Transcript No Slide Title

Blackbody Radiation
Photoelectric Effect
Wave-Particle Duality
SPH4U
Everything comes unglued
The predictions of “classical physics” (Newton’s
laws and Maxwell’s equations) are sometimes
completely, utterly WRONG.


classical physics says that an atom’s electrons should fall
into the nucleus and STAY THERE. No chemistry, no
biology can happen.
classical physics says that toaster coils radiate an infinite
amount of energy: radio waves, visible light, X-rays,
gamma rays,…
The source of the problem
It’s not possible, even “in theory” to know
everything about a physical system.

knowing the approximate position of a particle corrupts
our ability to know its precise velocity (“Heisenberg
uncertainty principle”)
Particles exhibit wave-like properties.

interference effects!
The scale of the problem
Let’s say we know an object’s position to an accuracy Dx.
How much does this mess up our ability to know its speed?
Here’s the connection between Dx and Dv (Dp = mDv):
h
DpDx 
4
That’s the “Heisenberg uncertainty principle.” h  6.610-34 J·s
“It is physically impossible to predict simultaneously the
exact position and exact momentum of a particle.”
Atomic scale effects
Small Dx means large Dv since Dv 
h
4 mDx
Example: an electron (m = 9.110-31 kg) in an atom is
confined to a region of size Dx ~ 510-11 m.
How is the minimum uncertainty in its velocity?
Plug in, using h = 6.610-34 to find Dv > 1.1106 m/sec
Example
The speed of an electron (m = 9.110-31 kg) is measured to
have a value of 5 x 103 m/s to an accuracy of 0.003 percent.
Determine the uncertainty in determining its position.
p  mv
m

  9.111031 kg   5.00 103 
s

kg  m
 4.56 1027
s
Dp  0.00003 p
kg  m 

  0.00003  4.56 1027

s 

kg  m
 1.37 1031
s
DxDp 
h
4
h
Dx 

4Dp
 6.63 10
34
J  s
kg  m 

4 1.37 1031

s 

 3.85 104 m
 0.385mm
Example
The speed of an bullet (m = 0.020 kg) is measured to have a
value of 300 m/s to an accuracy of 0.003 percent. Determine
the uncertainty in determining its position.
p  mv
m

  0.020kg   300 
s

kg  m
6
s
Dp  0.00003 p
 kg  m 
  0.00003  6

s 

kg  m
 1.8 104
s
DxDp 
h
4
h
Dx 

4Dp
 6.63 10
34
J  s
kg  m 

4 1.8 104

s 

 2.93 1031 m
Example
A proton has a mass of 1.67 x 10-27 kg and is close to
motionless as possible. What minimum uncertainty in its
momentum and in its kinetic energy must it have if it is
confined to a region :
(a) 1.0 mm
(b) An atom length 5.0 x 10-10m
(c) About the nucleus of length 5.0 x 10-15m
Example
A proton has a mass of 1.67 x 10-27 kg and is close to
motionless as possible. What minimum uncertainty in its
momentum and in its kinetic energy must it have if it is
confined to a region :
(a) 1.0 mm
Dx Dp 
h
4
Dp  mDv
6.63  10 J  s 

h
Dp 

4Dx 4 1.0  10 3 m 
34
 5.28  10 32
kg  m
s
6.63  1034 J  s 

h
Dv 

4 mDx 4 1.67  1027 kg 1.0 103 m 
 3.16  105
KE 
m
s
1 2
mv
2
1
m

 1.67 1027 kg   3.16 105 
2
s

 8.33 1037 J
2
Example
A proton has a mass of 1.67 x 10-27 kg and is close to
motionless as possible. What minimum uncertainty in its
momentum and in its kinetic energy must it have if it is
confined to a region :
(b) An atom length 5.0 x 10-10m
Dx Dp 
h
4
Dp  mDv
6.63  10 J  s 

h
Dp 

4Dx 4  5.0 10 10 m 
34
 1.06  10 25
kg  m
s
6.63 1034 J  s 

h
Dv 

4 mDx 4 1.67 1027 kg  5.0 1010 m 
 63.2
KE 
m
s
1 2
mv
2
1
m

 1.67 1027 kg   63.2 
2
s

 3.33 1024 J
2
Example
A proton has a mass of 1.67 x 10-27 kg and is close to
motionless as possible. What minimum uncertainty in its
Notice
that
consider
momentum
and in
its when
kineticwe
energy
musta itparticle
have if it is
(say
a proton),
that is confined to a small
confined to
a region
:
the Quantum
(c) About region,
the nucleus
of length Mechanics
5.0 x 10-15mrequires
that such a particle cannot have a precise
h
Dp momentum
 mDv
Dx Dp 
momentum
(or
even
of zero).
4
This
means
that
even
at
absolute
zero,
6.63this
1034 J  s 

34
h
6.63  10 J  s 

h
Dv 

Dp 

proton must
have kinetic
This
4 menergy.
Dx 4 1.
67 1027 kg 1.06 1020 m 
15
4Dx 4  5.0 10 m 
energy is called the “zero point
energy”,
6 m
kg  m
 6.32 10
 1.06  1and
0 20 there is no way to avoid this.
s
s
KE 
1 2
mv
2
1
m

 1.67 1027 kg   6.32 106 
2
s

 3.33 1014 mJ
2
Quantum Mechanics!

At very small sizes the world is VERY
different!
Energy can come in discrete packets
 Everything is probability; very little is absolutely
certain.
 Particles can seem to be in two places at same
time.
 Looking at something changes how it behaves.

