3-2 Classical E&M Light Waves In classical electromagnetic theory we express the electric and magnetic fields in a complex form. eg.

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1 Chapter-3 Wave Particle Duality 3- Classical EM Light Waves In classical electromagnetic theory we epress the electric and magnetic fields in a comple form. eg. Plane Wave E! B Poynting Vector! S = 1 µ 0! E! B = 1 µ 0 E 0 B 0 = E 0 = cb 0 1 E cµ 0 0 B E E * (,t) = E o e i(k!"t) B * (,t) = B o e i(k!"t) c = # = % ) " ) ' ( k * + ' ( % * + = " k = speed of light k φ The physically measureable electric field would be the absolute magnitude of this comple field E(,t) = Re E(,t) { } = E 0 cos(k! "t!" # # ) The phase φ of the EM field is defined as the argument of the eponential or cosine. We determine the velocity of the wave by sitting at a point of constant phase.! = k " #t = cons tant d! = kd " #dt = 0 v phase = d dt = # k = c V phase = phase velocity of the wave = c for light wave # Energy density u = 1! 0 E(,t) " Energy density # energy % volume ' ( p = u c " momentum density along direction of motion 3-4 Photoelectric Effect and Einstein Equations From the photoelectric effect Einstein predicted that light consisted of energy packets called photons. E = h! =!" E = Nh! = N!" for each photon for N photons Einstein s theory of special relativity states that E = p c + m c 4 and E = pc for massless particles photons (m = 0) p = E c = h! c = hc " or p =!k momentum of a photon 1

2 3-5 Classical Interference of Light Waves Young s double slit eperiment displayed the classical wave interference of light rays. The quantum physcists saw a connection. I(y) = I 0 e ik 1 e ik! 1, = k 1, X y d Δ θ D X 1 I o, λ=π/k Beam N photons I(y) = I 0 e ik 1 e ik = (I 0 e!ik 1 e!ik )(I 0 e ik 1 e ik ) = I 0 e ik( 1! ) e!ik( 1! ) = I 0 cos( k") = I 0 (1+ cos( k")) # = k" I(y) = 4I 0 cos % k" ' ( ) = 4I k d y ' cos 0 % D ( ) with " d * y D 3-6 Quantum Interference of Light If the incident light source is attenuated to allow single photons to approach the slits the same intensity pattern is seen over a long eposure. We do not know which slit the photon passes through! But the observable pattern is one of a double slit. We conclude that the photon field can be represented by a quantum state! 1 = " 1 e ik 1 +" e ik, with probability P(y) =! 1 (y) = " 1 e ik 1 + " e ik = " (1+ cos( k#)) Feynman Path Approach (Feynman Lectures vol3) Ψ > < φ 1 Ψ> < φ Ψ>! F =!< "1! >! "1 > + < "! >! " >!# " #!# "# A1 The photon may pass thru either of the slits, thus there are two probability amplitudes contributing to the final state! F (position on screen). A

3 3-7,8 Quantum and Classical Particles Classical The momentum and energy of a non-relativistic classical particle is given by E = p + V and p = mv m Quantum- Matter waves DeBroglie suggested that all matter could be thought of as quantum in nature and suggested a characteristic wavelength for matter waves: p = h! =!k and E =! k m + V The phase velocity of the matter wave is be defined as before V phase =! k =!! p!k = E p = m = p p m = 1 v?? How can this be understood that the quantum velocity of a matter wave is one-half its classical velocity? 3-9 Wave Packets and Recipe for Finding Ψ(,t) Generalized particle wave packets can be epressed as a series of plane waves of varying wave numbers. There will eist a relationship of ω on k, ω(k), called the dispersion relationship. The series can be epressed as a Fourier integral also.!(,t) = " 1 e i(k 1 #t) + " e i(k #t) + " 3 e i(k 3 #t) +.. Fourier Series!(,t) = +% "(k) e i(k #t) dk Fourier Integral Epansion #%!(,0) = +% "(k) e i k dk t = 0 #% We can determine Ψ(,t) if Ψ(,0) is known! First we must take the inverse Fourier transform of Ψ(,0)!(k) = 1 #(,0) e ik d " % Then insert!(k) back in to the Fourier Integral Epansion above and walla! +% "(,t) = #!(k) e i(k % t) dk Group and Phase Velociy of Simple Two Wave Packet! = Acos(k 1 " # 1 t) + Acos(k " # t)!!!!!!!cos ( ) + cos(% ) = cos + % " % ( ) cos( )!!!!=!Acos 1 ( k + k 1 ) " (# + # 1 )t ' ( )!cos 1 ( k " k 1 ) " (# " # 1 )t ' ( )! 3

4 Group! Packet Phase!### "###!##"##!=!Acos( k avg! " avg t)!cos #k! #" % t ' ( )!!!!!!!!!!!!!!!!!!!!!!v ph = " avg!!!!!!!!!!!!!v gr = #" k avg #k V ph = ( ) / ( ) / = 0.9!m / s!!!!!!!v gr 7.4! 8.4 = =.5!m / s! 8.8! Gaussian Wave Packet Let us assume a particle at t=0 is represented by a wave packet centered at = 0 and with initial wave number k 0. The packet (particle) has a Gaussian spread of Δ around = 0.!(,0) = A e ik 0 e (" 0 ) /#X p 0 =!k 0 0 Gaussian Packet initial momentum initial position # = r.m.s. spread around = 0 After Fourier inversion an amazing thing happens, the Fourier distribution of momenta (k vectors) are spread about k 0 with a Gaussian spread Δk also! Only the Gaussian wave packet has this unique property! 4

5 Spacial Packet Momentum Packet!(,0) = A e ik 0 e (" 0 ) /#X (k) % e "ik 0 e "(k"k 0 ) /4#X Ψ (,0) φ (k) Δ Δ Δk Δk 0 k 0 k Additionally, we have an inverse relation between Δk and Δ!k! = 1/ The inverse relation is only true for an ideal gaussian shaped packets. In general!k! " 1/ 3-11 Heisenberg Uncertainty Relationship If we interpret the wave packet as representing a quantum particle then p =!k and!p! "! / Heisenberg first showed this relationship in his matri approach to quantum theory Evolution of the Wave Packet- finding Ψ(,t) After taking the Fourier integral of φ(k) we can obtain the time evolving wave packet.!(,t) = # "(k) e i(k % t) dk!(,t) 1 '(t) e+( 0 vgt)/'(t) (See Fitzpatrick!) Ψ(,t) v g = d% dk ' (t) = '(0) + a t ' ( wave group velocity wave packet spreading in time t t 1 t 0 t 3 5

6 Group and Phase Velocity Revisited Particles in quantum theory can be described by wave packets. The group velocity of the packet corresponds to the particle velocity. Given E = p m and p = mv classical v p =! k = E p = 1 p m = 1 v classical v g = d! dk = de dp = p m = v classical phase velocity group velocity 3-1 Heuristic Derivation of the Time Dependent Schrodinger Equation!(,t) = A e i( p! "E! t) Quantum Plane Wave d! d = A i( p p! )ei( " E t )!! = i( p! )! d! d = "( p! )! # E = p m = "! d! m d! = "i E t!! # E = i!! t "! d! m d +V! = i!! t Time Dependent Schrodinger Equation 3-13 Collapse of the Wave Function and Quantum Decoherence After making a measurement of a particle s state (position, momentum, spin, etc.) it sits In this state till decoherence occurs (memory loss). A second measurement very soon after t < t decoherence will yield the same result! This is called Collapse of the Wave Function. (Copenhagen Interpretation!) 6