Quantum
Mechanics
• Light has a particle
nature. This is most clearly
shown by the photoelectric
effect.
• Particles have a wave
nature. All of the wave
phenomena we have seen
apply to particles as well.
• Quantum principles
are well understood
and well accepted. But
they are pretty weird.
Why
“Quantum”?
Not Quantized.
Quantized.
The photon model
1240
λ (in nm)
1240
λ ( in nm ) =
E ( in eV)
E (in eV) =
First example of quantization.
Creating X rays
If an electron is accelerated through a 5.0 kV potential
difference, what is the maximum photon energy of the
resulting x ray? What is the wavelength?
One electron. One photon.
Photon Production
A particular species of bioluminescent
copepod (a small marine crustacean,
typically a few mm in length) emits blue
light at a peak wavelength of 490 nm. In a
typical flash lasting 2.4 s, the copepod
emits 1.4 x 1010 photons.
• What power does this correspond to?
• What is the intensity at a distance of
10 m?
hc
λ
h = 6.62 × 10 −34 J ⋅s
Ephoton =
P = ΔE Δt
I = Psource 4π r 2
Can You See a Single Photon?
At the wavelength corresponding to the maximum sensitivity
of the human eye, 510 nm, the limit of sensitivity of the darkadapted eye has been shown to be correspond to a 100 ms
flash of light of total energy 240 eV. (Weaker flashes of light
may be detected, but not reliably.)
a) What is the energy of a single photon at this wavelength?
b) How many photons does the flash contain?
c) If 60% of the incident light is lost to reflection and
absorption by tissues of the eye, how many photons reach
the retina?
The light from the flash covers well over 500 rod cells.
d) So, can you see a single photon?
Quantum
Concept #1:
EM Waves have a
particle nature
The Photoelectric Effect
Light
Window
I
Ammeter
Cathode
Anode
A
f
DV
I
0
f0
There is a threshold frequency.
Above it, electrons are emitted.
Below it, not so much.
Just Checking
In the photoelectric effect experiment, why does red light not
cause the emission of an electron though blue light can?
The photons of
red light don’t
have sufficient
energy to eject an
electron.
Red light contains
fewer photons
than blue, not
enough to eject
electrons.
The electric field
of the red light
oscillates too
slowly to eject an
electron.
The red light
doesn’t penetrate
far enough into
the metal
electrode.
The Photoelectric Effect
Light
Window
I
Intense light
Weak light
Ammeter
Cathode
Anode
A
DV
2Vstop
0
I
DV
Changing the accelerating voltage
changes the current.
But only within certain limits.
Just Checking
In the photoelectric effect experiment, increasing the
accelerating voltage from 3.0 V to 5.0 V does not increase the
current. How can we explain this result?
The resistance of
the tube changes as
well.
28.2
The electrons are
already at their
maximum speed.
The Photoelectric Effect
913
makes28.8
all the
FIGURE
A swimming pool analogy
g pool. Water3.0 V
Increasing the
electrons
reach
the
of
electrons
in
a metal.
m. To remove
voltage doesn’t
anode, so increasing
change the electron
orce of gravity, The minimum energy to remove a
voltage
drop causes
of water no
from the pool is mgh. kinetic energy.
er molecule—change.
e. Removing a
h
.
Water
Removing this drop
etal. To extract
takes more than the
ugh to escape.
minimum energy.
k function of
The
Work Function
e more
energy
work functions How much it “costs” to release an electron.
60 * 10-19 J.)
This varies with the electrode.
e 28.6. When
TABLE 28.1 The work functions
tic energy. An
for some metals
, so it emerges
Element
E0 (eV)
n energy E0 is
Potassium
2.30
ossible kinetic
ns. FIGURE 28.9
and the anode
= 0, there will
e anode, creat-
Sodium
2.75
Aluminum
4.28
Tungsten
4.55
Copper
4.65
Iron
4.70
Gold
5.10
Think About It.
