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A. La Rosa
Lecture Notes
APPLIED OPTICS
Lecture-3 QUANTUM THEORY of LIGHT
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3.1 Planck’s Hypothesis to calculate the atom’s average
energy W
To bring the theoretical prediction closer to the experimental
results Planck considered the possibility of a violation of the law of
equipartition of energy described above. The starting expression
would be expression (39) I (ω) 
1
ω 2 W , with the average
2 2
3 c
energy of the oscillator W no to have a constant value (as the
equipartition theorem predicted) but rather being a function of the
frequency, W (ω) , with the following requirements,
2 W ()    0
ω0
and
(51)
2 W ()    0
ω
For the statistical calculation of W ,
Planck did not question the classical Boltzmann’s
Statistics described in the section 2.1.E above;
That procedure would still be considered valid.
(52)
Planck realized that he could obtain the desired behavior expressed
in (51) if,
 rather than treating the energy of the oscillator as a
continuous variable,
that is the energy states of the oscillator should take
only discrete values:
0 , , 2, 3, …
(53)
 the energy steps would be different for each frequency
= ()
(54)
1
where the specific dependence of  in terms of  to be
determined
Incident
radiation
q

Planck postulated that
the energy of the
oscillator is quantized
Fig. 11 An atom receiving radiation of frequency ,
can be excited only by discrete values of energy 0 ,
, 2, 3, …
According to Planck, the classical integral expression to calculate the
 E / kBT

average energy Wclassical  E
g ( E )dE
 Ee

 E '/ k T
g ( E ' )dE '
e
0

would be
B
0
replaced by a discrete summation,

E e
n
WPlanckl (ω)  E 
 En / kBT
n0

E / k T
e n B
Planck
(55)
n0
where En = n() ; n= 1 2, 3, …
A graphic illustration can help understand why this hypothesis could
indeed work:
 First we show how classical physics evaluate the average energy.
E

classical
  E [ P( E )] dE = area under the curve EP(E )
0
2
E P(E)
P(E)
Classic calculation:
<E>= continuum addition
=Area (integral)
Boltzmann
distribution
Energy E
Energy E
Fig. 12 Schematic representations to calculate the average energy of
the oscillator under the classical approaches

 Using Plank’s hypothesis E
Planck
  E n P( E n )
n 0
 Case: Low frequency values of 
≈
For this case, Planck assumed
P(E)
small value
(56)
E P(E)
<E>=
Quantum calculation: =Area
Boltzmann
distribution
discrete addition
Energy E
~kB T
Energy E


Notice, the
value of
 E P( E )
n
will be very close to the
n
n 0
classical value.
It is indeed desirable that Planck’s results agree with the
classic results at low frequencies, since the classical
predictions and the experimental results agree well at low
frequencies (see Fig. 12.)
 Case: High frequency values of 
For this case, Planck assumed
3
≈
big value
(57)
P(E)
E P(E)
<E>=
Quantum calculation: =Area
Boltzmann
distribution
discrete addition

Energy E
~kB T
Energy E

Notice, the value of E
  En P( En ) will be very different that
Planck
n 0

the classical result (area under the curve.) In fact,
 E P(E ) tends
n
n
n 0
to zero as the step  is chosen larger.
This result is desirable, since the experimental result indicate that
E tends to zero at high frequencies.
It appears then that the Planck’s model (55) could work.
Now, which function () could be chosen such that:
it is small at low , and large at large ?
An obvious simple choice is to assume a linear relationship
() =  
(58)
where  is a constant of proportionality to be determined (when
fitting the model prediction to the experimental results.).
Replacing (58) in (55) one obtains,

WPlanckl (ω)  E 

En e
n0

e
 En / kBT
n0
Let  

 En / kBT
 ne n / k T
B

n0

 e n / k T
B
n0

kT
4

kT
WPlanckl (ω) 


ne  n
n0

 kT
n0


 e n

   e n 
 e n
n0
n0
Since d [ ln(u) ] /dx = (1/u)du/dx


WPlanckl (ω)  k T  ln [

If we define
 en  ]
n0
x  e   , then

 en   1  x  x
2
 x3...
n0
This series is equal to 1/(1-x)
=
1
1
=
1  x 1  e 



1
WPlanckl (ω)  k T  ln[ e n  ]  k T  ln[
  ]



1

e
n 0


ln[1  e  ]

e  
1
 kT 
 k T   



1 e
e 1
 kT 
WPlanckl (ω)  
1

(59)
e kT  1

 kT
Notice: WPlanckl (ω) 
0
WPlanckl (ω) 

 0

With this result, expression (39) becomes
I (ω) 
1
ω 2 WPlanck
2 2
3 c
(60)
5
I()
Experimental
results

Particle’s and wave’s energy quantization
 Historically. Planck initially (1900) postulated only that the energy
of the oscillating particle (electrons in the walls of the blackbody) is
quantized. The electromagnetic energy, once radiated, would
spread as a continuous.
 It was not until later that Plank accepted that the oscillating
electromagnetic waves were themselves quantized. The latter
hypothesis was introduced by Einstein (1905) in the context of
explaining the photoelectric effect, which was corroborated later by
Millikan (1914).
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