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Chapter 2 Particle properties of waves Electronics: particles charge , mass Wave? Electromagnetic wave: wave diffraction, interference, Polarization particle? Wave-particle Duality 2.1Emwaves Changing magnetic field current (or voltage) Maxwell proposed: changing electric field magnetic field Hertz created EM waves and determined the wavelength and speed of the wave, and showed that they both have E and B component, and that they could be reflected, refracted, and diffracted. Wave characteristic. 1 Principle of Superposition: When two or more waves of the same nature travel past a point at the same time, the instantaneous amplitude is the sum of the instantaneous amplitude of the individual waves. 2 Constructive interference Destructive interference same phase, greater amplitude different phase, partial or completely cancellation of waves Interference wave characteristic Young’s diffraction experiments: diffraction wave characteristic 3 2.2Blackbody radiation Is light only consistent of waves? Amiss: understand the origin of the radiation emitted by bodies of matter. Blackbody: a body that absorbs all radiation incident upon it, regardless of frequency. A blackbody radiates more when it is hot than it is cold, and the spectrum of a hot blackbody has its peak at a higher frequency than that of a cooler one. 4 Considering the radiation inside a cavity of absolute temperature T whose walls are perfect reflectors to be a series standing EM waves. L n* / 2 Density of standing waves in cavity G( )d 8 2d / c3 The higher ν, the shorter the wavelength, and the greater the number of possible standing waves. The average energy per degree of freedom of an entity that is a member of a system of such entities in thermal equilibrium at T is 1/2kT. K is Boltzmann’s constant=1.381*10-23J/k 5 An idea gas molecular has three degree of freedom: kinetic energy in three independent directions 3/2kT One dimensional harmonic oscillator has two degree of freedom: kinetic energy and potential energy. Each standing wave in a cavity originates in an oscillating electric charge in the cavity wall. Two degree of freedom. Classic average energy per standing wave kT Total energy per unit volume in the cavity in and u ( )d G( )d (8kT / c3 ) 2d increase +d Rayleigh-Jeans formula energy density increase with 2 . In the limit of infinitely high frequencies, u( )d goes to infinity. In reality, the energy density(and the radiation rate)falls to 0 as goes to infinity. Ultraviolet catastrophe 6 Plank Radiation Formula u( )d =(8πh/c3)( 3d )/(eh /kT-1) h is planck’s constant=6.626*10-34Js h >>kT eh /kT u( ) 0 No ultraviolet catastrophe. In general, ex=1+x+x2/2+……… When h << kT, 1/(eh /kT-1)~1/((1+(h /kT)-1)~kT/h u( )d ~(8πh/c3)( 3d )/( kT/h )~(8πkT/c3) 2d which is Rayleigh-Jeans formula. 7 How to justify the Plank radiation formula The oscillators in the cavity walls could not have a continuous Distribution of possible energy ε but must have only specific energies εn=nh n=0,1,2…… An ocillator emits radiation of frequency when it drops from one energy state to the next lower one, and it jumpsto the next higher state when it absorbs radiation of . Each discrete bundle of energy h is called a quantum. With oscillator energies limited to nh , the average energy per oscillator in the cavity walls turn out to be not kT as for a continuous distribution of oscillator energies, but ε=h /(eh /kT-1) average energy per standing wave 8 2.3 Photoelectric effect Some of photoelectrons that emerge from the metal surface have enough energy to reach the cathode despite its negative polarity Current When V is increased to a certain value V0, no more photoelectrons arrive. V0 correspond to the max photoelectron kinetic energy. Three experimental finding: (1)No delay between the arrival of the light at the metal surface and the emission of photoelectrons. (2) A bright light yields more photoelectrons than a dim one, but highest electron energy remain the same. 9 (3)The higher the frequency of the light, the more energy the photoelectrons have. At the frequencies smaller than 0, which is a characteristic of the specific metal, no more electrons are emitted. Quantum theory of light Einstein proposed Photons. The energy in light is not spread out, but is concentrated in small packets. Each photon of light of frequency has the energy h . Einstein proposed that energy was not only given to em waves in separate quanta but was also carried by the waves in separate quanta. 