CHAPTER 40 : INTRODUCTION TO QUANTUM PHYSICS

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CHAPTER 40 INTRODUCTION TO QUANTUM PHYSICS
40.2) The Photoelectric Effect
Light incident on certain metal surfaces caused
electrons to be emitted from the surfaces
photoelectric effect
The emitted electrons photoelectrons
Figure (40.6)
A diagram of an apparatus in which the
photoelectric effect can occur
An evacuated glass or quartz tube contains a
metallic plate E connected to the negative
terminal of a battery and another metallic plate
C is connected to the positive terminal of the
battery
When the tube is kept in the dark the ammeter
reads zero no current in the circuit
When plate E is illuminated by light by light
having a wavelength shorter than some particular
wavelength that depends on the metal used to make
plate E a current is detected by the ammeter
a flow of charges across the gap between plates E
and C
This current arises from photoelectrons emitted
from the negative plat (the emitter) and
collected at the positive plate (the collector)
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Figure (40.7)
A plot of photoelectric current versus potential
difference ?V between plates E and C for two
light intensities
At large values of ?V the current reaches a
maximum value the current increases as the
intensity of the incident light increases
When ?V is negative (when battery in the circuit
is reversed to make plate E positive and plate C
negative) the current drops to a very low value
because most of the emitted photoelectrons are
repelled by the now negative plate C
Only those photoelectrons having a kinetic energy
greater than the magnitude of e?V reach plate C
where e the charge on the electron
When ?V is equal to or more negative than - ?V
s (the stopping potential) no photoelectrons
reach C and the current is zero
The stopping potential independent of the
radiation intensity
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The maximum kinetic energy of the photoelectrons
is related to the stopping potential through the
relationship
(40.7)
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Successful explanation by Einstein
Assumed that light (or any other electromagnetic
wave) of frequency f can be considered a stream
of photons
Each photons has an energy E hf
Figure (40.8)
In Einsteins model a photon is so localized
that it gives all its energy hf to a single
electron in the metal
According to Einstein the maximum kinetic
energy for these liberated photoelectrons is
Photoelectric effect equation
(40.8)
Where ? work function of metal ( the minimum
energy with which an electron is bound in the
metal and is on the order of a few electron
volts) Table (40.1)
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Final confirmation of Einsteins theory
Experiment observation of a linear relationship
between Kmax and f Figure (40.9)
Figure (40.9)
The intercept on the horizontal axis the cutoff
frequency below which no photoelectrons are
emitted, regardless of light intensity
The frequency is related to the work function
through the relationship fc ? / h
The cutoff frequency corresponds to a cutoff
wavelength of
(40.9)
c speed of light
Wavelengths greater than ?c incident on a
material having a work function ? do not result
in the emission of photoelectrons
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40.3) The Compton Effect
Compton and his co-workers the classical wave
theory of light failed to explain the scattering
of x-rays from electrons
Classical theory
Electromagnetic waves of frequency fo incident on
electrons should have two effects (Figure
(40.10a))
Radiation pressure should caused the electrons to
accelerate in the direction of progagation of the
waves
The oscillating electric field of the incident
radiation should set the electrons into
oscillation at the apparent frequency f
f the frequency in the frame of the moving
electrons
Frequency f is different from the frequency fo
of the incident radiation because of the Doppler
effect Each electron first absorbs as a moving
particle and then reradiates as a moving particle
exhibiting two Doppler shifts in the frequency
of radiation
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Because different electrons will move at
different speeds after the interaction
depending on the amount of energy absorbed from
the electromagnetic waves the scattered wave
frequency at a given angle should show a
distribution of Doppler-shifted values
Comptons experiment
At a given angle only one frequency of
radiation was observed
Could explain these experiment by treating
photons not as waves but as point-like particles
having enegy hf and momentum hf/c and by assuming
that the energy and momentum of any colliding
photo-electron pair are conserved
Compton effect adopting a particle model for
wave (a scattering phenomenon)
Figure (40.10b) the quantum picture of the
exchange of momentum and energy between an
individual x-ray photon and an electron
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• Figure (40.10b)
• In the classical model the electron is pushed
  along the direction of prpagation of the incident
  x-ray by radiation pressure.
• In the quantum model the electron is scattered
  through an angle ? with respect to this direction
  a billiard-ball type collision.

Figure (40.11a)
A schematic diagram of the apparatuse used by
Compton
The x-rays, scattered from a graphite target
were analyzed with a rotating crystal spectrometer
The intensity was measured with an ionization
chamber that generated a current proportional to
the intensity
The incident beam consisted of monochromatic
x-rays of wavelength ?o 0.071 nm.
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Figure (40.11b) the experimental
intensity-versus-wavelength plots observed by
Compton for four scattering angles (corresponding
to ? in Fig. (40.10)
The graphs for the three nonzero angles show two
peaks
At ? gt ?o
At ?o
The shifted peak at ? is caused by the
scattering of x-rays from free electrons, and it
was predicted by Compton to depend on scattering
angle as
(40.10)
Compton shift equation
Where me the mass of the electron
Compton wavelength ?c of the electron
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The unshifted peak at ?o (Figure (40.11b)) is
caused by x-rays scattered from electrons tightly
bound to the target atoms
This unshifted peak also is predicted by Eq.
