Matter WavesThornton and Rex, Ch. 5Matter WavesEM waves also behave like particles(photons).1924 - de Broglie asked:Can particles also behave like waves?He suggested a relation betweenwavelength and momentum:l = h/pNo experimental evidence for thisexisted.But with his matter wave idea, De Brogliecould “derive” Bohr’s quantizationcondition.An electron orbiting an atom at radius r:Assume stationary states correspond tostanding waves of the electron.I.e, an integral number of wavelengthsmust fit into the circumference:2pr = nl = nh/p = nh/mvorL = mvr = nh/2p = n hCircumference = 2prn=6 orbitDe Broglie’s wave idea fitted naturallywith Bohr’s atomic model.Encouraged with this success, de Brogliepresented his ideas in his PhD thesis.With no experimental evidence for theidea, de Broglie’s professors wereskeptical of this radical concept. Onesent a copy of the thesis to AlbertEinstein. Einstein replied that the ideascertainly appeared crazy, but they wereimportant, and the work was sound.De Broglie received his PhD in 1924.A few years later the wave nature ofelectrons was confirmed. In 1929 he wasawarded the Nobel Prize.Discovery of Electron Waves1925 - Davisson and Germer werescattering electrons from metals.On scattering electrons off crystallizedNickel, they saw peaks at certain angles.fi Interference spectra!Interference is constructive (i.e. peaks)when2 d sin q = n lfor integer n. (Bragg’s law)dd sinqqqll for the electron agreed withde Broglie’s formula.This was the first experiment to revealthe wave nature of matter.Wave/Particle DualityWave nature of light from the double-slit interference pattern:Pattern expected from particles is verydifferent:1122I1I2I1I2InterferenceNo InterferenceWhat happens at very low intensities?Photons hit at discrete points, graduallybuilding up the interference pattern.Does the photon go through slit 1 or slit2?Neither! (or rather, both!)What about electrons?They exhibit the same interferencepattern (although at smaller wavelengthsthan for visible light.)Bohr’s Principle of ComplementarityIt is not possible to simultaneouslydescribe physical observables in terms ofboth particles and waves.Bohr called the fact that all objects(light, electrons, etc.) have bothwave-like and particle-like properties complementarity.Generalities about light wavesA plane wave:y(x,t) = A cos[2p (x-ct) /l]Amplitude: AWavelength: lSpeed: cFrequency: n=c/lIt is convenient to rewrite:y(x,t) = A cos(kx-wt)Wave number: k = 2p/lAngular frequency: w=2p ny lcAWave relation: c = ln = w/kAll light waves have same speed c invacuum, independent of wave number k.fi Not true for matter waves.Planck: E = h n = hw/2p = h wEinstein/de Broglie: p = E/c = h/l = h kA periodic wave can be constructed froma sum of plane waves:fi Fourier Seriesy(x,t) = ∑ Ai cos(kix-wit)++ . . .=A wave packet can be constructed as acontinuous sum (integral) of plane waves.fi Fourier Transformy(x,t) = ∫ A(k) cos(kx-wt) dkGeneral fact about Fourier Transforms:The extent Dx of the wave y is inverselyrelated to the extent Dk of its FourierTransform A.yxAAAyyyDkDxDkDxDxDkWe can write this asDx Dk ≥ 1/2Multiplying by h and using p = h k gives: Dx Dp ≥ h/2Heisenberg uncertainty principleIt is impossible to know precisely theposition and the momentum of an objectat the same