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Chapter 40- Introduction to Quantum Physics
Attempts to apply the laws of classical physics to explain the behavior of matter on the atomic scale,
such as blackbody radiation, the photoelectric effect, the emission of sharp spectral lines by atoms in
a gas discharge,.. have been solved between 1900 and 1930s by the new theory called QUANTUM
PHYSICS. Like relativity, the quantum theory requires a modification of our ideas concerning the
physical world.
Blackbody Radiation and Planck’s Hypothesis
An object at any temperature emits radiation called thermal radiation. At low temperatures the
radiation is in the infrared region. As the temperature increases, the radiation shifts to visible
wavelengths. From a classical viewpoint, thermal radiation originates from accelerated charges near
the surface of the object. By the end of the 19th century, it had become apparent that the classical
theory of thermal radiation is inaccurate.The basic problem was understanding the radiation of
blackbody. A blackbody is an ideal system that absorbs all radiation incident on it. A good
approximation is the inside of a hollow object as shown:
The graph shows the distribution of energy as the temperature of the blackbody increases.
Here you see that the intensity increases as the temperature increases and the peak wavelength
shifts to smaller wavelengths. This behavior can be explained by the following relationship, called
Wien’s displacement law:
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max .T  0, 2898 102 m.K
Considering the classical model, the blackbody radiation is described by Rayleigh-Jeans equation:
I ( , T ) 
2 ck BT
4
In 1900 Planck discovered a formula for blackbody radiation that was in complete agreement with
experiment at all wavelengths:
I ( , T ) 
2 hc 2
 5 (ehc /  kBT  1)
where h constant is adjusted to fit the experimental data.
h=6.626x10-34 J.s
You can see that at low wavelengths Planck’s expression reduces to Rayleigh-Jeans equation.
Planck made two controversial assumptions:
1) The molecules can have only discrete units of energy, En
En = nhf
Where n is a positive integer called the quantum number and f is the frequency of the vibration of
the molecules. The energy of a molecule can have only discrete values, we say that the energy is
quantized.
2) The molecules emit or absorb energy in discrete packets called photons. The amount of
energy radiated from the molecule from one state to the adjacent state is hf. If the molecule
remains in one quantum state, no energy is absorbed or emitted.
These assumptions made the birth of the quantum theory.
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The Photoelectric Effect
In the latter part of the 19th century experiments show that light incident on certain metallic
surfaces caused electrons to be emitted from the surfaces. This phenomenon is known as
photoelectric effect and the emitted electrons are called photoelectrons.
In the below figure when light strikes the plate E, the photoelectrons are ejected from the plate.
Electrons collected at C constitute a current in the circuit.
Only electrons having a kinetic energy greater than eV reach C. When V is less or equal to Vs, the
stopping potential, the current is zero. The stopping potential is independent of radiation
intensity. The maximum kinetic energy of the photoelectrons is related to the stopping potential
through the relationship,
Kmax  eVs
Several features of the photoelectric effect cannot be explained by the classical physics.
1) If the frequency of the incident light is below the cutoff frequency, no electrons are emitted.
2) The maximum kinetic energy of the photoelectrons is independent of the light intensity.
3) The maximum kinetic energy of the photoelectrons increases as the light frequency
increases.
4) Electrons are emitted from the surface almost instantenously (less than 10-9 s after the
surface is illuminated.) Classicaly we would expect the electrons to require some time to
absorb the incident radiation.
A successful explanation of the photoelectric effect is given by Einstein in 1905. He assumed that
light of frequency f can be considered a stream of photons. Each photon has an energy E given by
the equation, E=hf.
Einstein received the Nobel prize in 1921 for his paper on electromagnetic radiation.
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hf    Kmax
 is called the work function of the metal. It represents the minimum energy with which an
electron is bound to a metal and is on the order of a few electron volts. If the kinetic energy
of the electrons ≤ work function, the electrons are never ejected from the surface. If the light
intensity is doubled the number of photons is doubled which doubles the number of
photoelectrons emitted. However their kinetic energy remains the same because it only
depends on the light frequency.
The slope of this curve gives
h,the Planck constant.
The Compton Effect
In 1906, Einstein predicted that a photon of energy E travelling in a single direction carries a
momentum E/c=hf/c. In 1923, Compton and Debye carried Einstein’s idea of photon a
farther. They realized that the scattering of X-ray photons from electrons could be explained
by treating photons as point-like particles having energy hf and momentum hf/c and
assumed that the momentum of photon-electron pair is conserved in a collision.
The below figure shows the quantum picture of the transfer of momentum and energy
between a X-ray photon and an electron. The quantum picture explains the lower scattered
frequency f, because the incident photon transfers some of its energy to the recoiling
electron.
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The below figure shows the diagram of the experiment that Compton used. X-rays were
scattered from a graphite target. The scattered angle was measured by a rotating crystal
spectrometer and the intensity was measured in an ionization chamber. The incident Xray was monochromatic. The  ' represents the shifted wavelength.
Compton shift equation:
 ' 0 
h
(1  cos  )
mc
m is the mass of the electron, h/mc is the Compton wavelength of the electron and has a
value of 0.00243nm.
We can derive this equation by applying the conservation of energy and momentum.
Applying conservation of energy gives,
hc
0

