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Quantum Qualities
Quantum Qualities
frequency, wavelength and momentum of
photons and electrons
1
Quantum Qualities
Quick Trip Way Down Memory Lane
In 1833, Michael Faraday observed evidence of electrons
when he observed a gas discharge glow in an
evacuated container. The container was a sealed
glass tube with two electrodes.
In 1858, Julius Plucker proved the glow was charged.
The glow cast a shadow and responded to a
magnetic field. Casting a shadow meant the
glow was comprised of rays, and responding
to a magnet meant the glow was charged.
In 1876, Eugen Goldstein named the glow “cathode rays”.
2
Quantum Qualities
Quick Trip Way Down Memory Lane
In 1883, Thomas Alva Edison observed that a sealed light bulb
deposited carbon inside the bulb and modestly called
it the Edison Effect.
In 1885, Johann Balmer, a Swiss schoolteacher developed an
equation that describes the photographs of emission
light spectra of the elements (rainbow light patterns
with dark spaces among the colors).
In 1896, Jean-Baptist Perrin captured cathode rays and
measured their charge, proving that the rays
carry charge.
In 1890’s, Max Planck used spectroscopic data to create a
blackbody radiation law.
3
Quantum Qualities
Quick Trip Way Down Memory Lane
In 1897, J. J. Thompson proved that:
• cathode rays were particles.
Hear J. J. Thompson
describe the relative size of
an electron.
• the charged particles emitted by a
heated cathode were the same as
cathode rays.
• cathode rays acted as charged particles
in both electric and magnetic fields.
• Later he measured the charge to mass
ratio of the charged particles, later
determined to be electrons.
From the soundtrack of the film,
Atomic Physics Copyright J.
Arthur Rank Organization, Ltd.,
1948,
4
Quantum Qualities
The 19th Century Version of a Diffraction Experiment
It was harder back then to get Edison’s bulbs to glow
and certainly harder to keep them from burning out.
But, it was still possible to direct a light beam to
a prism and then see the diffracted light on a
screen on the other side of the grating.
5
Quantum Qualities
Today’s Version of a 19th Century Diffraction Experiment
It was harder back then to get Edison’s bulbs (discharge tubes)
to glow and certainly harder to keep them from burning out.
-1000
0
+1000
On
Voltage
Off
l1
Current
Voltage Generator
Diffraction
Grating
Discharge Tube
Filter
l3
Screen
But, it was still possible to direct a light beam (from a
discharge tube) to a prism (grating), and then see the
diffracted light on a screen on the other side of the grating.
6
Quantum Qualities
Developing a Model: Matching Observations to Theory
Johann Balmer found a mathematical series that predicts the
wavelengths of the spectra lines when hydrogen is the gas in the
glowing bulb.
k
l  364.56 2
nm
k 4
2
657 nm
486 nm
410 nm
434 nm
375 nm
He put different integers in the “k” position of the equation and then
noticed that his answers matched the wavelengths of the spectra lines.
7
Unfortunately, Balmer did not
have any idea why the equation
worked or why there were
colored lines in the first place.
Quantum Qualities
Blackbody Radiation
Max Planck performed his “blackbody radiation”
experiment to figure out why the spectra lines occurred.
Light Source
Any object that absorbs all the light
that strikes it is called a blackbody.
8
Quantum Qualities
Blackbody Radiation
l1
l3
Screen
A blackbody will emit
(radiate) light as it heats up
Max Planck used this spectroscopic data
to create a blackbody radiation law.
9
Quantum Qualities
Blackbody Radiation
l1
l3
Screen
• Planck believed that the separation of light into discrete lines
meant that light energy was quantified as discrete energy levels.
• If an unbroken spectrum had been observed, it would have meant
a continuous energy distribution.
• Planck started with Balmer’s result and “quantized” the
photon emissions from various blackbody radiators.
10
Quantum Qualities
Blackbody Radiation
Plank’s experiments produced a description of energy emitted
from the blackbody as a function discrete frequency values.
Frequency can be calculated from the wavelength, l:
f = c/l = (speed of light wave)/(wavelength of light)
Planck’s Radiation Hypothesis states that:
The emitted or absorbed energy was quantized
with the smallest possible value being when n =1
and that:
En = n x h x f
All other permitted values of the energy are integral multiples of h f.
