Quantum Theory of Light - Introduction to Modern Physics - Lecture Notes, Study notes of Physics

Download Quantum Theory of Light - Introduction to Modern Physics - Lecture Notes and more Study notes Physics in PDF only on Docsity! 1 2. Quantum theory of light in the 1900s, unsolved problem with classical statistical mechanics (the works of Maxwell, Clausius, Boltzmann, Gibbs (American!) – calculating intensity of radiation at a given wavelength emerging from a heated cavity - deeper questions is what it light, is it a wave (Maxwell) or is it a stream of particles/corpuscles (Newton) Solution by Max Planck: Annalen der Physik IV, Folge 4, 553-563 (1901) “On the distribution law of energy in the normal spectrum” Approaching form a new angle what light may be, setting the scene for Einstein) Maxwell: from theory light is just as any other electromagnetic wave a disturbance traveling through space Heinrich Hertz electromagnetic waves produced by electrical oscillations travel trough space, behave just as light waves (reflect, refract, focuses, polarize, interfere) in every other aspect, (60 cm wavelength)! After that daring scientists speculated: results from extremely high frequency electric oscillators in matter. What are these oscillators, nobody knew, Planck called them resonators – assumption was the frequency of the light wave is equal of the frequency of the oscillators highest electrical oscillation produced 5 108 Hz, but visible (500 nm) light has frequency 6 1014 so it is extrapolated that the same physics should apply over 6 more orders of magnitude, this is indeed a very daring proposition! docsity.com 2 Blackbody radiation Observation: quite independent of the material, shape one gets graph Fig. 2.3 Challenge: Why is that so? predict the radiation intensity at a given wavelength and temperature Kirchhoff’s theorem: ef = J(f,T) Af body in thermal equilibrium with radiation, pretty much like a hot liquid in a container, same temperature everywhere as liquid (radiation) and container (cavity walls) have exchanged energy many times, stationary state ef power emitted per unit area Af fraction of incident power absorbed per unit area and frequency J(f,T) universal function, same for all bodies regardless of material, shape,… Black body Af = 1 for all f - appears pitch black ef = J(f,T) only a function of temperature and frequency, perfect radiator (power) docsity.com 5 an approximation for these constants A and ß, and the new formula fits everywhere from experimental data (Rubens at al.), values for h = 6.55 10-27 erg sec, kB = 1.346 10-16 erg/grad (Boltzmann’s constant) today's best values are h = 6.6260755(40) 10-34 Js k = 1.380658(12) 10-23 J/K i.e. Planck’s curve fitting results where too low by only ˜ 1.15% and 2.51%, respectively. Quantum of Energy Planck assumed walls of cavity are made up of billions of resonators, all vibrating at different frequency, emitted radiation has to be at different frequencies giving a curve analogous to Fig. 2.3 Now classical mechanics says: oscillator can have any value of frequency and energy, changes its amplitude by any incremental amount when energy is radiated off –resulting in Wien’s law Planck: total energy of resonator with frequency f is always an integral multiple of an energy quantum hf (it’s kind of like saying: you can not sip your wine at will, you have to drink it in units of glasses that are defined by the size of particular glass used) docsity.com 6 Eresonator = n hf n = 1,2, ….usually a very high number Emission when resonator drops one energy level ∆E = h f = one quantum of energy it’s like a bank with $ 1 000 000, can only give out physically money in multiples of 1 cent, = one quantum of money Planck’s law also avoids the ULTRAVIOLET CATASTPOPY inherent in Rayleigh-Jeans Law, which works well for low frequencies Another great triumph of Planck’s formula: quantifying Stefan’s constant etotal = 4 0 Tdfe f σ=⋅∫ ∞ σ = 5.67 10-8 Wm-2K-4 finding a physical meaning for σ etotal = ∫∫∫ ∞∞ ∞ − = − =⋅ 0 3 32 44 0 5 2 0 1 2 )1( 2 ),( 4 dx e x hc Tk d e hc dTu c x B Tk hc B π λ λ π λλ λ where we substituted x = Tk hc Bλ now ∫ ∞ = −0 43 151 π dx e x x docsity.com 7 etotal = 4 32 45 15 2 T hc kB ⋅ π and have the meaning of σ that’s how nature is - there are only a very few fundamental constants, all the other constants are derived from these fundamental constants Photoelectric effect and Light quantization facts about photoelectric effect: ultraviolet light, short wavelength, high frequency, i.e. high energy (E = h f) impacting on a metal surface lead to the ejection of electrons (photoelectrons) from that metal with a range of velocities, kinetic energy ½ m v2 (nonrelativistic) There is a maximal kinetic energy KEmax (velocity) of photoelectrons which does not depend on the intensity (I) of the exciting light, but KEmax ~ f KEmax easily measured KEmax = ½ m vmax2 = e Vstop (independent of intensity, I) Photocurrent ~ I - to be expected classically Linear relationship of KEmax to f - very strange, hinting at Planck’s relation E = h f - not explained classically docsity.com 10 Compton Effect 1906 Einstein: a photon