crucial proof that energy is quantized in atomic
systems.
The positively and negatively charged plates apply to an electron a nearly constant
electric force (if the plates are large compared to distance between them; the display is
not to scale). We see the work done going into raising the kinetic energy. The kinetic
energy is plotted as a function of distance. The slider on the right adjusts the amount
of electric force applied, and thus how fast electron is moving when it hits a mercury
atom.
Your job: “run” the experiment for various values of the electric force. What do you
see?
Pay attention to observations vs. hypotheses (inferences you can make about the
system based on your observations). Discuss within group, then between groups.
Finally, pick one recorder from table to write the observations and hypotheses on
wallboard.
We observe: At lower KE, nothing happens, electron goes straight through. At a
critical KE (about 5 eV), electron drops to zero KE when it collides. At higher KE, the
difference before and after collision is still 5 eV.
We conclude: The first excited energy level for Hg must be 5 eV above its ground
state.
NOTE: If below critical, not enough KE to cause it to jump to higher level. If above
critical, atom can absorb some KE from electron—enough to make “quantum jump”,
no more—and electron has some KE left over.
If the electron has less than 4.9 eV of kinetic energy at the time it hits the mercury
atom, nothing happens. But if the kinetic energy is greater than 4.9 eV when the
Chapter 8
10
electron hits the mercury atom, the electron can be observed to lose 4.9 eV. There are
only certain possible discrete bound-state energies possible for a mercury atom. The
first such state above the normal lowest-energy “ground state” evidently lies 4.9 eV
above the ground state energy. If the electron has less kinetic energy than 4.9 eV, it
cannot change the bound-state energy of the mercury atom. (The actual experiment
involved firing large numbers of electrons at large numbers of mercury atoms in a
low-density gas of mercury and observing the energy-loss effects shown in the
program.)
This is very different from what we observe with macroscopic systems such as
planetary orbits, where K+U for a bound state can have any value whatever – a
continuous range of possible bound-state energies. Discrete “energy levels” are found
in all kinds of atomic-sized systems. Point out that we have been talking about
“electronic” energy levels, but this isn’t the only kind of energy levels in atomic
systems, as we'll see later.
Discussion: Processes
Draw diagrams to represent the three key processes we will be considering:
1) Electron excitation (not absorption). As above, high-energy electron strikes an
atom and raises the atom to a higher energy state. The electron continues on, with
reduced kinetic energy. Show before and after, and energy level diagrams for the
atom. Emphasize the role of the energy principle: E f = Ei + Wext + Q , with Wext = 0 and
Q = 0 , because we’ll include all the objects inside the system.
Describe the process of exciting an atom to a higher (bound) state by hitting it with
an energetic electron.
Initial state
Final state
e–
Electron with
Ki
Ki
Ki
Ki
Chapter 8
e–
Ground state
atom
Excited
atom
Electron with
Kf
N=4
N=3
N=4
N=3
N=2
N=2
N=1
N=1
11
“excited” state
Energy of this electron is not
absorbed
“ground” state
Kf electron
Ki
emitted photon
If really high incoming energy, you ionize the atom.
DEsystem = E f , system - Ei, system = Wexternal + Q = 0
(m
electron
System is the whole universe
) (
)
c 2 + K f , electron + E f , atom - melectron c 2 + K i, electron + Ei, atom = 0
E f , atom - Ei, atom = K i, electron - K f , electron ³ 0
If Ki, electron < Egap, Ki, electron = Kf, electron
DEatom = -DK electron
2) Photon emission. Initially, atom in state above the ground state. Finally, atom in
lower state and there is a photon. Show before and after, and energy level diagrams
for the atom. Use the energy principle to find the energy of the photon. (The atom
recoils to conserve momentum, since the photon has momentum as well as energy; p
= E/c for a photon, but because of its large mass compared to the massless photon
almost all the energy change of the atom goes into the energy of the emitted photon.)
Initial state
Final state
Excited
atom
Lower state atom
(not necessarily
ground state)
Emitted
photon
DEsystem = E f , system - Ei, system = Wexternal + Q = 0
(E
f , photon
)
+ E f , atom - ( Ei, atom ) = 0
E f , atom - Ei, atom = -E f , photon
E f , atom = Ei, atom - E f , photon
3) Photon absorption. Initially, atom in ground state (this will be the case if the
collection of atoms is very cold). If photon energy is just right, it can be absorbed, and
the final state is the atom in an excited state, with no photon. Show before and after,
and energy level diagrams for the atom. Note the key difference from electron
excitation. Cannot destroy (or create) electrons, so there is an electron in the initial
Chapter 8
12
state AND in the final state. But when photons interact with atoms, the photon can be
destroyed (absorbed) or created (emitted). (We’re ignoring rare situations where a
high energy photon is destroyed AND a low energy photon is emitted; in these rare
cases the incoming photon energy doesn’t have to exactly match the difference in
energy levels). There has to be an exact “match” between the incoming photon and the
energy gap for the photon to be absorbed, unlike the electron excitation case.
Initial state
Incoming photon
Final state
Ground state atom
Excited atom
DEsystem = E f , system - Ei, system = Wexternal + Q = 0
(E
f , atom
) - (E
i, photon
)
+ Ei, atom = 0
E f , atom = Ei, atom + Ei, photon
VPython Demo: Line spectra
spectrum.py
Show the program 07_spectrum.py, which illustrates the experimental situation for
observing “spectra” and why we call them “line spectra” (line-shaped sources of light,
with different colors seen in different locations).
Emphasize the difference between energy levels and spectral lines.
Clickers
Discussion: Types of energy levels
quantum_oscillator.py (VPython Demo)
Chapter 8
13
We’ve been looking at “electronic” energy levels: the bound states of electrons
orbiting the nucleus of an atom. There are other types of microscopic bound systems:
Nuclear: Talk about briefly. Potential energy looks like a square well, energy levels
are roughly spaced by MeV, not eV as with electronic.
Diatomic molecule: in two ways:
 Rotation: the diatomic molecule only has specific values of rotational kinetic
energy. The rotational levels are on the order of 1e-5 or 1e-4 eV apart.
(Talk about microwave oven)
 Vibration: K_vibration + U_spring have discrete values
“Quantum harmonic oscillator” has a special properties: levels are evenly spaced
Difference between adjacent energy levels is: Spacing =
ks
m
= 1.05 ´ 10 -34 J × s
Show program quantum_oscillator.py, energy levels and visualization of quantum
harmonic oscillator.
Note the evenly spaced levels.
Spacing =
ks
m
= 1.05 ´ 10 -34 J × s
Advertise briefly as being in textbook:
Diatomic molecule (N2 show spectrum if available, energy bands).
Vibrational energy levels for molecules (typical energy spacing 10 -2 eV)
Rotational energy levels for molecules (typical energy spacing 10-4 eV)
Vibrational levels ~10-2 eV
NOT TO SCALE
Electronic levels ~1eV
Rotational levels ~10-4 eV
Chapter 8
14
Nuclear (typical energy spacing 106 eV)
Hadronic energy levels (typical energy spacing 108 eV), the ++
Advertise discussion of laser action at end of chapter.
Clicker B
Chapter 8
15