11
Quantum Qualities
Blackbody Radiation
Remember that:
hf is h multiplied by frequency and
has units of energy (Joules).
En = is the dependent variable assigned to the
energy absorbed or emitted
n = an integer
-34
x
h = 6.63
10
Joule·seconds
f = frequency of light associated with the heat energy
En = n x h x f
12
Quantum Qualities
Quick Trip a Shorter Distance Down Memory Lane
In 1905, Albert Einstein extended Planck’s concept to light
and used it to explain the photoelectric effect.
In 1911, Robert Millikan measured the charge of
an electron.
In 1913, Niels Bohr proposed a model for the hydrogen
atom that combined ideas about the nucleus and
electrons that explained the spectra colors and the
Balmer mathematical series as well.
13
Quantum Qualities
Einstein and the Photoelectric Effect (1905)
(Shine a light on a metal and electrons
will escape from the metal surface.)
E escape = hf - hfw
Photoelectric Effect
Light Source
Photons
E escape is the energy an electron has when
it leaves the surface of a metal.
hf
is the energy of the light shining on the metal
hfw
is the energy used to move the electron away from
the metal atom. (The work done to free the
electron from the metal surface.)
Einstein said that when hf is greater than
hfw an electron will leave the metal surface.
14
Quantum Qualities
Millikan Oil Drop Experiment (1911)
(Put a charged oil drop in two force fields and
balance the forces so the oil drop does not move.)
He selected the gravitational field and the electric field.
Gravity will pull
oil drop down.
Fg = mg
Electrode
Electric field can
pull oil drop up.
Fe = -qE
-1000
0
+1000
On
Voltage
Off
Current
Voltage Generator
Remember that:
Fg is the gravity force pulling the charged oil drop down.
Fe is the electric force pulling the charged oil drop up.
m is the mass of the the charged oil drop up.
q is the amount of charge on the oil drop up.
g and E are the gravity and electric field strengths, respectively.
15
Quantum Qualities
Millikan Oil Drop Experiment (1911)
(Put a charged oil drop in two force fields and
balance the forces so the oil drop does not move.)
mg
qelectron   E

1.602x10
An oil drop
with a surface
charge
-19
Coulombs
-1000
0
+1000
On
Electrodes
Voltage
Off
Current
Voltage Generator
• Electric field between electrodes can be adjusted to suspend the drop.
• The mass of the oil drop is known.
• The oil drop floats because the force on the drop in the electric field
exactly counteracted the force on the drop in the gravity field.
16
Quantum Qualities
Photon
E=hxf
Bohr Hydrogen Atom Model (1913)
ΔE = Einitial - Efinal = h × f
h = Planck’s constant
f = frequency of the photon
• Electrons reside in orbits and can absorb specific quantities of energy to
move to another orbit.
• If an electron returns to its original orbit, the specific amount of absorbed
energy could be returned to the world as a photon of light.
• The wavelength and frequency of that light could be calculated using
the equation Balmer developed in 1885.
• The energy of that light could be calculated using the equation Planck
developed in 1900.
17
Quantum Qualities
Particle Wave Duality
• Photons, packets of light with energy = hf, are an example of
mass less particles.
• The particle wave duality of light can be shown by combining
Planck’s Law and Einstein’s hypothesis.
•
Planck stated that an atom can only emit and absorb
energy of specific values.
E = Einitial - Efinal = h  f
•
Einstein stated that the frequency times the wavelength of
a photon is equal to the speed of light.
l  f=c
c
Putting the two equations together yields E = h    .
l
18
Quantum Qualities
Particle Wave Duality
The particle-wave nature of
h  c
photons can be described using E
=p  c
photon = h  f =
l
the work of Planck and Einstein.
This equation models the energy of a photon that is released
when an electron returns from an outer orbit to its original orbit
(closer to the nucleus).
It can be used to relate the energy of a photon to that photon’s:
•Frequency:
Ephoton = h x f
•Wavelength:Ephoton = (h x c)/l
•Momentum:
Ephoton = p x c
19
Quantum Qualities
Quick Trip not so Long Ago
Your Great Grandfather’s Time
By 1920, engineers and scientists understood that:
• photons existed
• if photons ever stopped moving, they would have no mass
(a photon’s rest mass equals zero).
