Physics 231
9
10
2013 1
Ch 8 Day 2
Fri., 11/1
8.4-.7 More Energy Quantization
RE 8.b
Mon. 11/4
Tues. 11/5
Wed., 11/6
Lab
Fri., 11/8
9.1-.2, (.8) Momentum and Energy in Multi-particle Systems
RE 9.a
HW8: Ch 8 Pr’s 21, 23, 27(a-c)
RE 9.b bring laptop, smartphone, pad,…
Practice Exam 2 (due beginning of class)
9.3 Rotational Energy Quiz 8
Review Exam 2 (Ch 5-8)
Exam 2 (Ch 5-8)
Equipment
o 23_antenta.py, oscillator.py, BallSpring.mov
Announcements
o Exam 2 in 2 weeks
Chapter 8 Energy Quantization
Last time
We made the observation that atomic spectra have very specific color bands in
them. Thinking about how light interacts with matter – that it delivers energy
when it shines on charged particles, and takes it way when particles shine
E atom
E light
we deduced that only having specific colors / photon energies in the spectrum
mean that the internal energies of atoms could change by only discrete steps.
Classically, that’s very surprising since we can imagine a contimuum of possible
K+U levels, yet it appears that only specific ones are actually ‘allowed.’
In the case of a Hydrogen atom, the allowed levels are
13.6eV
n = 1,2,3,…
K U H .n
n2
r
K+U4
K+U3
K+U2
Up-e
Elight=-(E1-E2)=10.2eV
K+U1
That explains why the only photons we see emitted have energies that fit
13.6eV 13.6eV
ni2
n 2f
It’s worth emphasizing here that this is a classically unexpected result – nothing about
classical physics would predict that there should be only specific allowed K+U values for
the bound p + e- system. We’ll touch on why this is later (and you’ll get deeper into it in
Phys 233). We say that the atomic energy is “quantized” – only specific values are
allowed.
Elight
Ef
Ei
Physics 231
Ch 8 Day 2
2013 2
Now, Hydrogen may have a simple equation that describes the allowed energy levels, and
therefore the spectrum of light emitted, but as you saw when you looked at spectra of
other elements yesterday, they too have only specific bands in their spectra so only
specific allowed K+U levels.
Quantization everywhere. It turns out that the energy of any bound system is quantized.
Atomic. If you’ve got electrons bound to a nucleus to form an atom – only specific K+U
values are allowed.
Nuclear. If you’ve got protons and neutrons bound together to form a nucleus – only
specific K+U values are allowed.
Molecular Vibrational. Heck, if you’ve got two atoms bound together to form a
molecule – only specific K+U values are allowed for their vibrations.
Molecular Rotational. For that matter, even a molecule’s rotational energy is quantized.
It’s not just energetic constraints that lead to quantization, it’s any constraints (in this
case, that after a rotation of 2 , the molecule’s back where it started.)
There really should be no limit to this principle. For example, a mass on a spring is a
‘bound’ system, so its vibrational energy should be quantized. In a little bit, we’ll see
why, even if that’s true it’s undetectable.
Before we dig deeper into this, we’re going to get a little more familiar with the energy
that light caries. Last time I bandied around the word “photon”, and this time I want to
get a little more exact about what that is.
Ee
Ee
Frank-Hertz
o We’ve been focusing on the light radiated by excited atoms. The energy
of the light equals the change in energy of the atom. But how did we get
the atoms into excited states to begin with? We slammed electrons into
them. By conservation of energy, what must happen to the incident
electron’s energy?
 It must decrease by the same amount that the atom’s internal
energy increases.
o So, another way to experimentally learn about the atomic structure is to
monitor the electron energies, or equivalently, the strength of the electron
current passing through the gas.
o Let’s think of the simple case of one H atom in its ground state and one
free electron.
o Low accelerating voltage. We apply a low voltage and the electron
slowly accelerates as it approaches the H atom, as long as its energy is less
Expected
than E1-2=10.2 eV, the most it can do is rebound off the H atom. Since
Energy
the H atom is much more massive, this is similar to a ball bouncing off a
Detected
wall – the kinetic energy doesn’t change appreciably.
Energy o High accelerating voltage. But if the accelerating voltage is high
enough, by the time the electron reaches the atom, it has more than
10.2eV, so it can cause the atomic electron to excite up, and in the process,
detector
it looses 10.2 eV of energy. If the electron’s kinetic energy can be
ascertained at the end of its flight, it will be 10.2eV less than expected.
