Chapter5_8

# Chapter5_8 - t and ∆ k represents an uncertainty in the...

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PHY3063 R. D. Field Department of Physics Chapter5_8.doc University of Florida Inescapable Consequence of Wave Packets De Broglie postulated that the “pilot wave- packet” governs the motion of the electron. However, the electron is a “point like” elementary particle ( i.e. radius zero! ). So where exactly is the electron within the pilot wave-packet? The electron moves with speed v group somewhere within the pilot wave packet given by ( ) ( ) 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”. As a crude guess lets say that at fixed time t the electron must be somewhere in the region x = x n+1 - x n , where the position of the n th node ( i.e. zeros) of the wave-packet is given by π ω ) 1 2 ( + = n t kx n . Thus at fixed time t , k k t n t n x x x n n = + + + + = + π ω π ω π 2 ) 1 2 ( ) 1 ) 1 ( 2 ( 1 and hence π 2 k x . Here x represents an uncertainty in the electrons position at time
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Unformatted text preview: t and ∆ k represents an uncertainty in the electrons precise wave number. Due to the nature of wave superposition asking for a small uncertainty in the position ∆ x of the electron requires a larger ∆ k and hence a bigger uncertainty in the electrons wave number. Similarly at fixed position x , 2 ≈ ∆ ∆ t . Here ∆ t represents an uncertainty in the time and ∆ω represents an uncertainty in the electrons angular frequency. Again due to the nature of wave superposition asking for a small uncertainty in the time ∆ t requires a larger ∆ω and hence a bigger uncertainty in the electrons angular frequency. Wave Packet-3-2-1 1 2 3 x v group electron v group ∆ x Wave Packet-3-2-1 1 2 3 t electron ∆ t At fixed time t At fixed x Wave of what?...
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