Download in nm 1240 E in eV - Little Shop of Physics

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.
1
The photon model
1240
! (in nm)
1240
! ( in nm ) =
E ( in eV)
E (in eV) =
First example of quantization.
2
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.
3
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
!
h = 6.62 " 10 #34 J $s
Ephoton =
P = !E !t
I = Psource 4" r 2
4
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.)
Problems
837
es
ec-
gh
he
ith
lly
tiarhe
he
as
a
ce
of
his
ter
ny
ogy
a) What is the energy of a single photon at this wavelength?
52. How
| Ourmany
sun’sphotons
5800 K surface
temperature
gives a peak waveb)
does the
flash contain?
length in the middle of the visible spectrum. What is the mini-
c) Ifmum
60%surface
of the temperature
incident light
to reflection
for isa lost
star whose
emissionand
peaks at
absorption
by tissues
of 400
the nm—that
eye, howis,many
reach
some wavelength
less than
in thephotons
ultraviolet?
53. the
||| While
using a dimmer switch to investigate a new type of
retina?
incandescent light bulb, you notice that the light changes both
The light
from the flash covers well over 500 rod cells.
its spectral characteristics and its brightness as the voltage is
increased.
d) So,
can you see a single photon?
a. If the wavelength of maximum intensity decreases from
1800 nm to 1600 nm as the bulb’s voltage is increased, by
how many
does the filament temperature increase?
b. By what factor does the total radiation from the filament
increase due to this temperature change?
54. || The star Sirius is much hotter than the sun, with a peak
wavelength of 290 nm compared to the sun’s 500 nm. It is also
larger, with a diameter 1.7 times that of the sun. By what factor
does the energy emitted by Sirius exceed that of the sun?
55. | The photon energies used in different types of medical x-ray
imaging vary widely, depending upon the application. Single
dental x rays use photons with energies of about 25 keV. The
photon energy used for x-ray microtomography, a process that
allows repeated imaging in single planes at varying depths
within the sample, is 2.5 times greater. What are the wavelengths of the x rays used for these two purposes?
Ratios.
56. | A python can detect thermal radiation with intensity greater
than 0.60 W/m2. A typical human body has a surface area of
1.8 m2, a surface temperature of 30°C, and an emissivity
at infrared wavelengths. What is the maximum distance from which
a python can detect your presence? You can model the human body
as a point source of radiation.
57. | If astronomers look toward any point in outer space, they see
5
6
vity
m2,
5.65
of
ire?
ent?
polarizer with axis
from vertical is inserted between the first
two. What is the transmitted intensity now?
72. ||| A light-emitting diode (LED) connected to a 3.0 V power supply
emits 440 nm blue light. The current in the LED is 10 mA, and the
LED is 60% efficient at converting electric power input into light
power output. How many photons per second does the LED emit?
73. | A 1000 kHz AM radio station broadcasts with a power of
20 kW.you
Howhad
manytophotons
emit
What
pay does the transmitting
Whatantenna
you get
each second?
74. |||| The human body has a surface area of approximately
a
surface temperature of approximately 30°C, and a typical emissivity at infrared wavelengths of
If we make the approximation that all photons are emitted at the wavelength of peak
intensity, how many photons per second does the body emit?
meter
MCAT-Style Passage Problems
Electromagnetic Wave Penetration
Radio waves and microwaves are used in therapy to provide “deep
heating” of tissue because the waves penetrate beneath the surface of
the body and deposit energy. We define the penetration depth as the
7
Quantum
Concept #1:
EM Waves have a
particle nature
16/10/13 9:49 AM
8
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.
9
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.
10
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.
11
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.
The electrons are
already at their
maximum speed.
3.0 V makes all the
electrons reach the
anode, so increasing
voltage causes no
change.
Increasing the
voltage doesn’t
change the electron
kinetic energy.
12
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
13
ode. A further
e and thus does
Think
It.
n Figure
28.7bAbout
FIGURE 28.9 The effect of different
eaving the catha ball hits the
negative anode
ecreases as the
of Figure 28.7b
current ceases.
voltages between
Lightthe anode and cathode.
Window
UV
Ammeter
Cathode
Anode
A
Cathode
Anode
∆V = 0: The electrons leave the cathode in
all directions. Only some reach the anode.
energy to the
f Figure 28.9,
I
DV
energy as they
e for electrons
5.0 eV
photons strike an electrode with work function 3.0 eV.
nergy.
When
What
with Ka.max
, are is the kinetic energy of emitted electrons?
100%b.ofWhat
their potential is needed to reduce the current to zero?
Kmax , or
(28.2)
14
∆V 7 0: Making the anode positive creates an
electric field that pushes all the electrons
to the anode.
Just
Checking.
imum
kinetic
Monochromatic light shines on the cathode in a
photoelectric effect experiment, causing the emission of
electrons. If the intensity of the light stays the
hy dosame
electrons
but the frequency of the light is increased,
ed on classical
rons, so it was
the ∆V
emitted
electrons
6 0: Making
the anode negative repels the
, causing it to
electrons.