Demo: Frank_hertz.py (this is for mercury which has different energy levels)
Physics 231
2013 3
Ch 8 Day 2
Here are the quantized energy levels (K+U) for an atomic or molecular object, and the
object is in the "ground state" (marked by a dot). An electron with kinetic energy 6 eV is
fired at the object and excites the object to the –5 eV energy state. What is the remaining
kinetic energy of this electron?
-1 eV
-2 eV
1) 9 eV
2) 6 eV
3) 4 eV
4) 3 eV
5) 2 eV
-5 eV
-9 eV
6.1.1 Absorption spectra
A slight variation on this theme is adsorption spectra. Consider a distant
interstellar gas cloud. Without being wired up to a high voltage source such as we
have here, its atoms aren’t excited and they aren’t radiating. But think of what
we’d see if a distant star back lit it. On its way from the star to us, the light passes
through the cloud. The cloud is transparent to most colors of the light, but those
that correspond to transition between allowed energy levels get adsorbed. Thus
the light that reaches us is deficient in those specific colors. We see an adsorption
spectrum of the gas could.
Demo: 06_Spectrum.py Adsorber.
They Do. A collection of some atoms objects is kept very cold, so that all the objects are
in the ground state. Light consisting of photons with a range of energies from 1 to 7.5 eV
passes through this collection of objects. What photon energies will be depleted from the
light beam (“dark lines”)?
-1 eV
1) 2 eV, 5 eV, 9 eV
-2 eV
2) 3 eV, 4 eV
3) 0.5 eV, 3 eV, 4 eV
-5 eV
4) 4 eV, 7 eV
5) 3 eV, 4 eV, 7 eV
-9 eV
Photons: Energy and Frequency. We’ve been making use of the idea of
photons to help us probe the structure of atoms. Now we’ll turn the tables a little
bit and learn some more about the energy of photons. As we qualitatively argued
last time, the energy and frequency of a morsel of light should be related to each
other; we’ll see exactly what that relationship is. The experimental observation of
this relation was one of the impetuses for the formulation of quantum mechanics.
One of the experiments that showed us this relation was the Photo-electric Effect.
Physics 231
Light
eV
Ch 8 Day 2
2013 4
o Photo-electric effect.
 First, I should note that, while it’s convenient for you and I, in this
present day and age, to speak of the electromagnetic spectrum in
terms of energies, what people were originally able to measure was
wavelengths, and from that they could easily deduce frequencies.
We will now see how frequencies are linked to energies.
 “Ionizing” chunks of metal. To rip an electron out of a H atom’s
ground state, you have to provide 13.6eV of energy, one way or
another. Similarly, to strip an electron off a metal surface, there’s
a minimum amount of energy necessary, depending on the metal,
it’s usually around 4 or 5 eV. We’ll call this Efree. Well before
people understood the energy structure of the materials, they could
experimentally determine this minimum energy for different
materials.
 “Ionization” by light. Now light can deliver this kind of wallup;
shine energetic enough light on a metal surface, and electrons get
knocked off. You can observe this effect by applying a large
voltage difference in the vicinity of the metal, and monitoring the
resulting current, i.e., flow of electrons.
 Threshold Frequency. Around the turn of the 19th to 20th century,
people were first doing these experiments and were varying the
intensity of the light they shone on the metal as well as the color,
or frequency of light that they used. They made a surprising
discovery. There was a threshold color or frequency.
 If you shone light of too low a frequency, no matter how intense,
you never got a significant flow of electrons. You gradually dial
up the frequency (proceed from red toward blue), and nothing
changes until you reach some threshold frequency, f. Then all of a
sudden the current of electrons would jump up. Now, at or above
this frequency, you vary the intensity, and the current strength
varies in response.
 Explanation. Why did this happen? Well, knowing what you and
I know now, the light that was being shone on the metal can be
viewed as a rain of photons, each one delivering its own quantity
of energy. Imagine an atom in the metal surface, it needs to gain
Efree in order to be knocked free of the surface. Unless there’s an
extremely high intensity of light / high density of photons raining
down, it’s extremely unlikely that our atom will get hit by more
than one photon at a time, so if it’s going to lose its electron it’s
because it got hit by a single energetic enough photon.
