Phys 233
Day 4, Q4: Wave Matter
1
Thurs 9/13
Q4 Matter Waves, E-gun demo
RE-Q4
Mon. 9/17
Tues 9/18
Q5 QM Facts
HW2: Q3: S3, S5, R1; Q4: S5, S9, R2; Lab Notebook & Procedure
RE-Q5
Equipment
o Ppt.
o Excerpt of beautiful data showing bubble chamber & STM
o Electron gun set up
o http://phet.colorado.edu/en/simulation/davisson-germer
o Cloud chamber apparatus
I structured things a bit differently
Q4: Wave Nature of Matter
for this day. I showed the bubble
Q4.1 Subatomic Particles as Particles
chamber picture and asked them to
Q4.2 The de Broglie Hypothesis
talk with their neighbor about what
Q4.3 Preparing an Electron Beam
was going on. Next I showed them
Q4.4 The Davisson-Germer Experimentthe electron diffraction set up,
Q4.5 Electron Interference
turning it on, turning it up and
Q4.6 Matter Waves
asked them what was going on
(electron gun, reflection off
Bubble Chamber Picture in Hall
obstruction, interference pattern.)
o Q: Talk with your neighbor - Ignoring
the pretty
what’s
this a picture of
At some
point colors,
I showed
the slides
/ how was it made?
and demo of reflecting laser off
 A: A vat of super heated liquid
on the edge
vaporizing
CDhydrogen
and DVDjust
to illustrate
thatofwe
was exposed to a source of charged
particles, patterns
perhaps an
get interference
foraccelerator. The
volume was suddenly increased
by
a
piston
/
pressure
decreased and then
reflection as well as transmission.
the liquid began to boil. As the
charged
particles
zipping
through,
I also
mentioned
thatwent
we have
rings
their gentle nudges and tugs on
the
helium
atoms
seeded
the
boiling
instead of dots because our target
process. So bubble tracks traced
particles
is a where
powderthe
of charged
randomly
orientedwent. An
instant later and the boiling would
really
take
off
and
these
tracks would
crystals.
be obliterated, but for a snapshot,
they
showed
where
the
charged
particles
Then I turned the voltage up to
went. This was done in the presence
a magnetic
so,out
depending on
3kV andofasked
them tofield
work
the sign of the particle’s charge,
it
arced
one
way
or
another.
how far apart the atoms should be.
o Take-away: For this class, the take away
is that
sub-atomic
particles
Part way
through
I paused
and can be
thought of as, well, particles.
went over how they’d get the
Cloud chamber
wavelength (about 0.022nm).
Here’s an analogous, table-top system.
Then it was back to them to
determine the distance between
STM images in Hall
atoms given the distance from
o Q: What’s this a picture of?
crystal to surface and the distance
 A: It’s an STM image. Eachto‘ball’
essentially ring.
an atom.
first is
constructive
o Take-away: again, particles can be particle-like.
I came back and filled in the
derivation gaps on the board,
mentioned where DeBroglie’s
relation comes from. At the end I
did show the Buckyball
experiment.
Phys 233
Day 4, Q4: Wave Matter
2
Last time
Last time, we considered evidence that light, which is known to behave like a wave, also behaves
like a particle. When it interacts with matter, it does so discretely and delivers a discrete packet
of energy.
This time
Now, we’ll consider theory & evidence that matter, which is known to behave like a particle, also
behaves like a wave.
Q: With your partner at your table, discuss, qualitatively, what’s happening.
First, an electron gun is accelerating electrons, as they all start from about at rest and they all
accelerate through the same electric potential, they all leave the gun with about the same energy
& momentum, thus same wavelength.
Next, they glance off a powder of small graphite crystals. While each dust-speck of crystal is
indeed a crystal and so electrons would be expected to bounce off making simple, interference
patterns, the specks’ orientations are random, so the composite of all the electrons reflecting off
one crystal at one orientation, another crystal at another orientation,… is rings rather than spots
in our pattern.
Q: now let’s get quantitative.
Given the reading, get a value for the atomic separation.
Note: They may treat it using
d sin
= 2 d sin =path length difference =
d sin
d sin
+
for constructive interference
Phys 233
Day 4, Q4: Wave Matter
3
Q4.1 Subatomic Particles as Particles
First, some evidence that even atoms and sub-atomic matter can behave like particles.
And yet…
Q4.2 The de Broglie Hypothesis
Building up to de Broglie
o With Special Relativity, Einstein had proposed that E
2