Light
Window
Ammeter
Cathode
Anode
DV
A
I
5.0 eV photons strike an electrode with work function 3.0 eV.
a. What is the kinetic energy of emitted electrons?
b. What potential is needed to reduce the current to zero?
Just Checking.
Monochromatic light shines on the cathode in a
photoelectric effect experiment, causing the emission of
electrons. If the frequency of the light stays the
same but the intensity of the light is increased,
the emitted electrons
will be moving at a
higher speed.
there will be more
electrons emitted.
both A and
B are true.
neither A nor
B are true.
Just Checking.
Monochromatic light shines on the cathode in a
photoelectric effect experiment, causing the emission of
electrons. If the intensity of the light stays the
same but the frequency of the light is increased,
the emitted electrons
will be moving at a
higher speed.
there will be more
electrons emitted.
both A and
B are true.
neither A nor
B are true.
The Details.
Light of wavelength 400 nm illuminates a potassium
electrode (work function 2.3 eV).
a. What is the photon energy?
b. What is the energy of the emitted electron?
c. What is the stopping potential?
Light
Window
Ammeter
Cathode
DV
Anode
A
I
7. Metal surfaces on spacecraft in bright sunlight develop a net
electric charge. Do they develop a negative or a positive charge?
Explain.
8. Metal 1 has a larger work function than metal 2. Both are
What’s The Fizics?
Quantum
Concept #2:
Particles have a
wave nature.
Diffraction
and
Interference
Diffraction
Diffraction
and
Interference
Double Slit Interference Pattern
Viewing
screen
Incident laser beam
Longer wavelength
means bigger spacing.
Grating Interference Pattern
Screen
y
y2
m52
y1
m51
0
m50
2y1
m51
2y2
m52
Grating
u2
u1
Dr between these paths
is exactly 2l (m 5 2).
Appearance
of screen
L
Particles have a Wave Nature
λ=
h
h
=
p mv
De Broglie wavelength for a moving particle
Particle or Wave?
m
λ
Localized.
Wavelength of a squirrel
running at 3 m/s:
1x10-33 m
Smeared out.
Particle or Wave?
In a television set, an electron is accelerated by a
voltage of 150 V.
a. What is the kinetic energy of the electron?
b. What is the speed of the electron?
c. What is the De Broglie wavelength?
Does this matter?
Size of a
hydrogen atom
Orbitals
0.1 nm
Looking Deeper
Electron microscope view of pigment molecule.
Quantum
Concept #3:
The wave nature of
particles leads to
quantization.
Particles have a wave nature. So...
Particle:
L
m
v
Wave:
L
...the possible states are quantized.
The Crux of the Quantum Biscuit
Photons have a particle nature.
Their energy is quantized.
It comes in chunks of a particular size.
Particles have a wave nature. Confining them restricts
them to certain energy states.
The energy of a confined particle is quantized.
It is restricted to certain values.
The wave nature of particles leads to quantized energy levels
for electrons in atoms. Only certain transitions are possible.
Energy
Energy
160 eV
n54
90 eV
n53
n54
160 eV
90 eV
40 eV
n52
40 eV
10 eV
0
n51
10 eV
0
DEsystem 5 |E3 2 E1|
5 80 eV
DEsystem 5 |E1 2 E2|
5 30 eV
n53
n52
n51
Ground
state
Energy levels for a particle in a
0.10-nm-long box
Possible transitions for a
system with these energy levels
2
En =
1 ⎡ hn ⎤
h2 2
=
n
8mL2
2m ⎢⎣ 2L ⎥⎦
n = 1, 2, 3, 4...
What is the maximum photon energy that could be
emitted by the quantum system with the energy level
diagram shown below? The minimum?
The Details.
Light of wavelength 400 nm illuminates a potassium
electrode (work function 2.3 eV).
a. What is the photon energy?
b. What is the energy of the emitted electron?
c. What is the stopping potential?
Light
Window
Ammeter
Cathode
DV
Anode
A
I
Ocean water is most transparent at wavelengths of 470 nm, so
bioluminescent creatures emit light at approximately this wavelength.
