3. Waves and particles. on the particle properties (photons) of electromagnetic radiation (light)

Transcription

1 3. Waves and particles on the particle properties (photons) of electromagnetic radiation (light)

2 3.1 Electromagnetic waves in a nutshell Historically, waves have been identified first: this was a big success of classical physics: diffraction phenomena, wave optics modern knowledge: electromagnetic radiation has also particle properties (wave-particle dualism) FU - Physik III - WS 2001/2002 I.V. Hertel 2

3 3.1.1 Maxwells equation, wave equation just to remind you of Maxwell s equations: divd free divb 0 rote B t D 0 E roth jfree D t B µµ 0 H wave equation: E 1 c 2 2 E t 2 1 with c c 0 0 µµ 0 n with µ 1 n (indexof refraction) FU - Physik III - WS 2001/2002 I.V. Hertel 3

4 3.1.2 Plane and spherical waves solutions e.g. plane waves E E cos k r t or E E 0 exp i k r t or spherical waves E E exp i kr t kr where k is the wave vector k 2 in vacuum: k E B H 0 0 ce 0 k is related to the angular frequency 2 : k /c or c (c phase velocity) k 2 / 2 /c 1 : 8065 cm 1 1 ev 1 cm 1 1Kayser FU - Physik III - WS 2001/2002 I.V. Hertel 4

5 3.1.3 Running waves, standing waves (resonance) k k k L L=m(λ/2) with m = 1,2,3,4 or k 2 2 L m FU - Physik III - WS 2001/2002 I.V. Hertel 5

6 3.1.4 Index of refraction in dielectrics for a monochromatic wave E cos kr t for simplicity often written: expi kr t x x given: 0 /n and k nk 0 for dielectrica(without proof): c 0, n 0 =1 (vacuum) c<c 0 n=1.5 (glass) n 1 for low res n 1; c c for high res n 1; c c 0 ù FU - Physik III - WS 2001/2002 I.V. Hertel 6

7 3.1.5 Index of refraction in metals for metals: 1 P 2 n n 1 in 2 (complexindexof refraction) expi kr t exp n 2 k 0 r expi n 1 k 0 r t damping term = absorption FU - Physik III - WS 2001/2002 I.V. Hertel 7

8 3.1.6 Energy density (w) and intensity (I) energy density in el. magn. field: w t 1 2 0E 0 2 cos 2 t µµ 0 H 0 2 sin 2 t w t w 1 2 0E 0 2 w 0 E 2 As 2 m 2 N V 2 m 2 J2 Jm 3 J m 3 Nm m 3 N m 2 P rad radiationpressure intensity I w c with I J m 2 s W m 2 I w c I 1 2 0cE 0 2 E 0 2 Pointing vector: S E H energy flux density and directionof flow ( k ) S I FU - Physik III - WS 2001/2002 I.V. Hertel 8

9 3.1.7 Intensity dependence on distance plane wave: none spherical wave (point source): I E 0 2 1/r FU - Physik III - WS 2001/2002 I.V. Hertel 9

10 3.2 The photo electric effect Basic facts 1.) N e I light E 2 for very small intensities: single electrons (photomultiplier) 2.) dependence on wavelength short wavelength (UV): gives electrons long wave length (red): no e-signal e g metal surface photocurrent ν 0 frequency ν FU - Physik III - WS 2001/2002 I.V. Hertel 10

11 3.2.2 Photo effect: contradiction to classical theory Why dependent on frequency? Cutoff cannot be understood classically! Electron energy classically it should be related to intensity ( I E 2 ) (high field high accelleration high frequency) FU - Physik III - WS 2001/2002 I.V. Hertel 11

12 3.2.3 The photoelectric effect: experiment 0,8 e 0,4 A U + - U / V 0,0-0,4-0,8 U Linear Fit of Data1_U measure U for current=0 eu= E kinmax 0,0 2,0x ,0x ,0x10 14 eu=a+b 1 ν=hν - W n / Hz h mess = B 1 = ev s W= A= -0,27 ev h literatur = ev s = Js FU - Physik III - WS 2001/2002 I.V. Hertel 12

13 3.2.4 Photoelectric effect: the result E kinmax =hν-w A W A =work function (solid state) ionisation potential (atoms, molecules) Planck s constant h= Js in electron volts = evs E photon = hν: energy quantum of photon e.g.: yellow light (sun, Na-street lamp) of 590 nm ν= Hz, E photon =hν= J =2.1 ev FU - Physik III - WS 2001/2002 I.V. Hertel 13

14 3.2.5 Photo effect: interpretation simple potential model (well) for quasi free electrons in metal: pot. energy hν E kinel W A electron lake ( e-see ) x FU - Physik III - WS 2001/2002 I.V. Hertel 14

15 3.2.6 Photon properties modern knowledge: particle properties (wave-particle dualism) photons ( ) energy packets: energy E h mass m 0 0 momentum angular momentum p k h/ (Vorgriff) (Vorgriff) extension? how big is a photon?... probability to find it is given by wavefunction (wavepacket) can be very limited in space or extended or FU - Physik III - WS 2001/2002 I.V. Hertel 15