moves with c all the time, is never at rest, carries a relativistic momentum p r = E/c = hf/c = h/? [kg m/s = Js/m = Ns] in Einstein’s own words: if a bundle of radiation causes a molecule to emit or absorb an energy packet hf, then momentum of quantity hf/c is transformed to the molecule, directed along the line of motion of the bundle for absorption and opposite the bundle for emission Debye/Compton (American) 1923 use this idea for explanation of scattering of X-ray photons by electrons X-rays When a beam of electrons is slowed down, e.g. by hitting a metal target, it produces electromagnetic radiation in the range of 0.1 nm wavelength (f ˜ 3 1018 Hz) , this radiation is very high in energy E = h f (˜ 2 10-15 J = 1,25 104 eV so it’s the same energy an electron would have if it were accelerated by an electrical force going through a potential of 12,500 V) and has a momentum of 6.7 10-24 kg m/s docsity.com 11 as an electron has rest mass of 9,108 10-31 kg this is a rather large momentum, so if an X-ray photon is hitting an electron at rest it will knock it about quite a bit so that the effect can be measured – this makes X-rays also dangerous to living organisms visible light, 550 nm, ˜ 5.45 1014 Hz, ˜ 3.6 10-19 J ˜ 2.25 eV (as you may have guessed from the work function of the photoelectric effect, p ˜ 1.2 10-27 kg m/s does not produce a strong effect we return to X-rays when we discuss: “Applied Modern Physics” -------------------------------------------------------------------------- classical prediction based on Maxwell’s wave theory of light: - incident wave of f0 should accelerate electron in direction of propagation of wave, electron should start oscillating and reradiate wave of frequency f’ smaller than f0 f’ should depend on intensity of wave (we had something like it for the photoelectric effect, hinting on classical premise the energy is not quantisized) don’t take fig. 2.23 in textbook literary it’s wrong so it can’t be shown correctly experimental result: shift of f = ∆ f = f0 - f’ of X-ray photon is independent on intensity (we had something like this for the photoelectric effect, hinting at energy quantization) so we have to expect that speed of light, Planck’s constant and rest mass of electron will be in the formula that explains effect docsity.com 12 experimental result ?’ – ?0 = )cos1( Θ−cm h e ? scattering angle of photon idea conservation of momentum components of momentum have to be resolved in x and y as scattered photon and electron head off in different directions for x-component: ppho + 0 = ppho’ cos ? + pele cos f for y-component: 0 + 0 = ppho’ sin ? - pele sin f replace ppho with hf/c , rearrange, multiply by c pele c cos f = hf – hf’ cos ? pele c sin f = hf’ sin ? each equation squared, adding the two of them together (with cos 2 f + sin 2 f = 1) pele2 c2 = (hf)2 – 2(hf)(hf’) cos ? + (hf’) 2 (1) on the other hand: pele2 c2 = KE2 + 2 KE m0elec2 with KE = hf – hf’ pele2 c2 = hf2 - 2(hf)(hf’) + (hf’)2 + 2 m0elec2 (hf-hf’) (2) docsity.com 15 really ? you are not supposed to translate the German in this graph, just assume it is either light or electrons, we will discuss this experiment in detail at the end of this session, after “applied modern physics” – it is the epitome of the quantum weirdness docsity.com 16 well, that’s looks more like it for light, it is Young’s famous 1808 experiment that proved that light is a wave, but wait a minute – we just get the same result for electrons – that’s strange ---------------------------------------------------------------- so is light kind of a wave that consist of particles, or is it kind of particles that are guided/piloted by a wave ? light as an electromagnetic wave: electric (E) and magnetic (B) field vectors (FV) are producing each other in close vicinity, (E) and (B) are perpendicular to each other and oscillate while traveling a distance called wavelength (?) from magnitude +FVmax to – FVmax and back to +FVmax. So the average FV is zero, how can such a wave have some effect on matter ??? How can sunlight produce a sunburn on human skin if the average effect, i.e. FV, is zero??? docsity.com 17 answer: FV are not physically relevant entities, after all, they are only models, and B can be considered to be the effect of a moving charge + relativity, i.e. does not really exist, to do something, burn human skin energy is needed, the longer one is exposed to this energy so larger the effect hence, we have to consider the intensity of the wave per unit area (I) (which has unit energy per area (W m-2) as the physically significant parameter for interactions with matter I = e0 c average (FV2) over both one ? and f cycle - that is a physically significant I in a particle description the intensity is energy divided by area I = N hf/area where N is the number of photons both descriptions of I must give same value for intensity N = 20 averageFVhf areac ⋅ε (over both ? and f cycle) if N is large, one observes a full blown interference pattern and one would describe it by the wave theory if N is medium, one observes a fuzzy interference pattern (and may not sure how to describe it properly) docsity.com