Quantum Mechanics Notes: Rayleigh Energy Identity Distribution

(11) Rayleigh energy density distribution In 1900, Rayleigh focused on understanding the nature of electromagnetic radiation He considered radiations consist of standing waves having temperature T. These standing waves are equivalent to harmonic oscullator, due to harmonic osaliations of large no: of electric charges at the walls of the cavity. Acc: to this the no: of waves having wavelength b/w a and a + da is, E,da = 8TKT2 dr. Rayleigh- - Jean Formula was found to agree with experimental graph only for longer wavelength regions and fails at shigh wavelength region (This failure was reffered as the ultraviolet catastrophe) (iii) Planck's Energy By analysing the interpolation b/w wein's rule and Rayliegh - Jean rule, Planck sweeded in 1900 in avoiding ultraviolet catastrophe and proposed accurate discription of black body radiation. Planck considered that the energy exchange between matter and radiation must be disacte.

He postulated that the energy of radiation ( of frequency v) emitted by the osullating charges must come only in integral multiples of ht E=nhr h= = 6.626x10 JS. -34 we This eq: represents planck's quantization rule for energy Based on quantum Theory Plank deduced the law of black body as E 8ThC 7-5 Add e hc/ AKT , This Formula fit well with experimental curves over entire range. weirs; E law law [energy density Planck's law a Planck's Quantum Hypothesis He assumed that the atoms in the cavity of the black body was filled with electromagnetic linear oscillators, each having a characterstic frequency of Oscillation . These oscillators emit electromagnetic energy into the cavity and reach equilibrium at some Particular temperature, when radiation would be absorbed as well as emitted by each oscillator. The problem of spectral distribution of radient energy in the cavity can be reduced to that of the average

energy of an oscillator at a certain temperature. Plank made 2 assumptions 1 The linear oscullators can vebrate with the integral values of 0, E , , 2 E , 3 E n E . E elementary quantum of energy (ii) Oscillators can emit or absorb (hv) in discrete units called quanta. when an osullator is in a quantised state it neither emit nor absorbs energy Applying the law of probability, E 1 average energy per oscillation, E = e&/KI 1 Here a is very small :. e -T - tends to E/KT and E/KT Shence average energy will become KT. So E should be made to be equal to a finite quantity greater than zero- plank observed that wein's and Rayleigh's formula both contained a Function of (AT). He assumed that E was proportional to 2 of the osullator. 800 = E = hr 1) 11 in I E = hr ny/KT e -1 The no: of modes of vibrations per unit Volume lying within the wavelength range a and at da = 8IT 2 -4d7 :. Energy Density distribution, dE If 8TH2 -4 da hr by/KT e /

But c=va :- E = 8Thcadd -5 e hc/AKT -1 This is plank's formula. This formula fits well with experimental curves as well When, hc/AKT>> T : planck's Formula reduced to wain's Formula When F/AKT < T : Planck's Formula reduced to Ray leigh -Jean's Formula. Evidences for Quantum Theory 1. Photoelectric emission It is the emission of electrons from the Surface of certain substance , mainly metals / when they are illuminated by electromag netic radiations like X- rays , UV and even visible light The electrons emitted are called photoeledrons. 0 Alkali metals like Li, Na,K etc emit photoelections when exposed to visible light UV ray eject election from Ln, Mg IR ray can eject photoelections from caesium 0 Einstein explained photo electric emission using quantum theory [ wave theory could'nt explain it] Acc: to Einstein, En energy is not emitted and 7 absorbed in discrete energy quanta, but also it propagates through space in definite quanta with the speed of light

Let the energy of the incident photon on photogenschive material be hv. The insident energy is completely given to one free electron. part of it is used to extract the free electron and the remaining is used to impart K.E to the emitted electron. Let K E = + mv2 m mass max 2 V velocity Let W is the work function of the metal which is the minico minimum energy required to eject an electron to the surface of the metal with zero velouty . Then, hr W+Lmv2 This relation is known as Einstein's photoetectric equation 2. Compton Effect. When a homo geneous beam of X- rays of sharp defined frequency v were incident on a light element like carbon or aluminium , the x-rays suffered a change of frequency on scattering The scattered beam contained two wavelengths, (i) One scattered beam had the same wavelength as the incident beam or primary beam. (11) The second beam had a wavelength longer than that of primary beam.

The change in wavelength due to the loss of energy of the incident x-ray due to elastic interaction is called Campton effect. This effect can be explained using quantim theory of radiation The whole process is considered as an elastic collision of x-ray photon with a loosly bound electron of the gratherer. The change. in wavelength, da = h (1-wso) mc h Compton wavelength E Compton 0 shift equation mc of free electron da Compton shift Compton effect provides an indirect verification of the quantum theory 3. Quantum theory of Specific heat capacity Specific heat capacity is defined as the quantity of heat required to raise the temperature of unit mass of a body through unit Kelvin. Acc: classical theory, heat capacity at constant volume, Cv=3 3R [ R universal gas constant] means all Solids have same value for the specific heat and are independent of temperature But speufic heat varies with temperature. This could not explained using classical theory in which it was assumed that a body absorbs heat continuously in intof indefinitely small amount Constain