(40.10) if the electron mass is replaced with the
mass of a carbon atom, which is about 23 000
times the mass of the electron
There is a wavelength shift for scattering from
an electron bound to an atom but it is so small
that it was undetectable in Comptons experiment
Comptons measurements were in excellent
agreement with the predictions of Equation
(40.10)
Derivation of the Compton Shift Equation
By assuming that the photon behaves like a
particle and collides elastically with a free
electron initially at rest Figure (40.12a)
The photon is treated as a particle having energy
E hf hc/? and mass zero
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In the scattering process the total energy and
total linear momentum of the system must be
conserved
Applying the principle of conservation of energy
to this process gives
hc/?o the energy of the incident photon, hc/?
the energy of the scattered photon, and Ke
the kinetic energy of the recoiling electron
Because the electron may recoil at speeds
comparable to the speed of light use the
relativistic expression Ke ?mec2 mec2
(40.11)
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Apply the law of conservation of momentum to this
collision noting that both the x and y
components of momentum are conserved
The momentum of a photon has a magnitude
p E/c and E
hf
p hf/c
Substituting ?f for c gives p h/?
Because the relativistic expression for the
momentum of the recoiling electron is pe ?mev
we obtain the following expression for the x and
y components of linear momentum, where the angles
are as described in Fig. (40.12b)
(40.12)
(40.13)
Eliminating v and ? from Eq. (40.11) to (40.13)
a single expression that relates the remaining
three variables (?, ?o, and ?)
the Compton shift equation
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40.5) Bohrs Quantum Model of the Atom
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Using these four assumptions
Calculate the allowed energy levels
emission wavelengths of the hydrogen atom
Electric potential energy of the system (Fig.
(40.15)) U keq1q2/r kee2/r
where ke is the Coulomb constant and the negative
sign arises from the charge e on the electron
The total energy of the atom which contains both
kinetic and potential energy terms
(40.20)
Newtons second law
The Coulomb attractive force kee2/r2 exerted on
the electron must equal the mass times the
centripetal acceleration (a v2/r) of the
electron
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The kinetic energy of the electron is
(40.21)
Substituting this value of K into Eq. (40.20)
the total energy of the atom is
(40.22)
The total energy is negative indicationg a
bound electron-proton system
Energy in the amount of kee2/2r must be added to
the atom to remove the electron and make the
total energy of the system zero
Obtain an expression for r, the radius of the
allowed orbits by solving Equations (40.19) and
(40.21) for v and equationg the results
(40.23)
n 1, 2, 3,
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The radii have discrete values they are
quantized
The result is based on the assumption that the
electron can exist only in certain allowed orbits
determined by the integer n
The orbit with the smallest radius Bohr radius
ao (corresponds to n 1)
(40.24)
A general expression for the radius of any orbit
in the hydrogen atom by substituting Equation
(40.24) into Equation (40.23)
(40.25)
Radii of Bohr orbits in hydrogen
Figure (40.16) the first three circular Bohr
orbits of the hydrogen atom
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The quantization of orbit radii immediately leads
to energy quantization
Substituting rn n2ao into Equation (40.22)
obtained the allowed energy levels of hydrogen
atom
(40.26)
n 1, 2, 3,
Inserting numerical values
(40.27)
n 1, 2, 3,
Only energies satisfying this equation are
permitted
Energy levels
Ground state the lowest allowed energy level (n
1, energy E1 13.606 eV)
First excited state the next energy level
(n 2, energy E2 E1/22 3.401 eV)
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Figure (40.17) an energy level diagram showing
the energies of these discrete energy states and
the corresponding quantum numbers n
The uppermost level corresponding to n ? (or
r ?) and E 0 represent the state for which
the electron is removed from the atom
Ionization energy the minimum energy required
to ionize the atom (to completely remove an
electron in the ground state from the protons
influence)
Figure (40.17) the ionization energy for
hydrogen in the ground state (based on Bohrs
calculation 13.6 eV
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Eqs. (40.18) and (40.26) to calculate the
frequency of the photon emitted when the electron
jumps form an outer orbit to an inner orbit
(40.28)
Frequency of a photon emitted from hydrogen
Because the quantity measured experimentaly is
wavelength use c f? to convert frequency
to wavelength
(40.29)
(40.30)
Identical to relationships discovered by Balmer
and Rydberg (Eqs. (40.14) to (40.17)
The constant kee2/2aohc Rydberg constant, RH
1.097 373 2 x 107m-1
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Bohr extended his model for hydrogen to other
elements
In general to describe a singel electron
orbiting a fixed nucleus of charge Ze (where Z
the atomic number of the element), Bohrs theory
gives
(40.31)
(40.32)
n 1, 2, 3,