Chapter 27: Quantum Physics

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AP Physics
Chapter 27
Quantum Physics
Chapter 27: Quantum Physics
27.1
27.2
27.3
27.4
27.5
Quantization: Planck’s Hypothesis
Quanta of Light: Photons and the
Photoelectric Effect
Quantum “Particles”: The Compton
Effect
The Bohr Theory of the Hydrogen
Atom
Omitted
Homework for Chapter 27
• Read Chapter 27
• HW 27.A: p.861-862: 16, 18, 19-27.
• HW 27.B: p.863- : 42, 43, 52, 54-58, 61, 63, 64.
27.1: Quantization: Planck’s
Hypothesis
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Some History on the Atomic Nucleus…
J.J. Thomson Model: After discovering the electron in
1897, Sir J.J. Thomson proposed a model of an atom in
1904. This model later came to be known by different
names such as the plum-pudding and watermelon models.
In this model the pudding was the positive charge of the atom and electrons
were embedded in it like plums. The total positive charge was equal in
magnitude to the total negative charge of the electrons. Hence the atom was a
neutral particle.
Rutherford’s Model:
In 1911, Ernest Rutherford performed an experiment to observe the scattering of
alpha particles by a thin gold foil. (Alpha particles consist of two protons and two
neutrons). Based on the plum pudding model, Rutherford expected very little
scattering because of the large momentum for alpha particles. He was surprised to
observe that some alpha particles scattered through large angles and in fact some
of them had back scattered. This was completely inconceivable on the basis of the
plum-pudding model.
This remarkable experimental result let Rutherford to revise the atomic model. He
could explain the result of his alpha scattering experiment by the nuclear model.
According to the nuclear model the positive charge of the atom and most of its
mass is concentrated in a very small volume at the center of the atom. This part of
the atom came to be called the nucleus of the atom. The electrons revolve around
the nucleus in orbits similar to the planets going around the sun.
This model has since been further refined but the basic idea of a tiny atomic
nucleus at the center of outer electrons still holds true.
The Electromagnetic Spectrum
Visible Light Spectrum
Max Planck (1858-1947) – A German physicist; considered to be the founder of
quantum theory, and thus one of the most important physicists of the twentieth
century. Planck was awarded the Nobel Prize in Physics in 1918.
One of the problems scientists had at the end of the
nineteenth century was how to explain thermal radiation.
thermal radiation – the continuous spectra of radiation emitted
by hot objects. Maximum intensity shifts to shorter
wavelengths (higher frequencies) with increasing temperature.
blackbody – an ideal system that absorbs and emits all
radiation that falls on it.
A blackbody can be approximated by a small hole
leading to an interior cavity in a block of material.
Intensity vs. Wavelength Curves for the Thermal Radiation from an Idealized
Blackbody at Different Temperatures
• The location of maximum intensity shifts to
shorter wavelengths with increasing
temperature.
• The wavelength shift obeys
Wein’s displacement law:
maxT = 2.90 x 10-3 m·K
where max is the wavelength of radiation (in
meters) at which maximum intensity occurs and
T is the temperature of the body (in kelvins).
Example 27.1: What is the most intense color of light emitted by a giant star of
surface temperature 4400 K? What is the color of the star?
Classical theory predicts the intensity of thermal
radiation is inversely related to the emitted wavelength.
I1
4
Thus the intensity of the radiation would become
infinitely large as the wavelength approaches zero. This
was known as the “ultraviolet catastrophe”. In contrast,
Plank’s quantum theory predicts the observed radiation
distribution.
Max Planck successfully explained the spectrum of blackbody radiation by
proposing a radical hypothesis. According to Planck’s hypothesis, the energy
of the oscillating atoms emitting the radiation have only discrete, or particular,
amounts of energy rather than a continuous distribution of energies. The energy
is
On Gold
E = hf
Sheet
where E is the energy
h is Planck’s constant (6.63 x 10-34 J·s)
f is the frequency of the oscillation
• According to Planck’s hypothesis energy is quantized, or occurs in only discrete
amounts. A more specific way to represent his hypothesis is
En = n(hf) for n = 1,2,3,…
•The smallest possible amount of energy occurs when n= 1.
E1 = hf.
• All other permitted values of energy are integral multiples of hf.
• The quantity hf is called a quantum of energy.