Firefly squid use ATP to provide the energy for this reaction. Metabolizing
one molecule of ATP releases 0.32 eV.
How many molecules of ATP must be metabolized to produce one photon
of blue light at 470 nm?
In a photoelectric effect experiment, light of
wavelength 620 nm shines on a cathode with a work
function of 1.8 eV.
• What is the speed of the emitted electron?
• What anode voltage will stop current in the tube?
Particles have a Wave Nature
λ=
h
h
=
p mv
De Broglie wavelength for a moving particle
. Electrons are accelerated from rest through an 8000 V potential
difference. By what factor would their de Broglie wavelength
increase if they were instead accelerated through a 2000 V
potential?
. Can an electron with a de Broglie wavelength of m pass
Electron moving
more slowly:
Wavelength is
longer.
Ratio
reasoning.
K = ΔU e
K = 12 mv 2
λ=
h
h
=
p mv
The wave nature of particles leads to quantization.
L
m
v
L
2
1 ⎡ hn ⎤
h2 2
En =
=
n
8mL2
2m ⎢⎣ 2L ⎥⎦
n = 1, 2, 3, 4...
Allowed energies for particle in a box
The wave nature of particles leads to
quantized energy levels for electrons in atoms.
Only certain transitions are possible.
Energy
Energy
160 eV
n54
90 eV
n53
n54
160 eV
90 eV
40 eV
n52
40 eV
10 eV
0
n51
10 eV
0
DEsystem 5 |E3 2 E1|
5 80 eV
DEsystem 5 |E1 2 E2|
5 30 eV
n53
n52
n51
Ground
state
Energy levels for a particle in a
0.19-nm-long box
Possible transitions for a
system with these energy levels
What energy photons could be emitted by the quantum
system sketched below?
Electrons of the bonds along the chain of carbon atoms in
this dye molecule are shared among the atoms in the
chain, but are repelled by the nitrogen-containing rings at
the end of the chain. What is the longest wavelength of
visible light this molecule will absorb?
0.85 nm
If the length of the chain is increased, how will this affect
the wavelength of the light absorbed by the dye?
Ratio
reasoning.
2
1 ⎡ hn ⎤
h2 2
En =
=
n
8mL2
2m ⎢⎣ 2L ⎥⎦
n = 1, 2, 3, 4...
Changing Scale
The diameter of a typical atomic nucleus is about 10 fm.
(1 fm is 1x10-15 m.)
What is the kinetic energy, in MeV, of a proton with a de
Broglie wavelength of 10 fm?
Heisenberg uncertainty principle
∆ x ∆px Ú
h
4p
Δx
Electrons & Atoms
An electron is
associated with a
particular atom. This
limits it to an
uncertainty in position
of about 1 nm—it’s
somewhere within this
range.
What uncertainty in
speed does this imply?
One Photon, One Electron
66. ||| A silicon solar cell behaves like a battery with a 0.50 V terminal voltage. Suppose that 1.0 W of light of wavelength 600 nm
falls on a solar cell and that 50% of the photons give their energy
to charge carriers, creating a current. What is the solar cell’s
efficiency—that is, what percentage of the energy incident on
the cell is converted to electric energy?
67. |||| What is the kinetic energy in eV of an electron whose de Bro-
r they provide us with a basis for understanding these elusive but most fundaconstituents of nature. This two-sided point of view is called wave–particle
A spherical virus has a diameter of 50 nm. It is contained
over two
hundred
years, scientists
and nonscientists
alike felt
the clockwaves
obey
the
principle
of superposition
and 0.0001
exhibit
interference.