Another Consequence of
Heisenberg’s Uncertainty
Principle

A quantum particle can never be in a state of rest,
as this would mean we know both its position and
momentum precisely

Thus, the carriage will
be jiggling around the
bottom of the valley
forever
Blackbody Motivation
•
•
•
The black body is importance in thermal radiation
theory and practice.
The ideal black body notion is importance in studying
thermal radiation and electromagnetic radiation
transfer in all wavelength bands.
The black body is used as a standard with which the
absorption of real bodies is compared.
Blackbody Radiation
Hot objects glow (toaster coils, light bulbs, the sun).
As the temperature increases the color shifts from Red to
Blue.
The classical physics prediction was completely wrong! (It
said that an infinite amount of energy should be radiated by
an object at finite temperature.)
Maxwell Boltzmann Distributions
In 1859 Scottish physicist James Clerk Maxwell developed his theory on the
kinetic theory of gasses that explained the macroscopic properties of
pressure, temperature and volume.
O.K. James how
does this kinetic
theory of gasses
relate temperature to
pressure.
If we accept the idea
we heat a
Ithat
thinkwhen
gas consists
of
gas, that
billions
and heating
billions of
causes
the molecules
fast randomly
moving
to molecules
move faster
and
that
thus bang
into other
the
bounce
off each
walls
thethe
container
as
wellofas
walls of
more
frequently.
the container.
Maxwell Boltzmann Distributions
Maxwell statement was a bold one. He claimed that the macroscopic
properties of a gas (that was easily measured in a laboratory) could be
predicted by a microscopic model that consisted of billions and billions of gas
molecules.
1. The molecules act like tiny
spherical marbles. With the
diameter of the marbles much
My last statement uniquely
smaller than the distance
allowed me to apply a
between them.
I had
to make
4
branch
of mathematics
2. The collisions between the
assumptions
beforetoI prove my
called statistics
molecules where elastic (no
was comfortable
with
theory was quantitatively
myconsistent
kinetic model.
with the physical
energy was lost).
properties of gasses that
3. In between the collisions the
where measured in the
molecules moved according to
laboratories.
Newton’s Laws (constant speed
and straight line).
4. The initial position and velocity of
each molecule was random.
Maxwell Boltzmann Distributions
602,214,179,300,000,000,000,000
You used statistics
and thus used the
averages. Why did
you not calculate the
motion of the
molecules using my
Laws?
Man, I just couldn’t.
There are just too
many gas molecules,
the task would be
impossible. Even in a
small sample called a
mole, there are
6x1023 molecules.
Temperature is actually the measure of microscopic mean
square velocity (average velocity multiplied by itself). Maxwell’s
theory is the prediction of the probable velocity distribution of the
molecules. That is it gives the range of velocities.
Maxwell Boltzmann Distributions
Because all atoms of an element have roughly the same mass, the kinetic energy of
identical atoms is determined by velocity (KE= ½mv2)
How did you go
about
measuring
these average
values?
I devised an apparatus
that allowed me to
determine the kinetic
energies and thus the
velocities
Maxwell Boltzmann Distributions
Fraction of
molecules
Molecules hit disk here first
Kinetic energy 
Molecules hit disk last
The the
slower
movingof the dots on a
faster
moving
If we
plot
intensity
The
resulting
(lower
Kinetic
Energy
(higher
Kinetic
graph
we
get
a
graph
of fraction of
disk
looksmolecules)
molecules)
start hits
Energy
atoms/molecules
vs.
like
this.
the
disk
later
atkinetic energy:
start
hits the
disk
3 11
earlyaround
at around
Maxwell Boltzmann Distributions
Why is the
graph skewed?
Fraction of
molecules
Kinetic energy

This curve is characteristic of all molecules
 The curve is elongated due to how atoms collide, and to the units of the graph
 Recall all particles are in motion. An average speed will be reached.
 The graph is skewed because 0 is the lower limit, but theoretically there is no upper
limit
• More than that the graph is skewed because the x-axis has units of energy not
velocity.