Since photons do move:
• their mass is related to the energy needed for movement as
defined by Einstein’s famous equation!
• their mass is “relativistic” mass, m using E = mc2
• they have momentum, p , like all moving objects with
mass using Pr = mrc
20
Quantum Qualities
Quick Trip not so Long Ago
Your Great Grandfather’s Time
In 1923, Louis De Broglie understood the properties of photons:
• all photons obey Planck’s quantum energy equation:
E
Photon
= n(h)(v)
• all photons move and have a relative mass connected to
the energy, Ephoton, they possess.
• They, like all moving objects with mass, have momentum, pr
pr = (mr ) c
• Some photons can be seen by humans because they have
a wave frequency (color) that is detected by human eyes
and/or human instruments.
21
Quantum Qualities
Quick Trip not so Long Ago
Your Great Grandfather’s Time
In 1923, Louis De Broglie understood the properties of photons:
He also believed that:
• unlike photons, most things have a rest mass.
• all things including photons have wave properties.
Momentum and frequency values with corresponding wavelengths.
Thus, a baseball’s mass is not zero when it is not being thrown
but it does have a wave frequency and a wavelength associated
with it when it is in motion.
h Planck’s constant
l any object =
(m object) (vobject velocity )
22
Quantum Qualities
Quick Trip not so Long Ago
Your Great Grandfather’s Time
In 1923, Louis De Broglie understood the properties of photons:
He also believed that:
• since the mass of a baseball (0.15 kg) is so large, the
wavelength associated with a moving baseball is too
short for humans to detect.
• since the mass of an electron (9.11 x10-31 kg) is so small,
the wavelength associated with a moving electron is long
enough for human instruments to detect.
h Planck’s constant
l any object =
(m object) (vobject velocity )
23
Quantum Qualities
Quick Trip not so Long Ago
Your Great Grandfather’s Time
In 1926, Erwin Schrodinger:
• wondered why De Broglie’s relationship only quantitatively worked
when the moving particles where in a force-free environment.
• developed a general equation that:
• described electrons moving with a wave motion because
they were under the influence of a force environment
generated by the positive charge of the nucleus.
• gives the same momentum and frequency (wavelength)
values that would be obtained by De Broglie if the electron
was moving without the influence of the atom’s nucleus.
Schrodinger’s equation is simply called the “wave equation”.
Unfortunately, it only works really well for the hydrogen atom.
24
Quantum Qualities
The Mechanics of Electron/Atom Behavior
When De Broglie’s ideas were first applied to electron
behavior, the topic was called quantum mechanics.
When Schrodinger’s ideas were applied to electron
behavior, the topic was called wave mechanics.
25
Quantum Qualities
The Mechanics of Electron/Atom Behavior
In 1927, Werner Heisenberg suggested that:
• the difficulties with quantum mechanics and/or wave
mechanics is that there is no way to know both the
position and speed of an electron at the same time.
• a probability perspective turns out to be a better way to
describe electron behavior within an atom.
Heisenberg’s approach is known today as both:
statistical quantum mechanics
and/or
statistical wave mechanics.
26
Quantum Qualities
Quantum Qualities
The major step forward in the understanding of light, electrons
and atoms was the steady development of the ideas that:
• energy comes into and goes out of atoms but it does so
in packets or quanta.
• energy packets, quanta, differ from each other by
multiple integer values of Planck’s constant.
• each packet of energy can be associated with a frequency
(wavelength).
• each packet of energy can interact with an electron
within an atom.
• an electron will change orbits or even leave the
atom if it interacts with the right packet of energy.
27
Quantum Qualities
Equation Summary of our Historical Trip
In 1885, Johann Balmer:
An equation that described the photographs of
light spectra (rainbow light patterns with dark
spaces among the colors).
In 1900, Max Planck:
The quantum equation that describes the
relationship between the energy of a wave and
the frequency of that wave.