Eph< Efree : no current.
 But if Eph>or = Efree, then one photon is energetic enough to free
the electron.
 So, at this threshold frequency, f, we must have Eph = Efree; just
enough energy to free an electron. So, experimenters were able to
determine the energies for photons of different frequencies by
performing this experiment on different surfaces, with their
different values of Efree. Dial up the frequency until electrons start
Physics 231
Ch 8 Day 2
2013 5
coming off, note that frequency, and then you’ve got a (f, Eph).
Plotting Eph vs. f for all the different experiments gave a line:
Eph = hf.
Where h = 6.6×10-34Js = 4.1×10-15eVs: Plank’s Constant.
This is often written in terms of the angular frequency, w,
where
f. Then this extra factor of 2p is adsorbed in a
new constant  h / 2 “h-bar.”
o Eph =  .
o While it was Einstein who pieced this together, the
constant is named for Plank because he had
previously arrived at the same energy-frequency
relation based on a (more complicated) analysis of
another experiment with light.
Ex.: How much energy does one photon of Red light, with a frequency of 4.36×1014Hz
have? 2.88×10-19J = 1.8 eV.
o Demo / Ex. He-Ne laser. I’ve a little laser here. Inside is a gas tube not
too unlike the one we were just looking at. This tube happens to be filled
with He and Ne. This combination of gasses would radiate a spectrum
that is a combination of that of He and that of Ne. However, the gas tube
is made to help encourage only the red line to be radiated. A red photon
has an energy of about 1.8eV. This is a 0.95mW laser; that’s the rate at
which light energy is radiated. How many photons are emitted per
second?
E ph 1.8eV 1.8 (1.6 10 19 ) J 2.9 10 19 J


Etotal
9.5 10 4 J / s
t
Etotal Ptotal t 9.5 10 4 J / s1sec
N ph
3.3 1015 / sec.
19
E ph
E ph
2.9 10 J
Note: if you’re eye is sensitive to as few as 2 or 3 photons, you
can imagine that a flood of photons such as this could overwhelm
it and do damage.
Ptotal
Simple Harmonic Oscillators.
o Now that we’ve got this experimental foothold, we can use it to learn more
about the simple harmonic oscillator. Recall that we’d first deduced a
qualitative relationship between light’s frequency and energy by
imagining the energy transmitted between two oscillating charged
particles. Now let’s do a thought experiment looking more closely at one
of those oscillating charged particles, that will teach us something about
all oscillating objects (charged or not).
o Mass on a spring. Imagine we have a mass on a spring, we pluck it and it
oscillates back and forth. The frequency of its oscillation is determined by
Physics 231
o
o
o
o
o
Note: evenly
spaced.
Ch 8 Day 2
2013 6
the mass, m, and the spring stiffness, ks. Independent of the amplitude of
oscillation, the spring oscillates with angular frequency
ks
.
m
Its kinetic and potential energy go like
2
 K U K 12 k s s
U eq
at any given instant; or taking the instant when the spring is fully
extended, the full amplitude of the mass’s displacement, then when
2
 K = 0 at that instant and E 12 k s S max
U eq .
 To make things simple, let’s just focus on the term associated with
its oscillation:
2
Eoscillator 12 k s S max .
Of course, in practice, the spring-mass system isn’t isolated, so as the
mass oscillates back and forth of course it slowly looses energy and Smax
gets smaller and smaller.
Charge on a spring. To really illuminate this process let’s imagine the
bobbing mass has an electric charge. Then we have an oscillating charge,
oscillating at frequency . This necessarily radiates light of the same
frequency, and thus energy Eph =  . By conservation of energy,
E oscillator E photon  . Okay, now the oscillator has less energy, it
ks
.
m
Again, it sheds a photon, and again it drops energy by  . This goes on
and on until it reaches bottom, or as close to bottom as it can; that is, if it’s
less than  from the bottom, then it doesn’t have enough energy to
radiate of a photon, and it’s stuck!
o Energy Structure. The energy structure we’ve mapped out is
E oscillator n
E o , n = 0,1,2,… What we’ve found is that our simple
harmonic oscillator has discrete allowed energy levels. Comparing this
2
with Eoscillator 12 k s S max tells us that it has discrete allowed oscillation
amplitudes.
oscillates with a smaller amplitude, but same frequency,
o Demo: 06_oscillator.py
o Drop the charge & Generalize. Now imagining a charge on our mass
was just a convenient foot in the door to discovering the energy structure;
but whether there’s a charge along for the ride or not, a simple harmonic
oscillator should have this energy structure. This should apply to all
simple harmonic oscillators: our 1 kg mass on our 3 N/m spring, or a H
molecule with its two 10-27 kg mass protons and 100 N/m spring constant.