Same data, different axes..
v=1,
v=2,
v=3,
velocity
KE=1
KE=4
KE=9
Maxwell Boltzmann Distributions
Molecules with “most probable speed”
Distribution of Kinetic
Energy at temperature = T1
No. of molecules with
K.E.  Ea  rxn occurs!
Area under curve  total no. of molecules
Maxwell Boltzmann Distributions
Ludwig Boltzmann, building upon the work of Maxwell formalized the Theorem of
the Equipartition of Energy.
That is: the average kinetic energy in the translational motion of a molecule
should equal the average kinetic energy in its rotational motion.
I also looked at what
When a system
happens to the
reaches thermal
particles when the
equilibrium, all the
energy of the system
energy will be
decreases. This gave
equally shared
a new interpretation of
among all degrees
the Second Law to
of freedom.
Thermodynamics.
Maxwell Boltzmann Distributions
The Second Law
of
Thermodynamics?
S  k log W 
k – Known as Boltzmann’s
constant (1.38x10-23 J/K)
W – probability that a
particular arrangement
will occur
When a system starts to fall
apart, the atoms in the
system become more
disordered and the entropy
increases. This disorder can
be measured as the
probability of that particular
system. That is, it is defined
as the number of ways the
system can be assembled
from its collection of atoms.
Boltzmann created the field of Statistical Mechanics, a tool that where the properties of
macroscopic bodies are predicted by the statistical behaviour of their microscopic parts.
Black Body Radiation
Max Planck
When an object is heated it
releases energy in the form
of a broad spectrum of
electromagnetic waves.
A black body is an ideal body which allows the whole of the incident
radiation to pass into itself (without reflecting the energy) and absorbs
within itself this whole incident radiation. This propety is valid for
radiation corresponding to all wavelengths and to all angels of incidence.
Therefore, the black body is an ideal absorber and emitter of radaition.
The blackbody will then radiate at a wavelength that is related to its
absolute temperature. One should picture a hot oven with an open door
emitting radiation into its cooler surroundings or, if the surroundings are
hotter, one pictures a cool oven with an open door taking in radiation
from its surroundings. It is the open oven door, which is meant to look
black—and hence absorbs all colours or frequencies. This gives rise to
the term black body.
Black Body Radiation
We noticed that the
dominant
When
wavelength (the
measurements
highest peak on the
were made of the
curve) shifts to a
radiation leaving
lower wavelength
the opening of the
(or higher
oven, it was
frequency) when
discovered that
the temperature
the intensity (or
increases.
brightness) of the
radiation leaving
the oven
corresponded to
the Wavelength of
the radiation.
Black Body Radiation
This means that in
ideal conditions,
the radiation
depends only on
the temperature.
So any substance
(metal, glass,
coal, you, me)
that is at a
temperature of
4227 oC (4500K)
will glow orangeyellow in colour
Black Body Radiation
Hey, Max! Did you notice
the shape of your graph
looks very similar to the
shape of my graph
Fraction of
molecules
Kinetic energy

Then, maybe we can interpret the
electromagnetic waves jostling
around inside the oven statistically
like the way you did with the jostling
molecules inside a container.
Black Body Radiation
Predicted by
Theory
Observed in
Nature
The
theory
predicted
that the
amountinofradiated
But the
predicted
continual
increase
radiation
emitted
in a given
frequency
range
energy with
frequency
(dubbed
the "ultraviolet
should
be proportional
to the number of modes
catastrophe")
did not happen.
in that range.
When my colleagues tried this they came up
with an intensity equation, that agreed well
at the large wavelengths (low frequencies) ,
but at shorter wavelengths the intensity
went off the scale so that it predicted an
infinite intensity at the ultraviolet or higher
knew
region ofNature
the spectrum.
better!
Black Body Radiation
Yes, the model
You are right, the amount of
says if you
radiation predicted by the
opened
the oven
What
went
theory would be unlimited,
towrong?
look inside you
and as the temperature rose
would be
the intensity would even
vapourized buy a
increase. “Is it hot in here?”
UV blast.
The Rayleigh-Jeans model
applied the statistical model to
waves, rather than the
particles that were used in
Maxwell’s model. You can fit a
infinite number of waves at
higher and
frequencymodel
The higher
Rayleigh-Jeans
into a container.
Thisnot
lead
to the
certainly did
match
the model being
used
data.
incorrectly.
Black Body Radiation
Planck introduced the idea of electric
oscillators inside the cavity along with the
notion that all possible frequencies being
present.
Planck compared the
data with another
theory predicted by
his friend Wien (which
better matched the
data) and tried to
consolidated them
both.
This blackbody problem intrigued
also expected
the collected
average by
me. II looked
at the data
frequency
to increase
when
numerous
scientists.
I came
tothe
the
temperature
thewhat
ovenever
increased
conclusion of
that
the
as the hotter
walls
proper
formula
is, itcaused
should these
only
electric temperature
oscillators toand
vibrate
faster
contain
radiation
and faster (or
untilwavelength),
thermal equilibrium
frequency
plus a
wasconstant
achieved.
universal
or two.
Black Body Radiation
Then what
happened
I looked at Wien’s
formula and included
the requirement of the
inclusion of all
frequencies.
Eureka
Black Body Radiation
c1 and c2 where
constants chosen by
Planck to make the
equation fit the
experiments.
c1 f 5
E
e
  c2 f 
 T 1


To calculate the probabilities at various
arrangements, Planck divided the energy of
the oscillators into finite chunks.
Energy is
Discontinuous
Planck ended up
rejecting a classical
approach and
followed Boltzmann's
statistical approach .
So the total energy was written as E=ne
(where e is arbitrarily small amount of
energy) in which e=hf where h is some
small constant and f is the frequency.
The graph fits the experimental data
That’s
it. The
energy
exactly. I don’t
know
whatsmall
it means,
But,
why doe
packets
DID
NOT
GO
but boy am I happy. this TO
work?
ZERO, They came in quantum
chunks given by e=hf
h  0.000 000 000 000 000 000 000 000 006 626
Black Body Radiation
n=6
n=5
E=nhf
n=4
n=3
n=2
n=1
1
2
3
4
5
6
7
Frequency
8
9
10
11
There
are
modes
lowerfailed
frequencies
We
can
now
see
whywhatsoever
theavailable
classical
theI
I had
nomore
reason
toattheory
propose
this,inbut
to
absorb
or
thepossible
correct
energy
packet.
This
way
high
frequency
region
of the black
curve.
accepted
it. emit
It is
not
for anbody
absorber
toHere
absorb
we
don’tenergy
turn
into
briquette
looking
at
the
quanta
so large
that
onlywhen
a few
energy
or energy
emit
in aare
a charcoal
continuous
range,
but
rather
it must
an the
open
fire. chunk
modes
areor
excited,
radiation
emitted
drops
to
absorb
emit inthus
discontinuous
called
Quanta
zero and no ultraviolet catastrophe occurs.
Blackbody Radiation:
First evidence for Q.M.
Max Planck found he could explain these curves if he
assumed that electromagnetic energy was radiated in discrete
chunks, rather than continuously.
The “quanta” of electromagnetic energy is called the photon.
Energy carried by a single photon is
E = hf = hc/
Planck’s constant: h = 6.626 X 10-34 Joule sec
E = nhf,
n=1, 2, 3, 4
Questions
A series of light bulbs are glowing red, yellow, and blue.
Which bulb emits photons with the most energy?
Blue! Lowest wavelength is highest energy.
The least energy?
E = hf = hc/
Red! Highest wavelength is lowest energy.
Which is hotter?
(1) stove burner glowing red
(2) stove burner glowing orange
Hotter stove emits higher-energy photons
(shorter wavelength = orange)
Colored Light
Visible Light
Which coloured bulb’s filament is hottest?
1) Red
2) Green
3) Blue
4) Same
max
Coloured bulbs are identical on the inside – the glass is
tinted to absorb all of the light, except the color you see.
Photon
A red and green laser are each rated at
2.5mW. Which one produces more
photons/second?
1) Red
2) Green
3) Same
# photons Energy/second
Power
Power