In 1905, Albert Einstein:
l  364.56
28
2
k 4
2
nm
Ephoton = h  f
E
An equation that connects light energy to
electrons leaving an atom.
k
= h f - hf
w
escape
Quantum Qualities
Equation Summary of our Historical Trip
In 1911, Robet Millikan:
An equation that related the charge of an
electron to BOTH its mass and the electric field
its suspended in.
qelectron =
-mg
E
In 1913, Niels Bohr:
An equation that explained the electron orbit
quantum nature of hydrogen atom as seen from
the hydrogen absorption spectrum and predicted
the wavelength of the lines in that spectrum.
E = Einitial - E final = h  f
In 1923, Louis De Broglie:
An equation that connects the wavelength of a
moving object to its mass and velocity.
29
l object
h
(mobject) (vvelocity)
Quantum Qualities
Two Example Problems
Objectives:
1. to show the use of quantum equations
2. to illustrate unit manipulations
1.) Calculate the wavelength and the frequency of one photon
of light that has an energy of 3.64 x 10-19 Joules.
2.) For a photon with a wavelength of 700 nm, calculate the
a) frequency, b) the energy, and c) the momentum of the photon.
30
Quantum Qualities
1.) Calculate the wavelength and the frequency of one photon
of light that has an energy of 3.64 x 10-19 Joules.
Knowns:
Energy = 3.64  10-19 J
Planck’s constant = h = 6.63  10-34 J·s
8 m
3.0

10
Speed of light = c =
s
Unknowns: Wavelength = l = nm
Frequency = f = s -1
Equations:
Ephoton = h  f =
h  c
l
31
=p  c
Quantum Qualities
1.) Calculate the wavelength and the frequency of one photon
of light that has an energy of 3.64 x 10-19 Joules.
Unknowns: Wavelength = l = nm
Frequency = f = s -1
h  c
Equations: E
=p  c
photon = h  f =
l
Ephoton =
h  c
l
h  c
l=
Ephoton
Ephoton = h  f
f=
Ephoton
h
32
Quantum Qualities
Wavelength Value Calculation for Example #1
Knowns:
Equations:
Solution:
Energy = 3.64  10-19 J
Planck’s constant = h = 6.63  10-34 J·s
8 m
3.0

10
Speed of light = c =
s
h  c
l=
Ephoton
f=
Ephoton
h
6.63  10 -34 J s  3.0  10 8 m/s
l=
3.64  10 -19 J
The wavelength of light that has 3.64
x 10-19 J of energy is l = 5.464 
33
10 -7 m
Quantum Qualities
Wavelength Unit Conversion for Example #1
Solution:
l = 5.464  10 -7 m
Wavelength is typically reported in units of nanometers.
Units:
Remember:
9
or
1 m = 1  10 nm
l = (5.464  10
-7
m)  (1  10
9
1  109 nm
1m
nm
546 nmnm
) = 546.4
m
Answer:
A photon with an energy of 3.64  10
Joules has a wavelength of 546 nm
34
-19
Quantum Qualities
Frequency Calculation for Example #1
Knowns:
Equations:
Solution:
Energy = 3.64  10-19 J
Planck’s constant = h = 6.63  10-34 J·s
8 m
Speed of light = c = 3.0  10
s
h  c
l=
Ephoton
f=
Ephoton
h
3.64  10-19 J
f=
6.63  10-34 J s
The frequency of light that has 3.64 x
14 1
-19
10 J of energy is f = 5.49  10
s
35
Quantum Qualities
Frequency Unit Conversion for Example #1
Solution:
Units:
f = 5.49  10
14
1
s
Frequency is typically given in units of Hertz.
1
Hertz = Hz =
second
f = 5.49  10
14
1
= 5.49  10 14 Hz
s
Answer: A photon with an energy of 3.64  10-19
Joules has a frequency of 5.49  1014 Hz.
36
Quantum Qualities
Summary for Problem #1
1.) Calculate the wavelength and the frequency of one photon
of light that has an energy of 3.64  10-19 Joules.
Answer:
A photon with an energy of 3.64  10-19
Joules has a wavelength of 546 nm
and a frequency of 5.49  1014 Hz.
?
Is this wavelength of light in the
visible part of the spectrum?
If so, what color is it?
Yes, a 546 nm light wave is in the visible
spectrum and will be green.
