Chapt. 18 Lecture notes (Part I)(2)

Chapt. 18 Lecture notes (Part I)(2) - Chapter 18...

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1 Chapter 18 Bose-Einstein Gases (Part I) 18.1 Black-Body Radiation—a perfect quantal gas 18.1.1 Introduction First, one may ask: What is black-body radiation? A perfectly black body radiation absorbs all radiation falling on it. This is very similar to the case that radiation enters a small hole in the wall of an opaque enclosure: after many reflections nearly all radiation is absorbed by the inner surface of the enclosure. (Fig. 18.1) All bodies emit electromagnetic (EM) radiation by virtue of their temperature but usually this radiation is not in thermal equilibrium. If we consider the radiation within an opague enclosure whose wall is kept at a uniform temperature T, then radiation and walls reach a thermal equilibrium state, in which the radiation has quite definite properties. To study this equilibrium-state radiation, one cuts a small hole in the wall of the enclosure. Such a hole, if sufficiently small, will not disturb the equilibrium in the cavity and the emitted radiation will have the same properties as the cavity radiation. The emitted radiation also has the same properties as the radiation emitted by a perfectly black body at the same temperature T as the enclosure. For this reason, cavity radiation is also called black- body radiation . To the end of 19 th century, black-body radiation was one of the most outstanding problems in classical physics. By that time, it had been known that both heat radiation and light radiation are essentially the same – EM wave . The nature of thermal EM radiation was a subject of intense interest—a central issue in physics in that time! In these studies, a central physical quantity is the density of radiation energy u( ) : u( )d is the radiation energy in the frequency range of to +d . In 1894, Wien found an empirical formula: , ) ( / 3 1 2 d e c d u T c (18.1) which agrees with the experimental data well at high frequency .
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2 (Fig. 18.2) In 1900, Rayleigh and Jeans applied classical EM theory and classical statistical physics to treat black-body radiation and obtained , 8 ) ( 2 3 d kT c d u (18.2) where c is the speed of light. This theoretical formula agrees with the experimental data well at low frequency . As  , however, u( )  . This is the so-called ultraviolet catastrophe—one of the biggest problems faced by classical physics ! Then in 1900, Max Plank obtained an excellent empirical formula . 1 ) ( 3 / 1 2 d e c d u T c (18.3) This formula agrees very well with the experimental data in the whole frequency range, both low and high . It can be easily seen from Eq.(18.3) that as  , u( ) 0. This removed the so-called ultraviolet catastrophe! At high , Eq. (18.3) reduces to Eq.(18.1). More remarkably, Plank did not stop there! Two months later, he found that Eq. (18.3) can be derived theoretically if one makes the following assumption: For EM radiation with a frequency , its energy can only be absorbed or emitted in the multiple of h , that is, E=nh , where n is an integer. In other words, the energy of EM radiation in emission and absorption is quantized. This is the concept (assumption) of
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Chapt. 18 Lecture notes (Part I)(2) - Chapter 18...

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