second
Energy/photon Energy/photon
hf
Red light has less energy/photon so if they both
have the same total energy, red has to have more
photons!
Wein‘s Law
max 
b
T
Wein Displacement Law
- It tells us as we heat an object up, its
color changes from red to orange to
white hot.
- You can use this to calculate the
temperature of stars.
The surface temperature of the Sun is
5778 K, this temperature
corresponds to a peak emission =
502 nm = about 5000 Å.
Wien’s Displacement Law
(nice to know)

To calculate the peak wavelength
produced at any particular temperature,
use Wien’s Displacement Law:
T · peak = 0.2898*10-2 m·K
temperature in Kelvin!
The Wave – Particle Duality
OR
Light Waves
Until about 1900, the classical wave theory of light
described
most observed phenomenon.
Light waves:
Characterized by:
 Amplitude (A)
 Frequency (n)
 Wavelength ()
Energy of wave is a A2
Waves or Particles ?
Physical Objects:
Ball, Car, cow, or point like objects called particles.
They can be located at a location at a given time.
They can be at rest, moving or accelerating.
Falling Ball
Ground level
Waves or Particles ?
Common types of waves:
Ripples, surf, ocean waves, sound waves, radio waves.
Need to see crests and troughs to define them.
Waves are oscillations in space and time.
Direction of travel, velocity
Up-down
oscillations
Wavelength ,frequency, velocity and amplitude defines waves
Particles and Waves: Basic difference in behaviour
When particles collide they cannot pass through each other !
They can bounce or they can shatter
Waves and Particles Basic difference:
Waves can pass through each other !
As they pass through each other they can enhance or cancel
each other
Later they regain their original form !
And then there was a
problem…
In the early 20th century, several effects were
observed which could not be understood using the
wave theory of light.
Two of the more influential observations were:
1) The Photo-Electric Effect
2) The Compton Effect
Photoelectric Effect
 Electrons are attracted to the (positively charged) nucleus by the
electrical force
 In metals, the outermost electrons are not tightly bound, and can
be easily “liberated” from the shackles of its atom.
 It just takes sufficient energy…
Classically, we increase the energ
of an EM wave by increasing the
intensity (e.g. brightness)
Energy a A2
But this doesn’t work ??
PhotoElectric Effect
 An alternate view is that light is acting like a particle
 The light particle must have sufficient energy to “free” the
electron from the atom.
 Increasing the Intensity is simply increasing the number
of light particles, but its NOT increasing the energy of each
one!
 Increasing the Intensity does diddly-squat!
 However, if the energy of these “light particle” is related to their
frequency, this would explain why higher frequency light can
knock the electrons out of their atoms, but low frequency light
cannot…
Nobel Trivia
For which work did Einstein receive the Nobel Prize?
1) Special Relativity
E = mc2
2) General Relativity Gravity bends Light
3) Photoelectric Effect Photons
4) Einstein didn’t receive a Nobel prize.
Photoelectric Effect
Light shining on a metal can “knock”
electrons out of atoms.
 Light must provide energy to overcome
Coulomb attraction of electron to nucleus

The Apparatus



When the emission of photoelectrons from the cathode occurs, they
travel across the vacuum tube toward the anode, due to the applied
potential. Even when the variable potential is dropped to zero, the
current does not drop to zero, because the kinetic energy of the
electrons is still adequate enough to allow some to cross the gas (thus
creating a current).
If we make the variable source of electrical potential negative then this
has the effect of reducing the electron flow. If the anode is made more
negative, relative to the cathode, a potential difference, the cutoff
potential, V0, is reached when the electrons are all turned back.
The cutoff potential corresponds to the maximum kinetic energy of the
photoelectrons. They do not have the KE to make it across the gap.
Classical physics prediction

Electrons can be emitted regardless of the incident frequency, though it
will take longer time for smaller incident wave amplitude.

There should be a time delay between the wave illumination and the
emission of electrons.

The higher the wave intensity, the higher electron energy, and thus the
higher the stopping voltage.
1

f
Modern physics explanation

The electromagnetic wave consists of many lumped energy particles
called photons.

