Notes: Quantum 2
Level 1: Compton Scattering
We are not going to spend a lot of time on this. Please don’t obsess over it. Please watch Ms. Twu do a short
lecture on it (she will cover everything you need to know about this experiment in 2:54), and then take a look at
the practice problem below. Notice that in order to solve the practice problem, you did not need to know a lot
about Compton scattering. It’s actually more of a kinematics questions. You need to have a qualitative
understanding of Compton’s scattering. Don’t obsess over the details.
Watch Twu: Modern Physics 11: Compton Scattering
https://sites.google.com/site/twuphysicslessons/home/modern-physics/modern-physics-page-2
AP Practice Problem: 2002B7
Solution
Level 2: Waves and Particles.
Einstein’s early work in relativity showed that light could be both a wave and a particle- and that it acted
differently depending on the experiment. We call this wave-particle duality, and it is well outside the realm of
things humans can easily visualize. We will not get into the details of this in this course, but we will look at some
of the repercussions of this work. Probably one of the most famous responses to Einstein’s work came from Victor
De Broglie in 1923.
The Wave Properties of Particles
This section comes directly from Knapp’s online textbook
In 1905 Albert Einstein showed that waves behaved like particles. This was very disconcerting – how could a
wave be both a particle and a wave? But what about the opposite thing? Could matter (which is what particles
are) exhibit wave-like properties? In 1923 Victor De Broglie (1892-1987) suggested just that. And he got Einstein
to support him. It seemed logical, if not symmetrical, to De Broglie that if waves had particle characteristics,
then particles ought to demonstrate some wave characteristics. Fair is fair!
The wavelength of a massive particle, like a baseball, would have matter-waves of such ultra-short wavelength,
that it would be impossible to detect. But the wavelength of a small, high-speed particle such as an electron
ought to be long enough that it could be measured. In 1927 an American physicist, Clinton Davisson (1892-1975)
did just that. He found that a beam of fast electrons could be diffracted and refracted. This was substantial
evidence for the wave-like behavior of matter.
All matter behaves as both a particle and a wave.
It seems odd that a photon could have momentum since it has zero mass. But according to special relativity,
photons do indeed have momentum.
The energy of a photon is of course:
E  hf

hc

The photon’s wavelength can be expressed as a function of its momentum:

h
p
This equation is provided on the AP Physics Test equation sheet.
So we now have an equation for the momentum of a photon.
Wavelength of Particle:
We can now develop an equation for the wavelength of a particle.
The momentum of a particle is:
p  mv
We now plug this value into the equation for the momentum of a photon and see what happens:

h
p

h
mv
This gives us an expression for the wavelength of a particle.

h
mv
You will be expected to be able to calculate the wavelength of a particle, but will not be given this equation. So
prepare thyself to derive the thing all on your own, but it’s a pretty simple thing to do, don’t you think?
Practice Problem
Find the wavelength of a 1.2 kg rock thrown at a speed of 22 m/s.
Solution
p  mv

6.63 x 1034
h
p

kg  m  m  s
s2
1.2 kg   22 ms 



h
mv
 0.25 x 1034 m 
2.5 x 1035 m
This is a very small wavelength, one that would be impossible to measure.
Practice Problem
Calculate de Broglie wavelength for an electron moving at 1.50 x 106 m/s.
Solution