10 Explanation of experiments: (1)Since em wave energy is concentrated in photons and not spread out, there should be no delay in the emission. (2)All photons of frequency have the same energy h . Changing the intensity of light only change the number of photoelectrons but not their energy. (3)The higher , the greater photon energy and so the more energy the photoelectrons have. νo corresponds to the min energy Φ for the electron to escape from the metal surface. This energy is called work function. Φ=hνo Photoelectric effect hν=kEmax+Φ h = kEmax+hνo kEmax=h(ν-νo). Photo energy E=(6.626*10-34Js/1.602*10-19J/eV )ν=(4.136*10-15)νeVs ν=c/λ E=1.24*10-6eVm/λ 11 What is light Wave model: light intensity Particle model: light intensity N E2 N(#of photons/sec.area) E2 .N is large interference pattern N is small a series of random flashes .if keep track of flashes for long time same as large N intensity of wave at a given place on the specific space the probability of finding photons. 12 Wave & quantum theory complement each other. photoelectric effect : Ephotons yes x-ray faster e΄ more x-ray # of e΄ increase Intensity of x-ray increase .for given accelerating V V .most of e΄ Ee΄ λ min λ min heat A few e΄ lose E in single collisions 13 x-ray .x-ray are em waves EM theory predicts that an accelerated electric charge will radiate em waves, and a rapidly moving e΄ suddenly brought to rest is certainly accelerated Bremsstranlung (“braking radiation”) x-ray at specific λ nonclassical * different targets give different characteristic x-ray * for the same V, λmin is the same for different materials λmin=(1.24 10-6)/V(m) 14 hνmax = Ve = hc/λmin λmin = hc/Ve = (1.24 10-6)/V Scattering by an atom (wave model) atom in E polarized distorted charge distribution electric dipole em wave with ν on atom polarization charge with ν oscillating electric dipole radiate em wave x-ray falls on a crystal will be scattered in all directions because of regular arrangement of atoms Bragg’s condition (2dsinθ=λ) 15 constructively interference Compton effect Loss in photon energy=gain in e΄ energy hν -hν΄ =kE for massless particle E= Pc (P=momentum) photon momentum P =E/c = hν/c 16 hν/c = (hν΄/c)cosΦ+ Pcosθ (parallel)……..(1) 0 = (hν΄/c)sinΦ – Psinθ () ……………….(2) (1)&(2)x c Pc(cosθ) = hν -hν΄cosΦ Pc(sinθ) = (hν΄)sinΦ P2c2 = (hν)2 – 2(hν)(hν΄)cosΦ +(hν΄)2 17 & E = KE + moc2 E= mo c 4 P 2c 2 2 (KE + moc2)2 = mo2c4 + P2c2 P2c2 = KE2 + 2moc2KE Because KE = hν - hν΄ P2c2 = (hν - hν΄)2 + 2moc2KE 2moc2(hν - hν΄) = 2(hν)(hν΄) (1 –cosΦ)……(3) (3)/2h2c2 moc/h(ν/c - ν΄/c) =(ν/c)( ν΄/c)(1 –cosΦ) moc/h(1/λ – 1/λ΄) =(1 –cosΦ)/( λλ΄) λ΄ - λ = (h/moc)(1 –cosΦ) = λc(1 –cosΦ) λ=Compton wavelength 18 Relativistic formulas Total energy E= mo c 2 mo= rest mass v2 1 2 c mo v Relativistic momentum P = When 1 v c2 mo = 0 (massless particle) & v<c How about v=c , mo= 0 E2 = 2 2 mo c 4 2 1 v 2 c , E –P c 2 2 2 P2= 2 mo c 4 = 2 1 v 2 c E=P=0 E=0/0 , P=0/0 (any values) 2 mo v 2 v2 1 2 c P2c2 = 2 mo v 2 2 c v2 1 2 c (1-v2/c2) = mo2c4 E2 = mo2c4 +P2c2 For all particles E= m0 c 4 p 2c 2 2 = Eo p 2c 2 2 Restriction of massless particles : E = Pc (mo=0) Total energy 2 2 mc =moc +KE = 19 mo c 2 1 v 2 c2 Pair production A photon give an e΄ all of its energy part of its energy a photon materialize into an e΄ & position 20 photoelectric compton (momentum is conserved with the help of the nucleus which carries away enough photon momentum) rest energy moc2 of electron or position is 0.51Mev pair production requires a photon energy 1.02Mev pair production cannot occur in empty space concervation of energy hν = 2mc2 momentum concervation P = mv hν/c = 2Pcosθ hν = 2mc2(v/c) cosθ v/c <1 & cosθ 1 hν < 2mc2 21 hν = 2Pc(cosθ) linear attenuation coefficient dI udx I I = Ioexp(-ux) absolute thickness x= ln( Io I ) u 22