The energy of each individual photon is given by the Joule
E  hf
Modern physics explanation

If N is the total number of photons incident
during time interval T, then the total incident
optical energy in Joules is: E  Nhf

The incident energy per second (power) is given
by: P  N hf Watt = J/Sec.
T

n=N/T is the number of incident photons per
second.
Modern physics explanation



Interaction (absorption / emission) between the
electromagnetic wave and matter occurs through
annihilation/creation of a quantized energy (photon).
In the photoelectric effect, each single absorbed photon
gives its total energy (hf) to one single electron.
This energy is used by the electron to:
 Overcome the attraction force of the material.
 Gain kinetic energy when freed from the material.
Modern physics explanation


Work function (): It is the minimum energy required by an
electron to be free from the attraction force of the metal
ions.
Some of the electrons may need more energy than the
work function to be freed.
Total Energy
+ve
Zero
-ve

The most
energetic
electrons in
the material
Modern physics explanation
Total Energy
+ve
Zero
-ve
hf

hf
The most
energetic
electrons in
the material
Modern physics explanation
The most energetic electron
outside the material
Total Energy
+ve
hf
1
2
mvmax
2
hf
Zero
-ve

Modern physics explanation

The electrons that need only the work function to
be freed, will have the greatest kinetic energy
outside the metal.
1 2
hf  mvmax  
2

The electrons requiring higher energy to be
freed, will have lower kinetic energy.
Modern physics explanation

Thus, there is a minimum required photon
energy (hfo) to overcome the work function of the
material (note f0 is called the cutoff frequency).
hfo  

If the incident photon energy is less than the
work function, the electron will not be freed from
the surface, and no photoelectric effect will be
observed.
If hf<
=
If f< fo
No photoelectric
current
Modern physics explanation

The most energetic electrons are stopped by the reverse
biased stopping potential Vo.
Kmax  hf  
1
2
mv max
 eVo
2
Modern physics explanation
1
2
hf  mvmax

2
hf  eVo  hf o
eVo  h f  f o 
Slope = h/e
h
Vo   f  f o 
e


The stopping potential doesn’t depend on the incident
light intensity.
The stopping potential depends on the incident
frequency.
Photoelectric Equation



Since the cutoff potential is related to the maximum
kinetic energy with which the photoelectrons are
emitted: for a photoelectron of charge e and kinetic
energy Ek, and retarding potential V0. Then we have
(loss is KE = gain in PE) : Ek=eV0.
Ephoton(hf)=Φ+Ek (Φ, the work function, is energy
with which the electron is bound to the surface, Ek is
the kinetic energy of the ejected photoelectron)
Ek=hf-Φ : This tells us that if f is small such that
hf=Φ, no electrons will be ejected.
Threshold Frequency

Photoelectrons are emitted from the
photoelectric surface when the incident light
is above a certain frequency f0, called the
threshold frequency. Above the threshold
frequency, the more intense the light, the
greater the current of photoelectrons
Threshold frequency

The intensity (brightness) of the light
has no effect on the threshold
frequency. No matter how intense the
incident light, if it is below the threshold
frequency, not a single photoelectron is
emitted.
Photoelectric Effect Summary
Each metal has “Work Function” (Φ) which is
the minimum energy needed to free electron
from atom.
 Light comes in packets called Photons



h=6.626 X 10-34 Joule sec
Maximum kinetic energy of released
electrons


E=hf
K.E. = hf – Φ
Photoelectrons are emitted from the
photoelectric surface when the incident light
is above a certain frequency f0, called the
threshold frequency.
Photoelectric Effect (Summary)
“Classical” Method
Increase energy by
increasing amplitude
electrons
emitted ?
No
No
No
No
What if we try this ?
Vary wavelength, fixed amplitude
electrons
emitted ?
No
Yes, with
low KE
Yes, with
high KE
No electrons were emitted until the frequency of the
light exceeded
Another
forfrom
a critical frequency, at which point electrons were symbol
emitted
the surface!
(Recall: small   large n) frequency
Photo-Electric Effect (Summary)
 In this “quantum-mechanical” picture, the energy of the light particle
(photon) must overcome the binding energy (work function, Φ) of the
electron to the nucleus.
 If the energy of the photon does exceed the binding energy, the
electron is emitted with a KE = Ephoton – Ebinding.
 The energy of the photon is given by E=hn, where the constant h =
6.6x10-34 [J s] is Planck’s constant.
“Light particle”
Before Collision
After Collision
Summary
If light is under your control: You can set the frequency (wavelength,
colour) and intensity. Your apparatus can count any ejected
electrons. You create a higher potential relative to the metal plate,
then the ejected electrons will be pulled into the collector and forced
into the ammeter circuit. If you are interested in the energy of the
ejected electrons, you would make the potential of the collector for
and more negative with respect to the surface and eventually you will
reach a voltage level where the ejected electrons can no longer
reach the collector. This potential is called the Stopping potential, Vo.
The maximum kinetic energy of the ejected electrons will then be:
KEelectron  qV0
By the definition of the eV, the Stopping Potential expressed in volts
will have the same numerical value as the electron energy expressed
in eV. That is a Stopping Potential of 2.7 V implies a maximum
electron energy of 2.7 eV
Summary
How does this explain the photoelectric effect? For our metal with 2.7
eV work function, then a single photon would need an energy of 2.7
eV to eject an electron. If you used red light (650 nm), then the
photons in the beam would have energy
hc  6.63 10  3 10 
 hf 