h
mv


34 kg  m  m
s
 6.626 x 10
2
s

 
m

9.11 x 1031 kg 1.50 x 106 
s


  0.485 x 109 m 

4.85 x 109 m
This wavelength we can measure. Countless experiments have been done which have shown the wave nature of
matter.
The reason that we do not notice matter behaving as a wave is because the wavelength’s of ordinary matter are
incredibly small, way too small too measure. It is only when we get down to the scale of subatomic particles that
this wave behavior becomes something that we can actually observe taking place.
Level 3: Atomic Models
Watch Ms. Twu teach atomic models. You don’t need to get too stuck up on the details. It’s just good to have a
well-rounded picture of the different experiments that built our understanding of the atom.
Video Lecture: Watch Ms. Twu in Modern Physics 17: Early Atom Models and Modern Physics
18: Rutherford’s Experiment and Planetary/Nuclear Model of Atom
https://sites.google.com/site/twuphysicslessons/home/modern-physics/modern-physics-page-3
Level 4: Emission Lines
A Rainbow in a Nutshell
Note: These notes are better experienced online due to the colors. It is not necessary, however.
In the past, we discussed the wavelength of various things- first as radiation and light (in our first section) and
later electrons and actual objects (see Level 2 of these notes). Let’s return to light for a moment, to review
something you may (or may not) remember from last year.
Radiation comes in many forms, classified by its wavelength (or frequency). Radiation just isn’t the deadly stuff
you hear about in popular culture- it can be heat, light, x-rays…
We can actually only see that very small section in the middle. This is the
type of electromagnetic radiation we call light. There exists all kinds of
electromagnetic radiation beyond our vision. If you turned off the lights, and
you could look around with special eyes, you would see that the room
glowing with creatures giving off tons of radiation- you emitting tons of
infrared (heat) radiation.
In terms of visible light, the wavelengths correspond to different colors (see
the table to the right). For example, blue light is around 460nm.
Emission Lines
Scientists noticed something really strange about gasses. If they
were heated up (or zapped with electricity, same difference), they would glow. If fact, they were
giving off a bunch of wavelengths (colors) all that once. If you used a special piece of plastic called a
diffraction grating, you could actually break these colors up. For example, if you take a tube of
hydrogen gas and heat it up, the tube will glow this pretty pinkish purple color (fuchsia, for the color
literate out there). It is pictured to the right.
If you were to look at this light through a diffraction grating, you would see that it is actually made
up of several colors (wavelengths) mixed together. You can see this below.
See the bright lines? This is what was making the hydrogen gas look pinkish-purpley (fuchsia…its called fuschia).
This is hydrogen’s emission spectrum, and these particular colors are unique to hydrogen. If we were to heat up
a different gas, it would give off entirely different colors. Each gas has its own characteristic spectra. It is like a
gas fingerprint. Some common spectra are pictured below.
Demo: At this point, you should ask to see the demo at the back of the room.
The fact that these gases gave off these very specific types of light was a real mystery to scientists. In the end, it
would actually help explain the exact nature of atoms- and unlock some of the secrets of physics.
Explaining Emission Lines
In this section, we are going to pretend
that electrons orbit atoms, even though
we know that that is technically not
true. It is good enough for what we need
to do with it. The model we are about to
discuss, by the way, is called the Bohr
Model of the atom.
Okay, so here is the idea. Around the
nucleus of an atom there are different
energy states an electron can hang out
it. The lower energy states are near the
nucleus. Higher energy states are further
away from the nucleus. The lowest
energy state is called the ground state.
Here is a really, really important idea:
Electrons only exist in these energy levels. They do not exist between them.
The levels are like levels of a house without stairs. You can be on the third floor, or you can fall down to the
second floor, but you can’t live on floor 2.6. You are in one energy state or another- not somewhere in between.
The electrons can go from one energy state to another. This is a lot like leveling up in a video game. If you earn
enough points, you can jump up to the next level. If you lose too many points, you can fall back down.
So imagine an electron in a hydrogen atom was at a pretty low state. Then, it was zapped with energy, so it is
now hanging out at a higher energy level. This is called an excited state. Electrons can do this for a little while,
but they don’t like to stay at higher energy levels. They will naturally give off the extra energy and fall back
down to a lower energy state. Where does the energy they give off go? Well, it is given off in the form of light. If
the electron fell several energy levels, it gave off a lot of energy, so the light it gave off was high frequency,
short wavelength. If it fell a short distance (maybe from the 3rd energy level to the 2nd energy level, for
example), it gave off less energy, so the light it emitted was lower frequency, longer wavelength). This is why
hydrogen gave off different emission lines- each line corresponded to a different drop in energy.
Video Lecture: Physicist Michael. Physicist Michael is going to talk you through this slowly and he will
use color. This is a slow paced lecture, but very clear and largely easy to follow. He’s a pretty good
teacher too. Pay special attention to when he talks about absorption. You will need this soon.
https://www.youtube.com/watch?v=TbFehcC4MHc
Energy Level Diagrams
Okay, we’ve talked about electrons jumping from one energy level to another. We could visualize that this way:
This can be kind of confusing when you are doing multiple jumps. Instead, we often use an easier method called
an energy diagram. Here’s an example.
The bottom line is the lowest energy level. The top one is