Only
the a
very fastest make it to the both A and
will
be
moving
at
nt. For light to
anode.
B are true.
higher speed.
there will be more
electrons emitted.
neither A nor
B are
24/10/13
1:57true.
PM
15
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.
both A and
B are true.
there will be more
electrons emitted.
neither A nor
B are true.
16
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
17
Metal surfaces on spacecraft in bright sunlight develop a net
electric charge. Do they develop a negative or a positive charge?
Explain.
What’s The Fizics?
18
Diffraction
and
Interference
19
Diffraction
Diffraction
and
Interference
20
Double Slit Interference Pattern
Viewing
screen
Incident laser beam
Longer wavelength
means bigger spacing.
21
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
22
Particles have a Wave Nature
!=
h
h
=
p mv
De Broglie wavelength for a moving particle
23
Particle or Wave?
m
!
Localized.
Smeared out.
Wavelength of a squirrel
running at 3 m/s:
1x10-33 m
24
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
25
Looking Deeper
Electron microscope view of pigment molecule.
26
Quantum
Concept #3:
The wave nature of
particles leads to
quantization.
27
Particles have a wave nature. So...
Particle:
L
m
v
Wave:
L
...the possible states are quantized.
28
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.
29
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...
30
What is the maximum photon energy that could be
emitted by the quantum system with the energy level
diagram shown below? The minimum?
31
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?
32
33
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?
34
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?
K = !U e
Electron moving
more slowly:
K = 12 mv 2
Wavelength is
longer.
!=
h
h
=
p mv
Ratio
reasoning.
35
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
36
The wave nature of particles leads to
quantized energy levels for electrons in atoms.
Only certain transitions are possible.
Energy
160 eV
90 eV
Energy
n54
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
37
What energy photons could be emitted by the quantum
system sketched below?
38
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
39
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
2m #" 2L &%
8mL
mL2
mL
n = 1, 2, 3, 4...
40
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?
41
Heisenberg uncertainty principle
∆ x ∆p
px ⁄
h
4p
!x
42
Uncertainty
If I know where you are,
I don’t know where you
are going.
!v "
"x large
1
!x
"x small
43
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?
44
“Beaming” Someone...
45
A spherical virus has a diameter of 50 nm. It is contained
wavesinside
obey the
principle
of superposition
and 0.0001
exhibit interference.
This
a long,
narrow
cell of length
m.
particle dichotomy seemed obvious until physicists encountered irrefutable
What
uncertainty
imply
velocity
the
ce that
light sometimes
actsdoes
like a this
particle
and,for
eventhe
stranger,
that of
matter
virus
mes acts
likealong
a wave.the length of the cell? Assume the virus has a
might
at
first
are both a wave and a particle, but
density think
equalthattolight
thatandofmatter
water.
a doesn’t quite work. The basic definitions of particleness and waviness are
ly exclusive. Two sound waves can pass through each other and can overlap
uce a larger-amplitude sound wave; two baseballs can’t. It is more profitable
lude that light and matter are neither a wave nor a particle. At the microscale of atoms and their constituents—a physical scale not directly accessible
ive senses—the classical concepts of particles and waves turn out to be simlimited to explain the subtleties of nature.
ough matter and light have both wave-like aspects and particle-like aspects,
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
mentary to each other, like a two-piece jigsaw puzzle. Neither the wave nor
ticle model alone provides an adequate picture of light or matter, but taken
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
46
Heisenberg uncertainty principle
over two hundred years, scientists and nonscientists alike felt that the clockniverse of Newtonian physics was a fundamental description of reality. But
particle duality, along with Einstein’s relativity, undermines the basic assumpf the Newtonian worldview. The certainty and predictability of classical
have given way to a new understanding of the universe in which chance and
inty play key roles—the universe of quantum physics.
2
h
2#
E = mc
!E!t "
ual nature of a buckyball Treating atomic-level structures involves frequent
ween particle and wave views. 60 carbon atoms can create the molecule
med at left, known as C60, or buckminsterfullerene. The scanning electron
ope image of a C60 molecule shown on the right is a particle-like view of the
e with individual carbon atoms clearly 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!
nance imaging
manent magt of electrons
ons also have
for magnetic
47
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.
But a quantum
compass is different.
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 . . .
. . . which correspond to
two possible orientations,
aligned with or opposite
the magnetic field.
Energy
E2 5 1mB
field
e the field
r
B
0
E1 5 2mB
µproton = 1.41! 10 "26 J/T
• 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?
48
ons also have
for magnetic
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 . . .
. . . which correspond to
two possible orientations,
aligned with or opposite
the magnetic field.
Energy
r
B
E2 5 1mB
field
e the field
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?
49
Quantum Weirdness: Non-locality
Two
places at
one time
Which slit did
the electron
go through?
Where is the electron?
50
Quantum Weirdness: Superposition
Many
things in
the same
place
51
Quantum Weirdness: Mixed States
Alive and
dead cats
Schrödinger’s Cat
52
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?
53
Relative intensity
Absorption band
Emission band
0
300 400 500 600
Wavelength (nm)
54