Note on this semi-classical derivation: Really, the motion of the charge is determined by the time evolution of the square
of the probability density, and that is constant for pure energy states – so no motion and no radiation. However, a system
isn’t really left in a pure excited energy state – perturbations by zero-point fluctuations in the field mean the real state of the
system is better described as a superposition of states, dominated by adjacent states. Such a mixed state does time evolve,
with a frequency f= E/h. Since, for the simple harmonic oscillator, the step size is the same between all adjacent states, the
frequency of emission is too. So, it’s not really accurate to say that a charge riding a simple harmonic oscillator in an
energy state will oscillate and thus radiate, but to the extent that all states above the ground state are fundamentally unstable
to perturbations by the field’s fluctuations, it is perhaps a more accurate picture anyway.
Physics 231
2013 7
Ch 8 Day 2
o Example. Like the notion that mass changes with internal energy, this is a
pretty radical claim that bares some proof. 1st: it shouldn’t predict
anything that we don’t see. We certainly don’t notice that a mass on a
spring has only specific allowed energies, it seems to be able to oscillate
with any energy / amplitude (until it breaks). Given our 1 kg mass on our
3 N/m spring, initially displaced 0.1m, how much energy has it got, and
how big is the step to the next energy level lower?
2
2
1
 Eoscillator 12 k s S max
0.015J
2 3N / m 0.1m
Eoscillator
E photon


ks
m

E
8.57 10 34.
E
S min 10 17 m

1/1000th the diameter of a proton!
1.05 10
34
Js
3N / m
0.01kg
1.29 10
o Okay, we’d never notice such a small fractional change in energy.
o Example. How about a H2 molecule, what’s the energy step size for its
vibrations? Roughly 10-27 kg mass protons and 100 N/m spring constant.
k
100N / m
o
Eoscillator E photon 
 s 1.05 10 34 Js
3.3 10 20 J
27
m
10 kg
ironically, the energy step size is much greater for this microscopic object
than for an every-day sized object.
o Solid. Recall our mass-spring model of a solid. We now add something
new to the picture: the atoms in the solid can only oscillate with specific
energies, thus the solid itself can only have specific internal energies. You
can see in Phys 233 and Phys 344 that this effects how the solid responds
to changes in temperature, i.e., its specific heat.

Low level approximation. If we look back at the shape of the
potential energy curve we’d imagined for inter-atomic bonds, we
recall that it’s nice and parabolic near the bottom. So if the bonds
are stretched only a little, then they respond like ideal springs and
all that we’ve just said follows including the stretch-independent
frequency and equal spacing of energy levels. However, for more
extreme stretches, the curve is not at all parabolic, and the
frequency does depend upon stretch and the spacing of energy
levels is not even. In fact, the spacing does the same thing as for
Note: near bottom, equally
the Hydrogen atom – get’s smaller and smaller.
spaced, like SHO
Spectra of Solids.
Near top more and more
closely spaced, like H.
o So a solid, made up of individual atoms (with their individual electronic
If one asks why, can say that
energy levels) bound to each other (with their vibrational energy levels)
spacing is related to potential
have more complicated spectra than those of simple atoms.
concavity through
o Reflection Spectroscopy. Since it’s rather hard to see through solids, and
d 2U
heating them up enough to get informative spectra often means vaporizing
E
k sp
dr 2
them (kind counterproductive in many cases), reflection spectroscopy is a
so less on concave, less
energy difference.
34
J
0.21eV
Physics 231
Ch 8 Day 2
2013 8
popular tool. Light is shone on the material, and the reflected light is
analyzed. One of the Mars probes was equipped with a reflection
spectrometer so that it could determine the mineralogical composition of
rocks found on Mars.
Different Spectral Ranges. So far, we’ve considered in detail the electronic orbital
energies, and the bond vibrational energies. Another freedom of course is rotation and,
you may now not be too surprised to learn, those are quantized too! Apparently
something can rotate at only specific rates.