Blackbody Radiation Applet
http://www.mhhe.com/physsci/astronomy/applets/Blackbody/frame.html
Check for Understanding
1. Which scientist is credited with the discovery of the electron?
a)
b)
c)
d)
e)
Albert Einstein
Count Rutherford
Robert Milikan
Max Planck
J.J. Thomson
Answer: e
2. Which scientist is credited with the discovery of the atomic nucleus?
a)
b)
c)
d)
e)
Answer: b
Albert Einstein
Count Rutherford
Robert Milikan
Max Planck
J.J. Thomson
Check for Understanding
3.
27.2: Quanta of Light:
Photons and the
Photoelectric Effect
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HO W T O R
photon – a particle of light
• Photons have no mass, but they can transfer energy to or from electrons.
Summary of Subatomic Particles
Name
Mass
Charge
Proton
1.67 x 10-27 kg = 1 amu
Positive
Neutron
1.67 x 10-27 kg = 1 amu
Zero
Electron
9.11 x 10-31 kg
Negative
Photon
0
Zero
• The mass of a proton and neutron equals one atomic mass unit, or amu.
• The electron-volt (eV) is a useful unit of energy for subatomic particles. One eV
is equal to the amount of energy needed to change the potential of an electron by
one volt.
1 eV = 1.6 x 10-19 J
The Photoelectric Effect
Towards the end of the 19the century, it had been experimentally observed that
when ultraviolet light was shone on a negatively-charged electroscope, the
charged leaves fell closer together; the electroscope discharged. This was the
beginnings of the path to understanding what we now call the photoelectric effect.
When light shines on any metal surface, the surface can release electrons. If light
were composed of waves, then eventually any wavelength of light should be able
to build up enough energy to knock an electron free. However, scientists had
discovered that only certain wavelengths worked with each metal and that
electrons were either emitted instantaneously, or never emitted. They had also
noticed that shorter wavelengths worked better than longer wavelengths.
The equation for the photoelectric effect was first explained by Albert Einstein in
1905.
On Gold
Sheet
Some observations …..
•This equation is actually just a restatement of conservation of energy.
•The intensity of the light source affected the number of photoelectrons ejected
from the surface since higher intensities permit more photons to strike the surface.
•The frequency of the light source affected the kinetic energy of each
photoelectron.
•Since each photon can be absorbed by only ONE photoelectron (that is, there is a
one-to-one correspondence), the energy of the photons directly affects the kinetic
energy of the released photoelectrons.
hf
The Experiment
• The electrons with the maximum KE can
be stopped from completing their journey
across the photoelectric tub if there is a
stopping potential set up to impede their
progress. The formula that relates the KE of
these photoelectrons to this stopping
potential is
KEmax = UE = qVstopping or eVo
where Vstopping (Vo) is the stopping potential
q (e) is the magnitude of the charge on an electron, 1.6 x 10-19 coulombs
• This formula is based on the fact that work is done on charged particles when they
cross through an electric field.
• The work done (qV) equals the change in each electron’s KE.
• Incident light on the photoelectric material in a photocell causes the emission of
electrons, and a current flows in the circuit.
• The voltage applied to the tube can be changed by means of a variable resistor.
• As the plots of current vs. voltage for the two intensities of monochromatic light
show, the current is constant as the voltage increased. However, for negative
voltages (by reversal of the battery polarity), the current goes to zero at a
particular stopping voltage, which is independent of intensity.
• As would be expected classically, the current is proportional to the intensity of
the incident light – the greater the intensity, the more energy there is to free
additional electrons.
• The minimum energy needed to free the electrons from the material is called the
work function (o).
• According to energy conservation, hf = Kmax + o , that is, the energy of the
absorbed photon goes into the work of freeing the electron, and the rest is carried
off by that emitted electron as kinetic energy.
• The threshold or cutoff frequency (fo ) 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
0 = hfo - o
hfo = o
fo = o
h
or
Ephoton = o
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·s.
(think: y = mx+b)
All the lines on this type of graph will
be parallel, only differing in their y-axis
intercept (-) and their x-axis intercept
(the threshold frequency).
f1
f2
f3
Photoelectric Effect Characteristics
(Table 27.1 in textbook)
Characteristic
Predicted by wave theory?
1. The photocurrent is proportional to the
intensity of the light.
yes
2. The maximum KE of the emitted electrons
is dependent on the frequency of the light
but not on its intensity.
no
3. No photoemission occurs for light with a frequency
below a certain cutoff frequency fo regardless of
its intensity.
no
4. A photocurrent is observed immediately when the
light frequency is greater than fo even if the light
intensity is extremely low.
no
• Albert Einstein received the Nobel Prize for
Physics in 1921 for his discovery of the Law of
the Photoelectric Effect.
