Chapter5_all - PHY3063 R. D. Field De Broglie's Pilot Waves...

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PHY3063 R. D. Field Department of Physics Chapter5_1.doc University of Florida De Broglie’s Pilot Waves Bohr’s Model of the Hydrogen Atom: One way to arrive at Bohr’s hypothesis is to think of the electron not as a particle but as a standing wave at radius r around the proton. Thus, r n π λ 2 = and 2 n r = with n = 1, 2, 3, … The orbital angular momentum is h n h n p n rp L = = = = 2 2 , which is Bohr’s hypothesis provided that p = h/ λ ! Pilot Waves: In his doctoral dissertation (1924) Louis De Broglie suggested that if waves can act like particles ( i.e. the photon), why not particles acting like waves? He called these waves “pilot waves” (wave of what?) and assigned them the following wavelength and frequency: h E f p h = = or k p E h h = = ω where k = 2 π / λ and ω = 2 π f . Phase Velocity of a Plane Wave: For a traveling plane ( i.e. monochromatic) wave given by ) sin( ) , ( t kx A t x = Ψ the position of the n th node ( i.e. zeros) is given by n t kx n = and the speed of the n th node along the x-axis is f k dt dx v n phase = = = Phase Velocity of a Plane Pilot Wave: For a plane De Broglie pilot wave we get c cp c m cp c cp E p E k v phase 2 2 0 2 ) ( ) ( + = = = = , which is greater than c for particles with non-zero rest mass! Plane Wave -1 0 1 024681 0 x v phase r λ = 2 π r/4 Bohr’s Postulate! Particles cannot be plane pilot waves!
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PHY3063 R. D. Field Department of Physics Chapter5_2.doc University of Florida Wave Superposition and Wave Packets Suppose we add two plane waves together one with wave number k and frequency ω and the other with wave number k+ k and frequency ω + ∆ω as follows ) ) ( ) sin(( ) , ( ) sin( ) , ( ) , ( ) , ( ) , ( 2 1 2 1 t x k k t x t kx t x t x t x t x ω + + = Ψ = Ψ Ψ + Ψ = Ψ Then + + = Ψ t x k k t x k t x 2 2 2 2 sin 2 2 cos 2 ) , ( where I used )) ( sin( )) ( cos( 2 sin sin 2 1 2 1 B A B A B A + = + Now suppose that k << 2k and << 2 ω so that () t kx t x P t kx t x k t x = Ψ sin ) , ( sin 2 2 cos 2 ) , ( . The term P(x,t) describes the “wave-packet”. The position of the n th node ( i.e. zeros) of the wave-packet is given by π ) 1 2 ( + = n t kx n and the speed of the wave-packet along the x-axis is dk d k dt dx v n group = = Group Velocity of a Pilot Wave-Packet: For a De Broglie pilot wave-packet we get c E cp dp dE dk d v group = = = , where I used h / E = , h / p k = , and ( ) E p c c m cp dp d 2 2 2 0 2 ) ( ) ( = + . Note that for pilot waves v phase · v group = c 2 . The pilot wave-packet travels at the speed of the particle! Wave Packet -3 -2 -1 0 1 2 3 x v group Wave Packet! Wave Superposition -2 -1 0 1 2 x
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PHY3063 R. D. Field Department of Physics Chapter5_3.doc University of Florida Re( Ψ ) A Ψ = Ae i(kx- ω t) Im( Ψ ) φ = kx- ω t t Im( Ψ ) = Asin(kx- ω t) A A Crest Trough A Ψ (0,t) = Ae -i ω t Distance r φ = kr- ω t x = 0 φ =- ω t A x = r Ψ (r,t) = Ae i( kr - ω t) Representing Waves as Complex Numbers We can use complex numbers to represent traveling waves. If we let ) ( t kx i Ae ω = Ψ then ) sin( ) Im( t kx A = Ψ is a traveling plane wave with wave number k = 2 π / λ , “angular” frequency ω = 2 π f , and amplitude A .
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This note was uploaded on 04/17/2008 for the course PHY 3063 taught by Professor Field during the Spring '07 term at University of Florida.

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Chapter5_all - PHY3063 R. D. Field De Broglie's Pilot Waves...

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