Depending on the quirks of the specific material of interest, and exactly what you want to
know, different ranges of its spectrum can be illuminating.
 Bond vibration – Infra-red. Infra-Red spectroscopy looks at the
transitions in vibrational energy levels – so it directly investigates
the bonds (very useful in determining how molecules are formed).
Vibrational

Bonding / electronic excitation. Visible spectroscopy looks at the
Rotational
levels
electronic levels involved in forming the bonds – which are often
levels
smeared into broad bands (conduction and valance bands of
metals).
Metal Mirrors. Indeed, metals are such good reflectors
because their bonding electrons are quite free to flow back
and forth at a wide range of frequencies, thus adsorbing and
re-emitting light over a broad spectrum.
Visible spectroscopy can also look at levels near these, and
they are perturbed from their free-atom form due to the
influence of the environment of other atoms, but remain
fairly discrete.
 Large Atom, near-nucleus states, UV. Finally, UV / X-Ray
spectroscopy can look at the deepest energy levels, which are
relatively unaffected by the atom’s environment (what other atoms
it’s bond to) and resemble those of free atoms.
o For more info see http://speclab.cr.usgs.gov/aboutrefl.html.
Rotation. Backing out from a solid, think of a much smaller group of atoms all
bound up – a molecule. This object is not only free to have electronic excitations,
Electronic
and bond vibrations, it can also spin. I won’t argue it out until a later chapter, but
levels
you might not be too surprised to learn that rotational motion is also quantized,
and that there are discrete allowed energies associated with rotational states.
These energies are on order of 10-4eV – recall that this is the Microwave range (as
in Microwave ovens).
Putting it all together: Diatomic Molecule: electronic + vibrational +
Rotational. Putting it all together, a diatomic molecule has three different kinds
of energy that are quantized: Electronic, spaced on the order of eV, Vibrational,
spaced on the order of 10-2 eV, and rotational, spaced on the order of 10-4eV.
Putting it all together, we have an energy structure something like this (picture on
left). Notice that the bigger-scale the motion, the smaller scale the energy steps
between allowed states. We already saw that trend when we considered the
microscopic and macroscopic mass on a spring.
Would
electron
riding a
spinning
dumbell
give the
right
spectrum?
For that,
E 12 I 2 an
d if we say
E  .
Qualitativel
y, it does
give
decreasing
frequencies
and energy
steps.
Physics 231
Ch 8 Day 2
2013 9
6.5 Nuclear energy levels: We started with the electrons outside the nucleus, and their
energy levels, and then proceeded out from there to the energy of atoms vibrating
against each other within a molecule to the molecule spinning. Now we’ll proceed
inward. Just as the electrons of an atom have different, specific, energy levels that
they can occupy, so do the protons and neutrons inside the nucleus. As we’ve already
discovered, the spacings are on order of MeV’s. If a photon of such high energy
strikes a nucleus, it can knock a proton out of its equilibrium location; the proton
subsequently falling back down radiates another photon in this range.
6.6 Hadronic energy levels: For that matter, looking even deeper, inside the protons and
neutrons are quarks. These have specific possible configurations; the corresponding
energies are hundreds of MeV’s apart.
6.7 Comparison of energy level spacings
The moral is that as long as there is a constraint such as a potential energy represents,
there are quantized energy levels.
Type of State
Hadronic (quark composites)
Nuclear (nucleon composites)
Electronic (atoms & molecules)
Vibrational (molecules)
Rotational (molecules)
periodicity upon full rotation).
Energy Scale
108eV
106eV
1 eV
10-2eV
10-4eV (here the “constraint” is the condition of
Physics 231
Ch 8 Day 2
Match the type of system or situation to the appropriate energy level diagram.
No
Response)
a.
vibrational states of a diatomic
molecule such as O2
d
No
Response)
f
electronic, vibrational, and
rotational states
of a diatomic molecule such as
O2
b.
No
Response)
rotational states of a diatomic
molecule such as O2
c
c.
No
Response)
hadronic (such as
+
)
g
d.
No
Response)
idealized quantized spring-mass
oscillator
e
e.
201310
Physics 231
No
Response)
Ch 8 Day 2
electronic states of a single atom
such as hydrogen
a
f.
No
Response)
nuclear (such as the nucleus of a
carbon atom)
b
g.
201311