 3.06 10
34
E photon
8
19

650 10
9
 1.91eV
1eV=1.60x10-19J
These photons will be absorbed, but they do not have enough energy
to eject electrons.
Often the photoelectric equation is illustrated on a graph of KE vs frequency. On this
graph, the slope ALWAYS equals Planck's constant, 6.63 x 10-34 J sec. It NEVER changes.
All lines on this type of graph will be parallel, only differing in their y-axis intercept (-f) and
their x-axis intercept (the threshold frequency).
The threshold frequency is the lowest frequency, or longest wavelength, that permits
photoelectrons to be ejected from the surface. At this frequency the photoelectrons
have no extra KE (KE = 0) resulting in
Energy (eV)
Slope= Planck’s constant, h
Curve for material
1
0 = hf – Φ
hf =Φ
Ephoton =Φ
Curve for material
2
fo (material 1)
Φ (material 1)
fo (material 2)
Frequency
(Hz)
Φ (material 2)
Note that red light has such a low frequency (energy) that it will never eject
photoelectrons - that is, the energy of a red photon is less than the work function of
Review
IfThe
suitable
light
is
allowed
to
fall
onwhich
plate
'P',
it will
photo
electrons
as
shown
Minimum
amount
energy
is
necessary
toout
start
photo
electric
emission
The
negative
potential
ofofthe
plate
'C'
at which
the give
photo
electric
current
becomes
in
The
photo Potential
electrons
are
attracted
by the
collector
'C' connected
thethe
isthe
called
Work
Function.
If the amount
of potential.
energy
of Stopping
incident
radiation
isisless
than
zero
isfigure.
called
Stopping
or
cut-off
potential
that to
value
+ve
terminal
of of
a battery.
The
glass
tube
is
evacuated.
When
the sufficient
collector
'C'halt
is kept
function
metal,
no
photo
electrons
are
emitted.
Threshold
frequency
isdifference
defined
asbetween
the
minimum
frequency
of
light which
can
ofwork
retarding
potential
two
plates
which
is incident
just
to
the
atIt+ve
potential,
the
photo
electrons
attracted
by
itable
and
aΦ=hf
current
flows in with
the
is energetic
denoted
by
Φ.emission
Work
function
ofare
a material
is just
given
by to
cause
photo
electric
i.e.emitted.
this
frequency
is
eject
most
photo
electrons
0. electrons
circuit
is additional
indicated
by
the
galvanometer.
isdenoted
awhich
property
of material.
Different
materialsby
have
out
them
energy.
It is denoted
f0. different values of work function.
It Itisgiving
by
"Vo"
Question

What happens to the rate electrons are emitted
when increase the brightness?


more photons/sec so more electrons are emitted.
Rate goes up.
What happens to max kinetic energy when
increase brightness?

no change: each photon carries the same energy as
long as we don’t change the color of the light
Photoelectric Effect: Light Frequency

What happens to rate electrons are emitted
when increase the frequency of the light?


as long the number of photons/sec doesn’t change,
the rate won’t change.
What happens to max kinetic energy when
increase the frequency of the light?

each photon carries more energy, so each electron
receives more energy.
Question
Which drawing of the atom is more correct?
This is a drawing of an electron’s p-orbital probability
distribution. At which location is the electron most
likely to exist?
1
2
3
Question
You observe that for a certain metal surface illuminated with
decreasing wavelengths of light, electrons are first ejected
when the light has a wavelength of 550 nm.
a) Determine the work function for the material.
b) Determine the Threshold Potential when light of
wavelength 400 nm is incident on the surface
Question
You observe that for a certain metal surface illuminated with
decreasing wavelengths of light, electrons are first ejected
when the light has a wavelength of 550 nm.
a) Determine the work function for the material.

hc


8 m
 6.63 10 J  s   3 10 s 

550 109 m
 3.62 1019 J
It is quicker is we
use hc=1240eV nm
34
 2.25eV

hc

1240eV  nm

550nm
 2.25eV
Question
You observe that for a certain metal surface illuminated with
decreasing wavelengths of light, electrons are first ejected
when the light has a wavelength of 550 nm.
b) Determine the Threshold Potential when light of
wavelength 400 nm is incident on the surface
KE  E photons  

hc


1240eV  nm
 2.25eV
400nm
 0.85eV

Question
Suppose you find that the electric potential needed to shut
down a photoelectric current is 3 volts. What is the maximum
kinetic energy of the photoelectrons.
The given potential is the stopping potential V0
U  qVo
 1.6  1019 C   3V 
 4.8 1019 J
 3eV
This is the maximum kinetic energy of the photoelectron
Question
If the work function of the material is known to be 2eV, what is
the cut-off frequency of the photons for this material.
The cutt-off frequency is the frequency above which
electrons can be freed from the material. That is, the
frequency of radiation whose energy is equal to the work
function
E  hf c  
fc 

E  hf c  

fc 
h
2eV
4.14 1015 eV s
 4.83 1014 Hz
or

h
2 1.6 1019 J

6.63 1034 J  s
 4.83 1014 Hz
So is light a
wave or a
particle ?
On macroscopic scales, we can treat a large number of photo
as a wave.
When dealing with subatomic phenomenon, we are often deal
with a single photon, or a few. In this case, you cannot use
the wave description of light. It doesn’t work !
Is Light a Wave or a Particle?

Wave
Electric and Magnetic fields act like waves
 Superposition, Interference and Diffraction


Particle
Photons
 Collision with electrons in photo-electric effect

Both Particle and Wave !
Are Electrons Particles or Waves?
Particles, definitely particles.
 You can “see them”.
 You can “bounce” things off them.
 You can put them on an electroscope.