• His work ended the controversy as to whether
light had particle properties.
• By invoking the quantum nature of light he was
able to explain experimental results that his
predecessors could not explain with just the wave
model of light.
Einstein’s official portrait after
receiving his Nobel Prize in 1921.
Problem-Solving Hint
• Start with the formula
E = hf
• Recall from wave theory that the frequency of a wave is related to the wavelength
by the formula
v = f
• For light, the velocity is c, 3 x 108 m/s, so we can instead write c = f
• This means we can rewrite the equation for the energy of a photon to read
E = hc where hc = 1.24 x 103 eV·nm (on your blue sheet)

• This is helpful because typically the wavelength in nm is given in a problem rather
than frequency.
•These formulas tell us that a photon with high frequency, and therefore with a
small wavelength, is higher in energy than a photon with low frequency and long
wavelength. So, gamma rays, for example, are a lot higher energy than radio
waves because gamma rays have a higher frequency.
Example 27.2: What is the photon energy of visible light having wavelength
632.8 nm?
Example 27.3: A metal has a work function of 4.5 eV. Find the maximum kinetic
energy of the emitted photoelectrons if the wavelength of light falling on the metal is
a) 300 nm
b) 250 nm
Example 27.4: When light of wavelength 350 nm is incident on a metal surface,
the stopping potential of the photoelectrons is measured to be 0.500 V.
a) What is the work function of the metal?
b) What is the threshold frequency of the metal?
c) What is the maximum kinetic energy of the photoelectrons?
Summary
• Thermal radiation, typically produced by hot objects, has a continuous spectrum.
• A blackbody is an ideal system that absorbs and emits all radiation that falls on it.
• Wein’s displacement law states that the wavelength of maximum intensity for
radiation from a blackbody is inversely related to its temperature.
• Classical theory states that
the wavelength of maximum
intensity for radiation from a
blackbody is inversely related
to its temperature.
• Planck’s constant (h) is the
fundamental proportionality
constant between energy and
frequency of thermal oscillators
as well as frequency of a light
wave and energy of the
corresponding photons.
f
fo
hf
KE
hf
hf
fo
hfo
Check for Understanding
1. A blackbody
a) absorbs all radiation incident on it
b) re-emits all radiation incident on it
c) emits thermal radiation in a continuous spectrum
d) all of these
Answer: d
2. The ultraviolet catastrophe is a consequence of
a) Planck’s Theory
b) Classical Theory
c) Einstein’s Theory
d) Rutherford’s Theory
Answer: b
Check for Understanding
3. Which is a true statement about the photoelectric effect?
a) Energy in the form of light can cause an atom to eject one of its
electrons.
b) The frequency of light must be above a certain value for the ejection to
occur.
c) An ejected electron has a KE of zero if the energy of the photon is equal
to the work function.
d) all of these
Answer: d
4. A photocurrent is observed when
a) the light frequency is above the threshold frequency
b) the energy of the photons is greater than the work function
c) the light frequency is below the threshold frequency
d) both a and b
Answer: d
Check for Understanding
5.
Check for Understanding
6.
Homework for Chapter 27.1-2
• HW 27.A: p.861-862: 16, 18, 19-27.
27.3: Quantum “Particles”: The
Compton Effect
Thomson
Millikan
Rutherford
Bohr
• In 1923, American physicist Arthur H. Compton (1892-1962) explained a
phenomenon he observed in the scattering of X-rays from a graphite block by
considering the radiation to be composed of quanta.
• His explanation of the observed effect provided additional convincing evidence
that, at least in certain types of experiments, light, and electromagnetic radiation
in general, is composed of quanta, or “particles” of energy called photons.
• When X-rays of a single wavelength were scattered by the electrons in metal
foil, the incident wavelength is increased in the scattered X-rays.
• The wavelength shift grew as the scattering
angle increased. The nature of the scattering
material did not contribute to the effect.
• This phenomenon came to be known as the
Compton effect.
• Compton theorized that an X-ray photon
colliding with and electron was like billiard balls
in an elastic collision. He reasoned that the
incident photon would transfer some energy and
momentum to the electron.
• After the collision, the energy and frequency of
the scattered photon should be decreased (E=hf)
and its wavelength increased ( = c/f).