the highest energy level.
On the left hand side, we see the different energies
marked in electron Volts (eV). Don’t worry about the
negative. We will get into why they are negative soon.
The amount of energy has to do with how much energy it
would take to get the electron to leave the atom. The
bottom level (closest to the nucleus) has the most energy
because it would more difficult to rip away an electron
close to the nucleus than one hanging out farther from the
nucleus.
The top line is zero because at that point, the electron has
complete left the atom. It won’t take any energy to steal
it.
The number on the right is the energy level. Notice the
ground state has n=1, the next level is n=2…
We use these diagrams to show transitions. We mark these using arrows. In the picture on the above, for
example, we see an arrow showing an electron going from energy level 4 to energy level 3. It is dropping n energy
so it is giving off light (a photon).
We can actually calculate the wavelength of the photon given off. Let’s say we were interested in an electron
that fell from energy level 4 (n=4) back to the ground state. What type of light would this energy transition give
off?
𝐸4→1 = 𝐸4 − 𝐸1
𝐸4→1 = −0.85𝑒𝑉 − (−13.6𝑒𝑉) = 12.8 𝑒𝑉
So, by dropping down between these energy levels, the electron will need to release 12.8 eV. What wavelength is
this?
𝜆=
ℎ𝑐
𝐸
If you don’t remember this equation, review Quantum 1 Notes
Plugging values in, we find the wavelength is 97nm. This is outside the wavelength of visible light- it is
ultraviolet. We would need special instruments to see the emission line created by this transition. If we could see
it, we would see a spectral line at 97 nm.
Where Hydrogen Spectral Lines Come From
Now let’s use this idea to work backward.
When we look at the light coming off hydrogen in
its excited state, we see a few emission lines.
Let’s look specifically at that 434 nm line.
What frequency of light is this? 6.9 x 1014 Hz. (Use
c=f𝜆)
How much energy does this light have?
2.86 eV (Use E=hf)
Where did this energy come from? From an
electron changing states. We can look at the
energy diagram of hydrogen to try to figure out
which state change produced this change.
Take a look at the different possible transitions. Which transition do you
think could have caused an emission of 2.86 eV? Take a moment and try to
figure out which it is before you read on.
The transition is from 52. The difference in these energy levels is:
𝐸5 − 𝐸2 = 𝐸𝑟𝑒𝑙𝑒𝑎𝑠𝑒𝑑
−0.54 − (−3.40) = 2.86 𝑒𝑉
This is where the pretty violet light comes from in the hydrogen spectrum.
Kinda cool stuff (I love this part of physics). Also, see why the energy
diagram is listed with negative energies? Makes it easier, right?
Practice: Finding Spectral Lines
The other wavelengths we noticed in hydrogen were 656.2 nm, 486.1 nm and 410.1 nm. Figure out what
transitions caused each of these spectral lines. You can check your answers in the diagram below.
Practice: Finding Emission Lines (with the answers)
AP Practice: Free Response with Spectral Lines
Solution
Level 5: Absorption and Ionization
Absorption
So far, we’ve looked at situations where we heat up a gas and look at the emission lines given off as the electrons
drop down from energetic states. The reverse can also be true. Let’s imagine that we send a full spectrum (all
different wavelengths) of radiation through a cool gas. What would happen?
This picture helps show what would
happen. We take a source of light that
gives off a fairly broad, continuous
spectrum (we can actually use black
bodies for this). In the picture, the light
source on the left is giving off a
continuous spectrum.
Then, this full spectrum light is passed
through the cool gas. As it passes
through, the atoms absorb the photons
and jump up to higher energy levels.
They only absorb photons with just
enough energy to allow them to jump
from one energy level to another. It
“stole” wavelengths with these exact
quantities of energy out of the
spectrum. Therefore, if you were to
stand on the other side of the gas, the full spectrum light would appear to have chunks missing out of it- black
bands where the energy had been absorbed by the atoms.
In the news, we often hear scientist talk about what distant stars are made of. Ever wonder how they do this? (I
know half of you just sarcastically said “no”). Stars are unique because they have a hot, dense core that acts like
a fairly good black body. They also have a layer of gases surrounding them. As full spectrum light leaves the
center of the star and passes through the gasses, some of the wavelengths are absorbed by the gasses in its
atmosphere. By looking at these absorption lines, we can tell what the star is made of. Here, for example, is
what the sun’s light looks like if you pass it through a diffraction grating.
Notice all the dark lines, these are caused by the different elements in the gasses (atmosphere) of the sun. We
can look at these lines to identify exactly what the atmosphere of the sun is made out of. The picture below
helps illustrate this process.
As you would expect, the dark lines on an abortion spectrum correspond to the bright lines on an emission
spectrum for any given element. See the example below.
Application Activity: Go to this site. Click on Classify Spectra Activity. Identify the elements in Stars
A, B, and C.
AP Practice Problem: Absorption Lines
Solution
Ionization
Let’s take a look at the energy levels of a hypothetical
element. Its ground state is -122.4 eV. What would happen
if the electron was given more than 122.4 eV by a photon?
It would actually escape the atom. We have a special name
for this. We call it ionization. We call the amount of
energy it takes to break the electron out of the atom and
set it free ionization energy. For this mystery element,
the ionization energy is -122.4 eV.
What would happen if a photon with an energy of 200 eV
hit the electron? Well it would take 122.4 eV to ionize the
electron (cause it to escape the atom). There would be
77.6 eV of energy left.
Is there a big difference between the ionization energy and the work function? No, not really. You can treat them
pretty much the same in this class.
AP Practice: Ionization
B2002B7 Solution