37
Quantum Qualities
2.) For a photon with a wavelength of 700 nm calculate the
energy, frequency, and the momentum.
Known:
Wavelength = l = 700 nm
Planck's constant = h = 6.63  10-34 J s
m
Speed of light = c = 3.0  108
s
Unknown:
Equations:
Energy
= Ephoton in J
Frequency = f
in s –1
Momentum = p
in kg-m/s
Ephoton = h  f =
h  c
l
38
=p  c
Quantum Qualities
2.) For a photon with a wavelength of 700 nm calculate the
energy, frequency, and the momentum.
Unknown:
Energy
= Ephoton in J
Frequency = f
in s –1
Momentum = p
in kg-m/s
h  c
Equations: E
=
h

f
=
=p  c
photon
l
Ephoton =
h  c
l
Ephoton = h  f
h  c
l
=p  c
p
f=
Ephoton
h
39
h  c
l c
p=
h
l
Quantum Qualities
Wavelength Unit Conversion
Known:
Unknown:
Wavelength = l = 700 nm
8 m
Speed of light = c = 3.0  10
s
Energy
= Ephoton in J
Frequency = f
Momentum = p
Units:
in s –1
in kg-m/s
Wavelength = l = 700 nm
Wavelength is usually reported in nm and the
speed of light is given in m/s.
(For this problem the wavelength will be converted to meters from
nanometers so the velocity can be used in meters/second.)
1  10 -9 m
700 nm = 700 nm 
= 700  10 -9 m
1 nm
40
Quantum Qualities
Energy Calculation
Known:
Wavelength = l = 700 nm
m
Speed of light = c = 3.0  10
s
Planck's constant = h = 6.63  10-34 J s
8
Unknown:
Equations:
Energy = Ephoton in Joules
Ephoton =
h  c
l
Ephoton =
f=
Ephoton
h  c
l
41
h
p=
h
l
Quantum Qualities
Energy Calculation
Known:
Wavelength = l = 700 nm
m
Speed of light = c = 3.0  10
s
Planck's constant = h = 6.63  10-34 J s
8
Equations:
Solution:
Ephoton =
h  c
l
m
6.63  10
J s  3.0  10
s
Ephoton =
7  10 -7 m
-34
8
Ephoton = 2.8  10 -19 Joules
42
Quantum Qualities
Energy Calculation
Units:
Energy is typically reported in units of Joules.
Answer:
A photon with a wavelength of 700 nm
has an energy of 3  10-19 Joules.
?
Is this wavelength of light in the
visible part of the spectrum?
If so, what color is it?
Yes, a 700 nm light wave is in the visible
spectrum and will be red.
43
Quantum Qualities
Frequency Calculation
Known:
Wavelength = l = 700 nm
Planck's constant = h = 6.63  10-34 J s
8 m
Speed of light = c = 3.0  10
s
Unknown:
Frequency = f
in s –1
Momentum = p
in kg-m/s
Energy
Equations: Ephoton =
=Ephoton in J
h  c
l
f=
Ephoton
h
44
p=
h
l
Quantum Qualities
Frequency Calculation
Known:
Wavelength = l = 700 nm
Planck's constant = h = 6.63  10-34 J s
8 m
Speed of light = c = 3.0  10
s
Equations:
Solution:
f=
f=
c
p=
l
3.0  10 8
7  10 -7
f = 4.3  10 14
m
s
m
1
s
45
h
l
Quantum Qualities
Frequency Calculation
Solution:
Units:
f = 4.3  10
1
s
Frequency is typically reported in units of Hertz.
1
Hertz = Hz =
second
f = 4.3  10 14
Answer:
14
1
= 4.3  10 14 Hz
s
A photon with a wavelength of 700 nm
14
has a frequency of 4  10 Hz.
46
Quantum Qualities
Momentum Calculation
Known:
Wavelength = l = 700 nm
Planck's constant = h = 6.63  10-34 J s
8 m
Speed of light = c = 3.0  10
s
Unknown:
Momentum = p
Energy
in kg-m/s
=Ephoton in J
Frequency = f
Equations:
Ephoton =
h  c
l
in s –1
f=
Ephoton
h
47
p=
h
l
Quantum Qualities
Momentum Calculation
h
Equations:
p=
Solution:
6.63  10 -34 J s
p=
7  10 -7 m
l
p = 9.5  10
note:
1 Joule of
energy
=
-28
This momentum unit is also
the same as the units of
energy divided by velocity.