mc 2 and p

mv , which
2
means you can relate the two as E 2
pc
mc2 , all three of which you’d used
in Physics 231
o As we learned last time, he also suggested that the observations of the
PhotoElectric effect could be explained if light delivered energy in units related to
its frequency by E hf .
o So, it’s fairly obvious that the amount of momentum that light delivers is
E2
E

h
2
mc 2
2
pc
hf
h
pc
pc
f
c
p
p
f .
Where the last step just makes use of c
Note: it’s nothing new that light can transfer momentum; you
probably saw that back in Phys 232. What’s new is that it comes
in units like this.
o de Broglie then proposed ‘what if this relationship held for massive objects too?’
Q4T.1 - Consider a beam of free particles that each have a certain (nonrelativistic) speed. If we
double this speed, what happens to the beam’s de Broglie wavelength?
o It increases by a factor of 2
o It increases by a factor of √2
o It remains the same
o It decreases by a factor of √2
o It decreases by a factor of 2
o Something else happens to the wavelength (specify)
Q4.3 Preparing an Electron Beam
Want to see interference. Of course, a very direct way of determining whether electrons
have wave lengths associated with them would be seeing if they displayed any
quintessentially wave-like behavior, such as making interference patterns.
The first step in that direction would be getting ‘monochromatic’ electrons to shoot at
two slits or something else that would produce an interference pattern.
Q: How is this done?
Phys 233
Day 4, Q4: Wave Matter
4
o A: wire up a cathode and an anode with a hole in it. Heat the Cathode until
electrons ‘boil’ off, then they’re whisked away in the field between the two.
Those that happen to hit the hole shoot out the other side with a well defined
energy / momentum / wavelength.
 Note: Electron guns get used in old style ‘cathode ray tube’ TV’s and
computer monitors like those around the room.
K
U 0

Kf 0 e V 0 Kf e V
 For electrons not moving near the speed of light,
K

1 2
p
2m
1 h
2m
2
1
hc
2
Alternatively, 2mc
2
h
2mK f
2
hc
2mc 2 K f
1 h
h
e V 0
2m
2me V
Demo: Electron Gun side of Diffraction Apparatus (crank up voltage only enough to get
beam, not see pattern.
Q4.4 The Davisson-Germer Experiment
Reflection.
o When we met interference patterns last Thursday, we were thinking about waves
coming up to a couple of slits and just small slices of them transmitting. As those
narrow transmitted waves rippled out, they interfered.
o Similarly, when waves strike a surface with just a few reflectors, then you get just
two small slices of them reflected. Again, that allows the waves that ripple out to
interfere.
o Same criteria for constructive interference
 d sin nc n
Where d is the distance between the two reflectors.
o Demo: shine laser on CD and reflect at angles.
o A single atom is a pretty darn small reflector, so it doesn’t reflect a lot of
electrons / you wouldn’t get a big signal; however, if you had a lot of atoms all
lined up, like on a crystal surface, then you’d get a really big signal.
 Note: The book mentions the complication of reflecting from not just the
atoms on the surface, but those a few layers in too. That means you’d
need to get constructive interference from them too.
o Demo: http://phet.colorado.edu/en/simulation/davisson-germer
 Turn gun on, then can adjust brightness to see pattern best – note, our law
applies to far-field, so don’t worry about complication near the surface.
Field Trip to LEED image
Q: What’s this?
A: A diffraction or interference pattern like that of the Davisson-Germer experiment –
electrons are shot from an electron gun at crystal surface. As we talked about last time
with light, the interference pattern that is produced reflects the surface’s geometry. Some
of the added detail here comes from the experimenters controlling the energies of the