16 3.2.7 Photon counting at low intensities with photomultiplier: single, isolated events can be registered photon counting statistics: Poisson distribution w N N N N! exp N <N> = total number of counts 0,3 0,25 <N>=2 <N>=5 0,2 0,15 0,1 0, FU - Physik III - WS 2001/2002 I.V. Hertel 16

17 3.2.8 Intensity in the photon picture I E ges A t N ph h A t N ph A t h photon current density photon energy I N ph A t h n ph c h photondensity velocity energy example: Ar ion laser h J 1mol photons in about 3h L 20 W L/ h photons/ s photon density: n ph photons/ cm 3 (c.f. gas: atoms/ cm FU - Physik III - WS 2001/2002 I.V. Hertel 17

18 3.2.9 Radiation pressure and photon momentum c.f. kinetic gas theory (however only one direction): radiation pressure was P ph w n ph h P ph photonfluxdensity photon momentum n ph c p ph number m 3 m s Ns N m 2 p ph h /c h/ k p ph k... experimentalproof later FU - Physik III - WS 2001/2002 I.V. Hertel 18

19 Photon momentum: relativistic considerations rest mass m 0 0 E ges m 0 2 c 4 p 2 c 2 p ph c E ph h E ges p ph c p ph h /c on the other hand: E ges h m ph c 2 m ph h /c 2 (at velocity c) FU - Physik III - WS 2001/2002 I.V. Hertel 19

20 3.3 Black body radiation Planck s radiation law 101 years of quantum mechanics (Planck s 1900 lecture on black body radiation) Nobel Prize 1918 Planck, Max Karl Ernst Ludwig Geboren Kiel gestorben Göttingen FU - Physik III - WS 2001/2002 I.V. Hertel 20

21 3.3.1 What is a black body? (schwarzer Körper, Hohlraumstrahler...) all radiation is absorbed emission is identical to radiation density inside the interesting quantity to measure is the spectral radiation density w ν (ν) FU - Physik III - WS 2001/2002 I.V. Hertel 21

22 3.3.2 Schematic of experiment for Planck s law detector black body radiator focussing lens prisma (dispersion) FU - Physik III - WS 2001/2002 I.V. Hertel 22

23 3.2.3 Facts about Planck s radiation law (black body radiation) w ν (λ) wavelength λ FU - Physik III - WS 2001/2002 I.V. Hertel 23

24 3.3.4 Planck s radiation law the formula w d 8 h 3 c 3 d exp h / k B T 1 derivation later properties now: FU - Physik III - WS 2001/2002 I.V. Hertel 24

25 3.3.5 Wien s (displacement) law: maximum of the distribution rewrite Planck's law for wavelength w d 8 hc 5 d exp hc/ k B T 1 dw/d! 0 for maximum of distribution max T const m K FU - Physik III - WS 2001/2002 I.V. Hertel 25

26 3.3.6 Radiation from the sun: surface T=5900K 1.2e+06 on the solar radiation 1e λ / nm FU - Physik III - WS 2001/2002 I.V. Hertel 26

27 3.3.7 Spectral distribution of solar radiation O FU - Physik III - WS 2001/2002 I.V. Hertel 27 µm

28 3.3.8 Number of modes per unit cell modes in the black body, cube L 3 for standing waves in 1D: k y m L m 0, 1, 2... or k 2 2 L m in 3D with k 2 / 2 /c : d 3 k k x 2 /L m x k k y 2 /L m y k x k z 2 /L m z with m x, m y, m z 0, 1, 2, 3,... k-space cell: k x k y k z 2 /L 3 k 2 L 3 (unit cell volum in k-space) FU - Physik III - WS 2001/2002 I.V. Hertel 28

29 3.3.9 Rayleigh-Jeans law; UV catastophy k y k 2 2 L m 3 k 2 L 3 (unit cell volum in k-space) k d 3 k number of modes in d 3 k : dm dk xdk y dk z 2 /L 3 L k2 dk d in -space dm L 3 2 c 3 d d k x integratedover all angles and 2 polarisations: dm L c 3 d density of modes (per Volume) dm L 3 w from (classical) equipartionlaw: 8 2 c 3 d each vib. mode energy kt w d 8 2 c 3 ktd Rayleigh-Jeans law! increases with (UV-catastrophy) FU - Physik III - WS 2001/2002 I.V. Hertel 29

30 Planck s quantitative derivation of radiation law Planck s hypothesis: energy discrete W nh (with n=0,1,2...) probabilityfor each energy: p W exp nh /kt averageenergy per mode: nh p W h exp h /kt 1 for a proof read e.g. Demtröder 3, S.77 with mode density dm/l 3 follows Planck s law w 8 h 3 c 3 d exp h /kt FU - Physik III - WS 2001/2002 I.V. Hertel 30