• He applied the principles of conservation of energy and momentum to develop
the formula for the Compton effect:
  = 1 - o = C (1- cos)
where
Compton wavelength
of an electron
o is the wavelength of the incident photon
1 is the wavelength of the scattered photon
C is the Compton wavelength of the electron
 is the scattering angle
C = h = 2.43 x 10-12 m = 2.43 x 10-3 nm
m ec
where
h is Planck’s constant
m is the mass of an electron
c is the speed of light
• Since the Compton shift is very small, it is only significant for X-ray and gammaray scattering where the wavelengths are on the order of C.
• Compton’s equation correctly predicted the observed wavelength shift, and
Compton was awarded a Nobel Prize in 1927.
• Einstein’s and Compton’s successes in explaining electromagnetic phenomena in
terms of quanta left scientists with two apparently competing theories of
electromagnetic radiation.
• Classically, the radiation is pictured as a continuous wave, and this theory
satisfactorily explains such wave-related phenomena as interference and
diffraction.
• Conversely, quantum theory was necessary
to explain the photoelectric and Compton
effects correctly.
• These two theories gave rise to a
description this is called the dual nature
of light. That is, light apparently behaves
sometimes as a wave and at other times as
photons or “particles”.
Example 27.5: X-rays of wavelength of 0.200 nm are scattered by a metal. The
wavelength shift is observed to be 1.50 x 10-12 m at a certain scattering angle
measured relative to the incoming X-ray.
a) What is the scattering angle?
b) What is the maximum shift possible for the Compton effect?
Summary
• The Compton effect is the wavelength increase of light scattered by electrons or
other charged particles.
• The dual nature of light means that light must be thought of as having both particle
and wave natures.
Check for Understanding
1. The Compton effect was first observed by using
a) visible light
b) infrared radiation
c) ultraviolet light
d) X-rays
Answer: d
2. The wavelength shift for Compton scattering is a maximum when
a) the photon scattering angle is 90°
b) the electron scattering angle is 90°
c) the shift is equal to the Compton wavelength
d) the incident photon is backscattered
Answer: d
Check for Understanding
3. A photon can undergo Compton scattering from a molecule as well as from a
free electron. How does the maximum wavelength shift for Compton
scattering from a molecule as a unit compare to that from a free electron?
a) max increases for the molecule
b) max decreases for the molecule
c) max doesn’t change
Answer: b. The Compton wavelength is inversely proportional to the mass of
the scattering particle. max = 2C = 2h  1
mc m
4. In Compton scattering, why does the scattered photon always have a longer
wavelength than the incident photon?
Answer: From energy conservation, the scattered photon has less energy
after scattering because the free electron receives part of the incident energy.
Since the energy of a photon is proportional to the frequency of the light or
inversely proportional to wavelength, the scattered photon always has a
longer wavelength.
27.4: The Bohr Theory of the
Hydrogen Atom
Answer: When the light waves strike the transparent material, a
chain of absorptions and reemissions occur through the material.
The time delay between each absorption and reemission
produces an average speed of light less than 3 x 108 meters per
second.
Spectra
•In the 1800s, much experimental work was done
with gas discharge tubes. (neon, hydrogen, mercury,
etc. vapor)
• Normally, light from an incandescent source (such
as a light bulb) exhibits a continuous spectrum.
• When a gas is excited by heat or electricity and the light it emits is separated into
its component wavelengths by a prism or diffraction grating, the result is a brightline, or emission, spectrum, such as these of a) barium and b) calcium. Each atom
or molecule emits a characteristic pattern of discrete wavelengths.
a.
b.
c.
• When a continuous spectrum consisting of all wavelengths is passed through a
cool gas, a series of dark lines is observed. Each line represents a “missing”
wavelength – a particular wavelength the gas has absorbed. The wavelengths
absorbed by any substance are the same ones it emits when excited. This
absorption spectrum of the Sun shows several prominent absorption lines; the
gases of the solar atmosphere produce it.
emission spectrum – or bright–line spectrum; a series of bright lines indicating which
wavelengths are being emitted.
• A discrete line spectrum is characteristic of the individual atoms or molecules
of a particular material.
• Emission lines can be used to identify a material with a spectroscope.
absorption spectrum – a series of dark lines superimposed on a continuous
spectrum.
• If white light passes through a relatively cool gas, certain frequencies or
wavelengths are missing, or absorbed.
• Absorption and emission lines for a gas occur at the same frequencies.
