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THE ORIGINS OF QUANTUM MECHANICS
1-BLACKBODY RADIATION.
Blackbody radiation is a common phenomenon that probably is
familiar to you. When you see stars of different colors, when you
observe an electric heating coil on a stove turn red, or when you observe
a lightbulb, you are observing blackbody radiation. A blackbody is a
device that converts heat into radiant energy. Heating an object to
different temperatures causes that object to radiate energy of different
wavelengths and therefore, different colors
E.g. Heating a steel
color changes
by increasing temperature
Color depends only on temperature Not on type of material
Color observed due to characteristic distribution of light emission at a
range of wavelengths
If you know the color you know the temperature
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The best laboratory black-body is a spherical cavity which is
constructed with insulating walls in one of which a small pinhole is
made.
Classical (Rayleigh-Jeans)
( , )
= 5000
= 4000
= 3000
When black-body heated it is observed to radiate a spectrum of
wavelengths having a characteristic energy density at each frequency.
(Draw the spectrum with )
Classical Explanation (Rayleigh-Jeans law)
They supposed that
1. Blackbody radiation is coming from standing electromagnetic
waves in the cavity that are in thermal equilibrium with the
vibrating atoms (or electrons) in the walls
2. The atoms in the blackbody are assumed to vibrate like harmonic
oscillators
3. according to the principle of equipartition of energy an oscillator
in thermal equilibrium with its environment should have an
average energy equal to kT
4. We already found out that (
=
) for standing waves
in sphere from classical physics
5. the energy density is
(
)
and so the relationship between the energy density and the frequency
of the radiation is (for unit volume)
Where
(
)
=
(
2
8)
is the energy density in units of energy per volume (e.g., J m-3),
is the frequency of emitted radiation,
T
is the temperature of the blackbody,
k
is Boltzmann's constant, and
c
is the speed of light.
This is known as the Rayleigh-Jeans law. A clear implication of this law is
that as the frequency becomes larger, the energy density increases as
the square of the frequency.
This is known as the "ultraviolet catastrophe", (why)
No maximum
energy density
Short wavelengths strongly excited
Everything glows at room
temperature?
Long Wavelengths Works OK.
CLASSICAL THEORY FAILS
new theory is required
Quantum Explanation (Max Planck)
Max Planck started from the standard assumption, that
1. blackbody could be modeled as a collection of oscillators
2. He then made an assumption that the oscillators could only take
on discrete, quantized energies, these energies being described
as
where
n
is the oscillator energy,
is a whole number (0, 1, 2, ...),
is the frequency, and
is some arbitrary constant having units of (energy
h
time).
3. With this assumption, and using Boltzmann distribution law ,
Planck derived
(
)
=
(
3
1)
at low frequencies
using the series expansion for the exponential
/
1+
we recover the Rayleigh-Jeans formula.
8
( , ) =
At high frequencies
/
( , )
=
1
8
/
the exponential dominates the cubic in and so goes to zero.
So using quantization is required, for correct behavior at high
frequencies. (Energy is quantized)
The constant h has come to be called Planck's constant, and its value is
6.626 x10-34 J s.
Expression of the energy density ( , ) in terms of wavelength
( , )
8
( , ) =
( /
1)
( , )
=
=
(
=
8
/
1)
Wien displacement law
He states that the wavelength distribution of thermal radiation
from a black body at any temperature has essentially the same shape as
the distribution at any other temperature, except that each wavelength
is displaced on the graph.
He states that if max is the wavelength at which ( , ) is a
maximum, and so
( , )
=
(
4
8
/
1)
=0
By differentiation of the above equation with respect to and
substitution by the constants value (h, c , k) it is found that
= 2.897 10
Example: calculate the temperature used to heat a black body to emit
radia on at 400 nm
2.897 10
2.897 10
=
=
= 7242.5
400 10
Stefan–Boltzmann law
The law states that the total energy radiated per unit surface area
of a black body per unit time (also known as the black-body irradiance or
emissive power), P, is directly proportional to the fourth power of the
black body's thermodynamic temperature T.
=
Where = 5.6697 10
By integration the Plank's law to get the total energy density so
8
( , ) =
=
( /
1)
=
8
15
The relation between the emissive power and the total energy density is
=
4
2
15
=
4
4
=
2
15
2
=
15
By substitution by the constants value ( , h, c , k) it is found that
= 5.670 10
=
Example: Calculate the intensive power of the above black body
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