hc
 Ke
'
The electron may recoil at speeds comparable with the speed of light, so we use the
relativistic expression
Ke   mc 2  mc 2
where   1 / 1   2 / c 2 .
Next, we apply the conservation of momentum to this collision. Note that both x and y
components of the momentum are conserved.
x component :
h
0

y component : 0 
h
cos    m cos 
'
h
sin    m sin 
'
After some algebra, the Compton shift equation can be obtained.
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The Particle-Wave Complemantarity
Is light in some sense simultaneously wave and particle?
Atomic Spectra
When light from a low-pressure gas discharge is examined with a spectroscope, a few bright
lines of pure color is observed on the dark background. Furthermore, as you can see from the
figure, the wavelengths that are contained in a given line spectrum are characteristic of the
element emitting the light. No two electrons emit the same line spectrum.
Bohr’s Quantum Model of the Atom
Bohr’s atom model contained a combination of ideas from Planck’s original quantum theory,
Einstein’s photon theory of light and Rutherford’s model of the atom. Using the simplest
atom, hydrogen, Bohr described what the atom structure must be. The basic ideas of the
Bohr theory are:
-
-
The electron moves at circular orbits around the proton under the influence of the
Coloumb force of attraction.
Only certain electron orbits are stable. These stable orbits are the ones in which the
electron does not emit energy in the form of radiation.
Radiation is emitted by the atom when the electron jumps from a more energetic initial
orbit to a lower orbit. The frequency of the photon emitted is related to the change in
the atom’s energy and is not equal to the frequency of the electron’s orbital motion.
The size of the allowed electron orbits is determined by a condition imposed on the
electron’s orbital angular momentum. The allowed orbits are those for which the
electron’s orbital angular momentum about the nucleus is an integral multiple of
 h / 2 .
mr  n
n=1,2,3,..
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Using these four assumptions we can calculate the allowed energy levels and emission
wavelengths of the hydrogen atom.
ke e 2  1 
En  
2a0  n 2 
n=1,2,3,..
Here a0 is the Bohr radius:
2
a0  
mke e2
It has a value of 0.0529 for hydrogen atom.
Inserting numerical values into this equation gives,
En  
13.6
eV
n2
n=1,2,3,..
The lowest allowed energy level, called the ground state, has n=1 and energy E 1=-13.6 eV. The
next energy level, the first excited state, has n=2 and energy -3.4 eV. The minimum energy
required to ionize the atom is called the ionization energy. It is 13.6 eV for hydrogen.
When the electron jumps from an outer to an inner orbit, it emits a photon of frequency:
f 
Ei  E f
h
ke e 2  1 1 

  
2a0 h  n 2f ni2 
We can convert the frequency to wavelength since c=fλ.
ke e 2  1 1 

  
 2a0 hc  n 2f ni2 
1
 1 1 
 RH  2  2 
n


 f ni 
1
RH is the Rydberg constant. RH = 1.0973732 x 107m-1.
Bohr extended his model for hydrogen to other elements in which all but one electron had been
removed. In general, to describe a single electron orbiting a fixed nucleus of charge +Ze (Z, the
atomic number (proton number)), Bohr’s theory gives,
a0
ke e 2  Z 2 
rn  ( n )
and En  

 for which n=1,2,3,..
Z
2a0  n 2 
2
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