The Description of the microscopic worldPrevious Lecture: Quantization of light, photonsPhotoelectric effectParticle-Wave dualismThis Lecture: More on Quantum mechanics Uncertainty Principle Wave functionsStart the atomThis Friday Honor lectureCatherine WoodwardBotany PhotosynthesisMTE 3 Wed Nov 28 5:30-7pm Ch 2103!Talk to me after this lecture and write us an email if you really need an alternate exam. Alternative possible times: Wed 6:30-8:00 and Thu 5:30-7:00!Contents:!Ampere’s Law (32.6)!Faraday’s Law (ch 33)!Maxwell equations (ch 34, no 34.2)!EM waves (34.6-7)!Polarization (34.8)!Photoelectric effect (38.1-2-3)!Matter waves and De Broglie wavelength (38.4)!Atom (37.6, 37.8-9, 38.5-7)!Wave function and Uncertainty (39)3Quantization of light!Light is made of ‘quanta’ called photons!quantum’ of energy: a photon carries the energy E=hf f = frequency of light!Photon is a particle, but moves at speed of light!!This is possible because it has zero mass.!Zero mass, but it does have momentum:!Photon momentum p=E/c!Photoelectric effect (1905) No matter how intense is the light: until the light wavelength passes a certain threshold, no electrons are ejected.Kmax = hƒ – !5Photon EnergyA red and green laser produce light at a power level of 2.5mW. Which one produces more photons/second?A. Red B. GreenC. Samefrequency of green light is larger than red oneRed light has less energy per photon so needs more photons! To extract photons from the bucket it is f that matters not how many photons!Quiz on photoelectric effectWhich of the following is not true of photoelectric emission?A. increasing the light intensity causes no change in the kinetic energy of photoelectronsB. the maximum energy of photoelectrons depends on the frequency of light illuminating the metalC. increasing the intensity of the light will increase the KE of photoelectronsD. Doubling the light intensity doubles the number of photoelectrons emitted6A. is true because the intensity is connected to the number of electrons not to the energy of each onesB. is true because Kmax fC. is false because Kmax depends on fD. is correctHow much is a quantum of green light?!One quantum of energy for 500 nm light (green)! E = hf =hc"=6.634 #10$34Js( )# 3 #108m / s( )500 #10$9m= 4 #10$19JWe need a convenient unit for such a small energy!1 electron-volt = 1 eV = |charge on electron|x (1 volt) = 1.602x10-19 JEnergy of an electron accelerated in a potential difference of 1 VIn these units, E(1 green photon) = (4x10-19 J)x(1 eV / 1.602x10-19 J) = 2.5 eVhc =1240 eV nmSwinging a pendulum: the classical pictureLarger amplitude, larger energySmall energyLarge energyd!Potential Energy E=mgd!E.g. (1 kg)(9.8 m/s2)(0.2 m) ~ 2 JoulesdThe quantum mechanics scenario!Energy quantization: energy can have only certain discrete valuesEnergy states are separated by !E = hf.f = frequencyh = Planck’s constant= 6.626 x 10-34 JsdSuppose the pendulum hasPeriod = 2 secFrequency f = 0.5 cycles/sec!Emin=hf=3.3x10-34 J << 2 JQuantization not noticeableat macroscopic scalesWave properties of particlesde Broglie wavelengthWavelength of an electron of 1 eV:Should be able to see interference and diffraction for any material particle!!Result: " = 1.23 nm Solve forIf m of particle is large ! small and wave properties not noticeableDe Broglie Question Compare the wavelength of a bowling ball with the wavelength of a golf ball, if each have 10 Joules of kinetic energy.A) "bowling > "golfB) "bowling = "golfC) "bowling < "golfThe largest the mass of the object the less noticeable arethe quantistic effects!Football launched by Brett Favre can go at 30m/s and m = 0.4kg! "=hp=6.6 #10$34Js0.4kg # 30m / s= 5.5 #10$35mDavisson-Germer experiment!Diffraction of electrons from a nickel single crystal.!Found pattern by heating just by chance. Nichel formed a crystalline structure.!Established that electrons are waves. 54 eV electrons ("=0.17nm) Bright spot: constructive interferenceDavisson: Nobel Prize 1937! D = D1+ D2= A sin(kr1"#t) + A sin(kr2"#t) == 2 A cosk$r2% & ' ( ) * sin krav"#t[ ]= 2 A cos+dx,L% & ' ( ) * sin krav"#t[ ]! I " A2Electron Interference and Diffraction!Intensity on screen and probability of detecting electron are connected: amount of energy in each stripComputersimulationphotographx! I( x) = C cos2"dx#L$ % & ' ( ) ! sin"+ sin#= 2cos"$#2% & ' ( ) * sin"+#2% & ' ( ) * ! P(x)dx =N(in dx at x) Ntot"Energy(in dx at x) /tNtothf /t==I( x)HdxNtothf" A(x)2dxWave function!When doing a light interference experiment, the probability that photons fall in one of the strips around x of width dx is!The probability of detecting a photon at a particular point is directly proportional to the square of light-wave amplitude function at that point!P(x) is called probability density (measured in m-1) !P(x)|A(x)|2 A(x)=amplitude function of EM wave!Similarly for an electron we can describe it with a wave function !(x) and P(x)| !(x)|2 is the probability density of finding the electron at x14xH15Wave Function of a free particle• #(x) may be a complex function or a real function, depending on the system•For example for a free particle moving along the x-axis #(x) = Aeikx –k = 2!/" is the angular wave number of the wave representing the particle– A is the constant amplitudeRemember: complex number imaginary unit16Wave Function of a free particle• #(x) must be defined at all points in space and be single-valued• #(x) must be normalized since the particle must be somewhere in the entire space• The probability to find the particle between xmin and xmax is:•#(x) must be continuous in space– There must be no discontinuous jumps in the value of the wave function at any point! P(xmin" x " xmax) = #(x)2dxxmaxxmin$Suppose an electron is a wave…!Here is a wave:…where is the electron?!Wave extends infinitely far in +x and -x direction "x! "=hpAnalogy with sound!Sound wave also has the same characteristics!But we can often locate sound waves!E.g. echoes bounce from walls. Can make a sound pulse!Example: !Hand clap: duration ~ 0.01 seconds!Speed of sound = 340 m/s!Spatial extent of sound pulse = 3.4 meters.!3.4 meter long hand clap travels past you at 340 m/sBeat frequency: spatial localization!What does a sound ‘particle’ look like?!One example is a ‘beat frequency’ between two notes!Two sound waves of almost same wavelength added.Constructive interferenceLarge amplitudeConstructive interferenceLarge amplitudeDestructive
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