Phys 233
Day 4, Q4: Wave Matter
5
electrons that can come back to the screen. If I recall, they can set up both an attractive
grid and a repulsive one.
Q4S.1 – (White boards) In the Davisson-Germer experiment described in example Q4.2, what
would the smallest nonzero angle (relative to the direction of the original beam) where reflected
electrons might constructively interfere if the kinetic energy of the electrons were 102 eV?
Is there another possible angle of constructive interference?
Q4.5 Electron Interference
Demo: Electron Gun (Now crank up beam strength and mention that there is a graphite powder
targer)
Q4T.6 - If the value of h were bigger, it would be easier to display interference effects in
macroscopic objects, true or false?
Q4S.2 - (White boards) A beam of electrons is created by accelerating electrons from rest
through a potential difference of 55 V.
a) What is the de Broglie wavelength of this batch of electrons? Express your result in
nanometers.
b) Explain why it is not going to be easy to make two slits with a spacing that is roughly the
same size as this wavelength. (Hint: the size of a typical atom is 0.1 nm.)
c) Find the de Broglie wavelength of a beam of protons instead of electrons accelerated
through the same voltage difference (mc2 = 938 MeV for a proton). Compare with your
result in part (a). Is it going to be easier or harder to set up a two-slit interference
experiment for protons?
Q4.6 Matter Waves
A matter of scale The author makes a good example of light – low frequency light like
FM has such small energies per photon (~10-7eV), that it’s hard to have a single-photon
interaction cause anything detectable, but the wavelength is quite large ( ~3m) so it’s
easy to see wave phenomena. Visible light has both reasonable energies per photons, so
we can see single photon effects, and it has large enough wavelengths that we can see
diffraction and interference. Then Gamma rays, with their MeV’s of energy can easily be
seen to have individual effects, but their wavelengths are incredibly small, so it’s hard to
detect wave phenomena with them.
Q: What’s the picture on the left side of the book’s cover?
o A colorized, 3-D-ized STM image. The spikes represent the strong electron
densities around individual atoms of one type. The subtle ripples represent the
undulations in electron density, the electron waves that run along the metal
surface. Within the atoms have been circled with a radius to hold a circular
standing wave of electrons.
Phys 233
Day 4, Q4: Wave Matter
6
Q4S.7 - A buckeyball is a large molecule comprised of 60 carbon atoms arranged in a shape
something like a hollow sphere 0.71 nm in diameter. Imagine that we create a beam of
buckeyballs all moving at the same speed v. What is the maximum value that v can have if the
de Broglie wavelength of the buckeyball beam is to be at least 10 times the size of the buckeyball
(so that we might actually be able to display interference of the buckeyballs)?
Buckeyball experiment (in AJP) – uses a much larger speed!
Okay, mater can interfere like waves. Where does this lead us?
o Waves and Bohr atomic Model. One early success of this proposal was that it
nicely explained the Bohr model of the electron. If you recall from Phys 231,
based on the observed spectrum of Hydrogen, Bohr proposed that there were
allowed ‘stationary state’ orbits for electrons about atoms. Those orbits
corresponded with circular standing waves.
o Wavelength implies Wave implies Wave equation: Whenever there’s a wave,
there’s a differential equation for whom the wave function is a solution. For
example, in Phys 232 you probably saw how Maxwell’s equations could be
Phys 233
Day 4, Q4: Wave Matter
7
combined into one differential equation for E and another for B, and those are
solved by sinusoidal functions. So, when Schrodinger was giving a talk about de
Broglie’s hypothesis, someone ask him ‘if massive objects have wavelengths and
thus waves associated with them, what’s the wave equation?’ That’s what set him
on the path to proposing Schrödinger’s Wave Equation which we’ll eventually get
to and, for one thing, helped replace Bohr’s model with a better understanding of
electron orbitals…and all of chemistry.