Chapter 30 - Quantum Physics1 Blackbody Radiation and Planck’s Hypothesisof Quantized EnergyA blackbody is an idealized object that absorbs all incident radiation. A blackbodyat a fixed temperature T is a perfect emitter of radation. The distribution of theamount of radiation given off by a blackbody of temperature T versus the frequencyof the radiation f is shown in the figure. This distribution is the same for anyblackbody, regardless of the material from which the blackbody is made. It onlydepends on the temperature.1) As the temperature increases, the total amount of energy radiated (area underthe curve) increases. (Hotter objects radiate more energy.)L. Whitehead 1 Phys 13022) As the temperature increases, the peak frequency (the frequency at which theblackbody emits the most energy) increases. The relationship between the peakfrequency and temperature is given by Wien’s displacement law:fpeak= (5.88 × 1010(sK)−1)T (1)Kelvin (K) is a unit of temperature. TK= TC+ 273.15, where TKis the temperaturein Kelvin and TCis the temperature in Celsius.This relationship means that the color of a hot object is related to its temperature!(A hot stovetop appears red.) An object that glows red is at a lower temperaturethan an object that glows blue, because fred< fblue.The sun has a surface temperature of 5800 K. What is the peak frequency of radiationemitted by the Sun?fpeak= (5.88 × 1010(sK)−1)T= (5.88 × 1010(sK)−1)(5800K)= 3.4 × 1014HzThis is in the infrared portion of the EM spectrum.Blackbody radiation was understood experimentally in the late 1800’s, but a the-oretical explanantion for the results couldn’t be found based on classical physics.German physicist Max Planck was finally able to derive an expression for the fre-quency spectrum by making this assumption:The radiation energy of a blackbody at frequency f must be an integral multiple ofa constant h times the frequency.En= nhf (2)n = 0, 1, 2, 3, ...We say that the energy is “quantized.” It is not a continuous spectrum; the energycan only take on certain values.Planck’s constant: h = 6.63 × 10−34Js.Classically, energy can be any value from zero to infinity. For classical waves (likesound waves), the energy of the wave depends only on the amplitude, not the fre-qency. This was the beginning of “quantum physics.” Because of the quantization,L. Whitehead 2 Phys 1302energy can only change in small discrete amounts, equal to hf.2 Photons and the Photoelectric EffectSome physicists didn’t really buy that a blackbody really had quantized energy,because they thought of light as a wave (remember Young’s double slit experiment),and a wave can have any energy. Einstein took the idea seriously though, andproposed that light comes in bundles of energy called photons. Light with frequencyf consists of many photons which each have an energyE = hf (3)In this model, a beam of light can be though of as a beam of particles, each carryingenergy hf. Higher intensity means that more photons pass a given point in a givenamount of time, i.e. the photons are more tightly packed together. But each indi-vidual photon in the beam has the same energy.What is the energy of a photon of red light with frequency 4.60 × 1014Hz?E = hf= (6.63 × 10−34Js)(4.60 × 1014Hz)= 3.05 × 10−19JA single photon has a very small amount of energy compared to macroscopic objects.Suppose a light bulb converts 5.00 W of its power to red light (frequency 4.60 ×1014Hz). How many photons are emitted in one second by the lightbulb?P =E∆tP =Nhf∆tN =P ∆thf=(5.00J/s)(1s)3.05 × 10−19J= 1.64 × 1019photonsL. Whitehead 3 Phys 1302What is the intensity of the light 1 m away from the bulb?I =PA=5.00W4π(1m)2= 0.398W/m2= 0.398J/sm2How many photons are absorbed in one second by your eye (area 2.0 × 10−5m2) ifyou are standing 1 m away? Remember intensity is the energy absorbed by a givenarea in a given time.Peye= IAeyeE∆t= IAeyeNhf∆t= IAeyeN =IAeye∆thf=(0.398J/sm2)(2.0 × 10−5m2)(1s)3.05 × 10−19J= 2.61 × 1013photonsIf you get further away, the intensity will decrease, and so will the number of photonshitting your eye per second.The photoelectric effect is a process in which light hitting the surface of a metalcauses an electron to be emitted from the metal. The ejected electron is referred toas a photoelectron.L. Whitehead 4 Phys 1302With an apparatus as shown in the figure, light shines on a metal plate and pho-toelectrons are ejected. The photoelectrons are attracted to a positively chargedcollector plate, which creates an electric current in the circuit that can be measuredwith the ammeter.The minimum amount of energy necessary to eject an electron from a particularmetal is called the work function (W0). The work function is different for differentmaterials. The metal must be given energy E from the light beam greater than W0to eject an electron (E > W0). If the energy is less than that, no electron will beemitted. The maximum kinetic energy a photoelectron can have isKmax= E − W0(4)According to classical physics (basically, if you assume that light behaves like anyother wave):(1) A beam of light of any frequency (any color) can eject electrons, as long as theintensity is great enough.(2) The maximum kinetic energy of a photoelectron should increase as the intensityof the light increases.But when the experiment was performed, both of the above predictions were shownto be incorrect. The observed behavior:(1) The light must have a frequency greater than some minimum value to eject elec-trons. If the frequency of the light is below this cutoff frequency (f0), it will not ejectelectrons, no matter what the intensity is.(2) The maximum kinetic energy of the photoelectrons does not depend on the in-tensity, it depends only the frequency.In the photon model, one photon gets absorbed by the metal. If that absorbedenergy (the energy of the photon hf) is greater than the work function (hf > W0),an electron will be ejected. Thus the frequency of the photon must be at leastf0=W0h(5)for an electron to be ejected. This is the definition of the cutoff frequency f0.If the intensity of the light is increased, this means the number of photons hittingthe surface in a given time is increased. Since each photon can eject one electron,L. Whitehead 5 Phys 1302more photons in a given time means more photoelectrons in a given time.The maximum kinetic energy of the electron depends on how much of the photonenergy is left over in excess of what is