This
inside
a long,
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cell
of length
m.that
niverse
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Newtonian
physics
was
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But
particle dichotomy seemed obvious until physicists encountered irrefutable
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along
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Einstein’s
relativity,
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imply
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ce that light sometimes acts like a particle and, even stranger, that matter
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ly exclusive. Two sound waves can pass through each other and can overlap
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ween particle
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ough
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C
60 sent through a grating will produce a diffraction pattern!
ow us only one face at a time. If we arrange an experiment to measure a
ke property, such as interference, we find photons and electrons acting like
not particles. An experiment to look for particles will find photons and eleccting like
particles, not waves. These two aspects of light and matter are
nance
imaging
mentary to each other, like a two-piece jigsaw puzzle. Neither the wave nor
manent mag- where m = 1.41 * 10-26 J/T is the known value of the proton’s
ticle
model alone provides an adequate picture of light or matter, but taken
t of electrons magnetic moment. FIGURE 28.25 shows the two possible energy
r they provide us with a basis for understanding these elusive but most fundaons also have states. The magnetic moment, like a compass needle, “wants” to
constituents
for magnetic of nature. This two-sided point of view is called wave–particle
align with the field, so that is the lower-energy state.
But a quantum
compass is different.
FIGURE 28.25 Energy levels for a proton in a magnetic field.
over
years, scientists and nonscientists alike felt that the clockaligntwo
withhundred
a
. . . which
correspond of
to reality. But
niverse
NewtonianQuantum
physics was a fundamental
description
on. This of
isn’t
mechanics limits
two possible orientations,
particle
duality, along with
Einstein’s
relativity,
undermines
the
basic assumpntum physics
the proton to two
aligned with or opposite
There
only
f theare
Newtonian
worldview.
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and
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of classical
possible energies
...
the
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Energy
r
inty play key roles—the universe of quantum physics.
B
E2 5 1mB
field
0
eual
thenature
field of a buckyball Treating atomic-level structures involves frequent
ween particle and wave views. 60 carbon atoms can create the molecule
E 5 2mB
med at left, known as C60,1or buckminsterfullerene. The scanning electron
ope image of a C60 molecule shown on the right is a particle-like view of the
−26
µproton carbon
= 1.41×
10clearly
J/T
e with individual
atoms
visible. The C60 molecule, though we
e a picture of it—showing the atoms that make it up—also has a wave nature. A
C60 sent through a grating will produce a diffraction pattern!
• What is the photon energy corresponding to a spin flip
for a proton in a 1.0 T magnetic field?
• What frequency does this correspond to?
What type of EM wave is this?
nance• imaging
manent magt of electrons
ons also have
for magnetic
where m = 1.41 * 10-26 J/T is the known value of the proton’s
magnetic moment. FIGURE 28.25 shows the two possible energy
states. The magnetic moment, like a compass needle, “wants” to
align with the field, so that is the lower-energy state.
Changing
field, changing frequency.
FIGURE 28.25 Energy levels for a proton in a magnetic field.
align with a
on. This isn’t
ntum physics
There are only
orientations—
Quantum
mechanics limits
the proton to two
possible energies . . .
Energy
E2 5 1mB
field
e the field
. . . which correspond to
two possible orientations,
aligned with or opposite
the magnetic field.
r
B
0
E1 5 2mB
µproton = 1.41× 10 −26 J/T
If you increase the field from 1.0 T to 2.0 T, how does this
change the frequency of the rf (radiofrequency) wave
necessary to cause a spin flip?
Quantum Weirdness.
Quantum Weirdness: Non-locality
Two
places at
one time
Which slit did
the electron
go through?
Where is the electron?
From Chapter 12:
vrms =
3kBT
m
Assume 85Rb
m = 85 u
Quantum Weirdness: Superposition
Many
things in
the same
place
Quantum Weirdness: Mixed States
Alive and
dead cats
Schrödinger’s Cat
First, Back to the Rainbow
Primary Colors
Red
Green
Blue
Complementary Colors
Cyan
Not Red
Magenta Yellow
Not Green Not Blue
Fluorescence
A range of wavelengths
can excite electrons to
the upper band.
The electrons fall to
the lower edge of the
upper band.
The electrons then
jump to the lower
band, emitting photons.
Would you expect the absorbed or
the emitted light to have a longer
wavelength?
Relative intensity
Absorption band
Emission band
0
300 400 500 600
Wavelength (nm)