Chapter1_3 - = ∆ − ∆ n t kx n and the speed of the...

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PHY4604 R. D. Field Department of Physics Chapter1_3.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 ω 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 ) , ( ω ω ω where I used )) ( sin( )) ( cos( 2 sin sin 2 1 2 1 B A B A B A + = + Now suppose that k << 2k and ∆ω << 2 ω so that ( ) ( ) t kx t x P t kx t x k t x ω ω ω = 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
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Unformatted text preview: ( + = ∆ − ∆ 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 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 1 2 3 x v group Wave Packet! Wave Superposition-2-1 1 2 x...
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