How would know if electron was a wave?
Look for interference!
Young’s Double Slit w/ electron
d
Source of
monoenergetic
electrons
L
2 slitsseparated
by d
Screen a distance
L from slits
Electrons are Waves?

Electrons produce interference
pattern just like light waves.
Need electrons to go through both slits.
 What if we send 1 electron at a time?
 Does a single electron go through both
slits?

Electrons are Particles

If we shine a bright light, we can ‘see’
which hole the electron goes through.
(1) Both Slits
(2) Only 1 Slit
But now the interference is gone!
Electrons are Particles and Waves!

Depending on the experiment electron
can behave like



wave (interference)
particle (localized mass and charge)
If we don’t look, electron goes through
both slits. If we do look it chooses 1.
Electrons are Particles and Waves!

Depending on the experiment electron
can behave like



wave (interference)
particle (localized mass and charge)
If we don’t look, electron goes through
both slits. If we do look it chooses 1.
I’m not kidding it’s true!
Schroedinger’s Cat

Place cat in box with some poison. If we
don’t look at the cat it will be both dead
and alive!
Here
Kitty, Kitty!
Poison
Momentum of a Photon
Compton found that the
conservation of
momentum did hold for
X-ray scattering collisions
at an angle (Compton
effect)
p  mv
E
p   2 v
c 
E

c
hf

c
hf

f
h


The Compton Effect
In 1924, A. H. Compton performed an experiment
where X-rays impinged on matter, and he measured
the scattered radiation.
Louis de Broglie
Incident X-ray
wavelength
1
M
A
T
T
E
R
Scattered X-ray
wavelength
2
2 > 1
e
Electron comes flying out
Problem: According to the wave picture of light, the incident X-ray gives up
energy to the electron, and emerges with a lower energy (ie., the amplitude
is lower), but must have 21.
Quantum Picture to the Rescue
If we treat the X-ray as a particle with zero mass, and momentum p = E / c,
everything works !
Incident X-ray
p1 = h / 1
Electron
initially at
rest
Scattered X-ray
p2 = h / 2
2 > 1
e
e
pe
Compton found that if the photon was treated like a particle with
mometum p=E/c, he could fully account for the energy & momentum
(direction also) of the scattered electron and photon! Just as if 2 billiard
balls colliding!
Compton Scattering (nice to know)

Compton assumed the
photons acted like other
particles in collisions

Energy and momentum were
conserved

The shift in wavelength is
h
D    o 
(1  cos  )
me c
Compton wavelength
DeBroglie’s Relation
p=h/
 The smaller the wavelength the larger the photon’s momentum!
 The energy of a photon is simply related to the momentum by:
E = pc
(or,
p=E/c )
 The wavelength is related to the momentum by:  = h/p
 The photon has momentum, and its momentum is given by simply p = h / 
.
Quantum Summary