Js
m
(1 kg) (1 m) 2
2
(1 second)
48
= 1 kg m 2 s
2
Quantum Qualities
Momentum Calculation
Units: There are several popular choices for momentum units,
but the usual units are (mass)(velocity).
1
Js
=1
kg m2 s
m
s2 m
For metric calculations these units would be kg m/s.
1
kg m2 s
s
9.5  10
Answer:
2
-28
m
Js
m
= 1
kg m
s
= 9.5  10
-28
kg m
s
A photon with an wavelength of 700 nm
has a momentum of 1  10-27 kg m/s.
49
Quantum Qualities
Summary for Example Problems 1 and 2
1.) Calculate the wavelength and the frequency of one photon
of light that has an energy of 3.64  10-19 Joules.
Answer: A photon with an energy of 3.64  10-19
Joules has a wavelength of 546 nm
and a frequency of 5.49  1014 Hz.
2.) For a photon with a wavelength of 700 nm calculate the
energy, frequency, energy and the momentum.
Answer: A photon with a wavelength of 700 nm has
an energy of 3  10-19 Joules, and
a frequency of 4  1014 Hz, and
a momentum of 1  10-27 kg m/s.
50
Quantum Qualities
Two De Broglie Wave Equation Examples
Objectives:
1. to show the use of quantum equations
2. to illustrate unit manipulations
De Broglie’s equation connects the
wavelength of a moving object to the
mass and velocity of that moving object.
l object =
h
(mobject )(vvelocity )
3a.) What is the wavelength value associated with a 0.15 kg baseball
moving with a velocity of 30 meters/second?
-31
3b.) What is the wavelength associated with a 9.11 x10
kg electron
moving with a velocity of 1.47 x 10 7 meters/second?
51
Quantum Qualities
Wavelength Calculation
Knowns:
Planck's constant = h = 6.63  10-34 J s
Mass of baseball = m = 1.5  10-1 kg
Mass of electron = m = 9.11  10-31 kg
m
Speed of baseball = v = 3.0  101
s
7 m
Speed of electron = v = 1.4  10
s
Unknowns:
Wavelength of baseball = l b in nm
Wavelength of electron = l e in nm
Equations:
h
l=
(mobject)(vvelocity)
52
Quantum Qualities
Wavelength Calculation
Knowns:
h = 6.63  10-34 J s
 10-1 kg
1 m
vball = 3.0  10
s
h
Equations: l =
(mball)(vball)
melectron = 9.11  10-31 kg
7 m
velectron = 1.4  10
s
h
l electron =
(m e )(v e)
mball = 1.5
b
lb 
le 
6.64 x 10
(1.5 x 10
1
34
kg) (3.0 x 10 m/s )
1
6.64 x 10
(9.11 x 10
31
Js
34
Js
kg) (1.4 x 10 m/s )
7
53
meters
meters
Quantum Qualities
Wavelength Calculation
Equations:
lb 
le 
lb
6.64 x 10
(1.5 x 10
-1
-31
Js
1
kg) (3.0 x 10 m/s)
6.64 x 10
(9.11 x 10
-34
-34
Js
kg) (1.4 x 10
7
m/s)
-34
 1.5 x 10
meters
le 
6.64 x 10
(12.8 x 10
-34
-24
Js
kg m/s)
le 
54
6.64 x 10
12.8
-10
meters
Quantum Qualities
Wavelength Calculation
lb  1.5 x 10 meters
-11
= l e= 5.19 x 10 meters
-34
baseball wavelength =
electron wavelength
3a Answer:
A baseball does travel on a wave path like water
but going 30 meters per second the ball goes
through one complete wave after it has gone 1.5
x10-34 meters.
3b Answer:
An electron does travel on a wave path like water
but going 1.4 x 107 meters per second the electron
goes through one complete wave after it has gone
5.19 x10-11 meters.
55
Quantum Qualities
56