Introduction and Origins of the Quantum mechanics

Introduction and Origins of the

Quantum mechanics

I: Quantum theory of light

1. Black Body Radiation: Catastrophe Ultraviolet and Planck's Hypothesis
   

Cavity with a hole forming a black body

A: Definition: a black body is an ideal body capable of absorbing and emitting all radiation regardless of their wavelengths

When you heat a black body, it emits waves electromagnetic in wavelength range characteristic of its temperature.

To understand this, heating an iron rod

By heating the bar changes color

Consequence: The body emits light when it is heated and it changes color when the temperature increases.

• The heated bar changes from red to purple in color going through orange, yellow, green to purple.
• Each color is characterized by a wavelength l and subsequently by a frequency n.
• The visible spectrum wavelengths from ultraviolet to infrared range from 400nm to 700nm.

The spectral density inside the black body was studied and measured experimentally by placing a spectrograph in front of the hole of the latter.

B- Experimental results:

At each temperature T the spectral density U (l) has a maximum emission at a certain length lmax

spectral density U as a function of   l

Increase of temperature::::::::::::>>>>U (l) increases lmax towards short wavelengths

Experimentally, Wien showed that lmax and the temperature obey the law

Color variation as a function of temperature

Experimentally, Wien showed that l_max  and temperature obey the law

Color variation as a function of temperature

In other words to each temperature corresponds a color well determined by lmax

This length lmax is connected to the frequency l by the relation

c being the speed of light c = 3×108m/s

color change depending on temperature

C- Stefan Boltzmann's Law (1879)

Stephan showed that the power radiated by a black body per unit area (area under the spectrum

 curve) is proportional to T4

s is Stefan Boltzmann's constant

By plotting the power as a function of T4 and by measuring the slope of the line obtained gives the value of s 

Application of Stefan's Law

Estimation of the sun's radius and temperature

For this we consider as a black body (C.N.)

The total power of a black body is:

Ptot=S×P=S× sT4 (en Watt)

If we consider the sun as a spherical star with radius R and area S = 4p R2. The measurement of the total power emitted by the sun using a luxmeter made it possible to determine the radius and temperature of the suN

R_sun=6,957×108 m        Tsun=1 950 °C or           2223 K

D- Theoretical study of black body radiation

In 1900, Reyleigh and Jones (R-J) failed to theoretically explain this phenomenon. This classic model leads to an equation that disagrees with the experimental data in the ultraviolet domain

R-J considered that the atoms of CN behave like harmonic oscillators which radiate an energy density given by:

e(n,T)=N(n)<E>

e(l,T)=N(l)<E>

<E>Average value of the energy of a harmonic oscillator of frequency n

N(n) Harmonic oscillator number frequency n

N(l) Wavelength harmonic oscillator number l

The classical theory of statistical physics of R-J predicts for <E>, N (n) and N (l) the following expressions

kB= Boltzmann constant

Under these conditions, the energy density as a function of l and n
of planned by R-J will be given by:

The results predicted by classical theory presented by Rayleigh and Jones are incompatible with the experimental results:

• No maximum emission therefore no color
• The energy diverges for small wavelengths (the radiated energy is infinite) while the experimental curve indicates zero energy for small l/ large n

It's CN's ultraviolet disaster

First failure of classical physics

E- Corrections apportées par la Mécanique Quantique

To explain the behavior of C.N., Planck assumed that the exchange of energy between matter and radiation cannot be done in a discrete fashion.

Planck hypothesis (1900):

Matter and frequency radiation n can only be exchanged in amounts of energy equal to or multiples of hn

So:E = nhn = nhw

n= 1, 2, 3 ……..

the amount of energy e=hn  is called the quantum of energy

h = 6.626 10-34J.s is Boltzmann's constant

Application of Planck's theory to black body radiation

The Planck hypothesis associated with the methods of statistical physics leads to Planck's formula:

In terms of the frequency n :

In terms of wavelength l:

With the Planck hypothesis, the catastrophe of the U.V. is

fully lifted.

The Planck curve is completely in agreement with the experimental curve.