Particles act as waves and waves act as
particles

Physics is NOT deterministic

Observations affect the experiment
Four Quantum
Paradoxes
Paradox 1 (non-locality):
Einstein’s Bubble
Situation: A photon is emitted from an isotropic source.
Paradox 1 (non-locality):
Einstein’s Bubble
Situation: A photon is emitted from an isotropic source.
Its spherical wave function Y expands like an inflating
bubble.
Paradox 1 (non-locality):
Einstein’s Bubble
Situation: A photon is emitted from an isotropic source.
Its spherical wave function Y expands like an inflating
bubble.
Question (Albert Einstein):
If a photon is detected at Detector A, how does the
photon’s wave function Y at the location of Detectors
B & C know that it should vanish?
Paradox 1 (non-locality):
Einstein’s Bubble
It is as if one throws a beer bottle into
Lake Ontario. It disappears, and its
quantum ripples spread all over the
Atlantic.
Then in Copenhagen, the beer bottle
suddenly jumps onto the dock, and the
ripples disappear everywhere else.
That’s what quantum mechanics says
happens to electrons and photons when
they move from place to place.
Paradox 2 (Y collapse):
Schrödinger’s Cat
Experiment: A cat is placed in a sealed box containing a device that has a 50%
chance of killing the cat.
Question 1: What is the wave function of the cat just before the box is opened?
(Y 12 dead + 12 alive ?)
When does the wave function collapse?
Question 2: If we observe Schrödinger, what is his wave function during the
experiment? When does it collapse?
The question is, when
and how does the
wave function
collapse.
•What event collapses
it?
•How does the
collapse spread to
remote locations?
Paradox 3 (wave vs. particle):
Wheeler’s Delayed Choice
A source emits one photon.
Its wave function passes
through slits 1 and 2, making
*
interference beyond the slits.
*
The observer can choose to either:
(a) measure the interference pattern at
plane s1, requiring that the photon travels
through both slits.
or
(b) measure at plane s2 which slit image it
appears in, indicating that
it has passed only through slit 2.
The observer waits until
after the photon has
passed the slits to decide
which measurement to
do.
Paradox 3 (wave vs. particle):
Wheeler’s Delayed Choice
Thus, the photon does not
decide if it is a particle or a
wave until after it passes
the slits, even though a particle
must pass through only one slit and a wave must pass
through both slits.
Apparently the measurement choice determines
whether the photon is a particle or a wave retroactively!
Paradox 4 (non-locality):
EPR Experiments
Malus and Furry
An EPR (einstein Poldalsky Rosen)
Experiment measures the correlated
polarizations of a pair
of entangled photons, obeying
Malus’ Law [P(rel) = Cos2rel]
Paradox 4 (non-locality):
EPR Experiments
Malus and Furry
An EPR Experiment measures the
correlated polarizations of a pair
of entangled photons, obeying
Malus’ Law [P(rel) = Cos2rel]
The measurement gives the same result
as if both filters were in the same arm.
Paradox 4 (non-locality):
EPR Experiments
Malus and Furry
An EPR Experiment measures the
correlated polarizations of a pair
of entangled photons, obeying
Malus’ Law [P(rel) = Cos2rel]
The measurement gives the same result
as if both filters were in the same arm.
Furry proposed to place both photons in
the same random polarization state.
This gives a different and weaker
correlation.
Paradox 4 (non-locality):
EPR Experiments
Malus and Furry
Apparently, the measurement on the right
side of the apparatus causes (in some
sense of the word cause) the photon on
the left side to be in the same quantum
mechanical state, and this does not
happen until well after they have left
the source.
This EPR “influence across space time”
works even if the measurements are
light years apart.
Could that be used for FTL signaling?
Sorry, SF fans, the answer is No!
Four
Interpretations
of Quantum
Mechanics
The Collapse
Interpretation
Scientists who subscribe to the Collapse interpretation make a
choice. They believe that when you accept the electron’s wave
nature, you must give up on the electron’s particle nature.
In this interpretation, the electron leaves the source as a
particle that is governed by one set of laws, but then
“expands” into a spread-out wave as it passes through the
slits. The electron is now governed by new laws. However,
before we can measure this wavy, spread-out quantum
electron it “collapses” back into a particle and arrives at only
one of the many possible places on the screen.
The consequence of choosing the Collapse interpretation line
of thinking is that you must accept that an electron physically
changes from particle to wave and back again. These two
realities, including the laws that describe them, alternate
uncontrollably
The Pilot Wave Interpretation
The Pilot Wave interpretation avoids this unexplained collapse altogether.
Scientists who subscribe to this interpretation choose to believe that the
electron always exists as a classical particle and is only ever governed by one
kind of physical law, for both the familiar classical as well as quantum
phenomena. However, to account for the electron’s wave behaviour this
description requires the introduction of an invisible guiding wave.
In this interpretation, wave-particle duality is explained by assuming that
electrons are real particles all of the time, and are guided by an invisible
wave. The electron’s wave nature is attributed to this abstract wave, called a
Pilot Wave, which tells the electron how to move. To obtain the interference
pattern in the double-slit experiment, this wave must be everywhere and know
about everything in the universe, including what conditions will exist in the
future. For example, it knows if one or two slits are open, or if a detector is
hiding behind the slits.
The Pilot Wave interpretation embodies all of the quantum behaviour,
including all the interactions between classical objects like the electron, the
two-slit barrier, and the measuring devices. In contrast to the Collapse
interpretation where the collapsing electron wave was considered real, in the
Pilot Wave interpretation the wave is an abstract mathematical tool. This
interpretation has a consequence. The Pilot Wave interpretation, which was
invented to deal with an electron as a real physical object, suffers the fate of
The Many-Worlds Interpretation
Supporters of the Many Worlds interpretation, similar to the Pilot Wave
idea, choose to accept that electrons are classical particles. Then they go
even further, demanding that all elements of the theory must correspond to
real objects—unlike the collapsing electron or the Pilot Wave. Supporters
insist on only measurable, physical objects within the world. This world is
constantly splitting into many copies of itself.
When electrons demonstrate wave behaviour they exist in a superposition
of many different states. To Many Worlds supporters, who maintain the idea
of an electron as a classical particle, a parallel universe must exist for each
of the electron’s possible states. When the electron reaches the slits, it has
to choose which slit to go through. At that moment, the entire universe
splits into two. In one universe, the electron passes through the left slit as a
real particle. In the other universe it passes through the right slit as a real
particle. The consequence of accepting the Many Worlds interpretation,
with many quantum particles constantly facing similar choices, is the
requirement that our universe must be constantly splitting into an almost
infinite number of parallel universes, each having its own copy of every one
of us
The Copenhagen Interpretation
Advocates of the Copenhagen interpretation choose to limit their discussion directly
to the experiment and to the measurements on physical objects. Questions are
restricted to what can be seen and to what we actually do. They try to think about
experiments in a very honest way, without invoking extra theoretical ideas like the
on-off switching of the Collapse idea, or the guidance supplied by the invisible Pilot
Wave, or the proposed splitting into Many Worlds.
It is tempting to come up with mental pictures about what is happening that go
beyond the results of an experiment, and to try to interpret what is happening by
means of those hidden theoretical mechanisms. The previous interpretations
attributed the mysterious wave–particle duality to imaginative mathematics. In the
Copenhagen interpretation much of this mystery is attributed to what happens when
an experimenter enters the lab and interacts with the quantum mechanical system.
With the Copenhagen perspective, the mathematics only deals with the
experimenter’s information about measurement interactions with the quantum
mechanical system.
The consequence of accepting the Copenhagen interpretation is a fundamental
restriction on how much you can read into experimental results. We know that
electrons are particles when they are fired from the source, and we know that they
are particles when they hit the screen. What happens to electrons in the middle,
what they are “doing”, or what they really “are” is not possible to know. In the
Copenhagen interpretation these are unfounded questions. We may call an electron
a wave or a particle, but ultimately those names are no more than suitable models.
Let’s Compare