• Hydrogen was under study because it was the simplest atom with one proton and
one electron, and had a relatively simple visible spectrum.
The spectral lines of hydrogen.
• The spectral lines of hydrogen in the visible region is called the Balmer series.
• In 1913 an explanation of the spectral lines was given in A Theory of the
Hydrogen Atom by Danish physicist Niels Bohr (1885-1962).
Bohr’s Postulates:
1. The hydrogen electron orbits the nuclear proton in a circular orbit (analogous to
planets orbiting the Sun).
2. The angular momentum of the electron is quantized in integral multiples of
Planck’s constant, h.
L = nh , n=1,2,3,…
2
3. The electron does not radiate energy when it is in certain discrete circular
orbits.
4. The electron radiates or absorbs energy only when it makes a transition to
another orbit.
hf = Ef - Ei
• From these assumptions, Bohr showed that the electron can have only certain
sized orbits with certain energies.
• The energy of the electron in the nth orbit is En = -13.6 eV
n2
• The radius of the orbit is rn = 0.0529 n2 nm for n = 1,2,3,…
• n is an example of a quantum number, specifically the principle quantum
number.
• The n = 1 orbit is known as the ground state. Orbits with n > 1 are called excited
states.
• The energy of the electron in any state is En, and the energy needed to completely
free the electron from the atom in that state is –En, which is called the binding
energy.
Electron Excitation and Emission Simulation
http://micro.magnet.fsu.edu/primer/java/fluorescence/exciteemit/index.html
En
rn = 0.0529 n2 nm for n= 1,2,3,…
where
En is the energy level of the electron
n an integer (quantum number)
rn is the radius of the electron orbit
• The Bohr theory predicts that the hydrogen electron can occupy only certain orbits
that having discrete radii. Each allowed orbit has a corresponding energy level. The
lowest energy level (n=1) is the ground state; those above (n>1) are excited states.
• The orbits are shown at the left, with orbital radius plotted against the 1/r electrical
potential of the proton. The electron in the ground state is deepest in the potential
well, analogous to the gravitational potential well.
• An electron generally does not remain in an excited state for long. It decays, or
makes a transition to a lower energy level, in a short time. The time an electron
spends in an excited state is called the lifetime of the excited state.
• If an electron makes a downward transition for ni to nf state, a photon is released
and its energy is equal to the energy difference between the final and initial states:
E = Ef – Ei = 13.6
1 – 1 eV
nf2 ni2
• The wavelength of the photon is then
 = hc = 1.24 x 103 eV·nm
E
E
Hydrogen Spectrum
• For transitions with nf = 1, the spectrum series is called the Lyman series (all
ultraviolet).
• For transitions with nf = 2, the spectrum series is called the Balmer series
(visible if ni = 3,4,5, and 6).
• For transitions with nf = 3, the spectrum series is called the Paschen series
(infrared).
• In fluorescence, an electron that has been excited by absorbing a photon returns
to the ground state in two or more steps. At each step a photon is emitted each with
less energy (longer wavelength) than the absorbed light.
Some mineral are fluorescent.
Fluorescent pigment.
A fluorescent butterfly.
Example 27.6: What are the orbital radius and total energy of an electron for a
hydrogen atom in
a) the ground state and
b) the second excited state?
Example 27.7: The electron of a hydrogen atom makes a transition from the fourth
excited state to the first excited state. What are the energy and wavelength of the
emitted photon?
Summary
• The Bohr theory of the hydrogen atom treats the electron as a classical particle in
circular orbit around the proton, held in orbit by the electric attraction force.
• The angular momentum of the electron is quantized in integral numbers of
Planck’s constant, h.
•The ground
state of a
hydrogen atom
is the state of
lowest energy
for the atomic
electron (in the
smallest orbit,
n=1). Excited
states have
greater
energies.
Check for Understanding
1. In his theory of the hydrogen atom, Bohr postulated the quantization of
a) energy
b) centripetal acceleration
c) light
d) angular momentum
Answer: d
2. An excited hydrogen atom emits light when its electron
a) makes a transition to a lower energy level
b) is excited to a higher energy level
c) is in the ground state
Answer: a
Check for Understanding
Bohr was able to arrive at a formula for the energy of a single electron atom
(hydrogen or a helium ion) in the nth orbit:
En
Check for Understanding
Homework for Chapter 27.3-4
• HW 27.B: p.863- : 42, 43, 52, 54-58, 61, 63, 64.
Chapter 27 Formulas

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