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Unformatted text preview: 1 Physics 316 Solution for homework 3 Spring 2005 Cornell University I. EXERCISE 1 A. Replace the relativistic treatment of de Broglie which lead to p = ¯ hk and E = ¯ hω with E = p ( pc ) 2 + ( mc 2 ) 2 , by a nonrelativistic treatment in which E = p 2 2 m + mc 2 . This is obtained by making a second order expansion in v/c . Find ω as a function of k and compute the phase velocity and the group velocity. Let’s reproduce de Broglie’s treatment of the wave properties of matter. In this case, we will consider the nonrelativistic limit. Let’s first assume that a certain physical quantity associated with a particle of mass m oscillates, in the rest frame of the particle, at a frequency ω = mc 2 / ¯ h : ξ ∝ cos( ω t ) . We then look at this quantity in a frame moving at a velocity v . Using the Lorentz transformation of the time coordinate, we get: t → t vx/c 2 p 1 v 2 /c 2 . The nonrelativistic case corresponds to v ¿ c . Expanding this expression in a Taylor series in v/c up to second order in v/c , we get t → • 1 + 1 2 v 2 c 2 + O µ v 4 c 4 ¶‚ ‡ t vx c 2 · = µ 1 + 1 2 v 2 c 2 ¶ t v c 2 x + O µ v 3 c 3 ¶ . (1.1) Hence, we have: ω t → ωt kx, with ¯ hω = mc 2 + 1 2 mv 2 , ¯ hk = mv. (1.2) Those are the usual nonrelativistic definitions of energy ( E ) and momentum ( p ). We can easily express ω as a function of k , since we know that p = ¯ hk and ω = E/ ¯ h : ω = 1 ¯ h µ mc 2 + p 2 2 m ¶ = ω + ¯ hk 2 2 m . (1.3) Now, the phase and group velocities can be readily calculated: v p = w k = ¯ hk 2 m + ω k = v 2 + c 2 v (1.4) v g = dω dk = ¯ hk m = v. (1.5) We found that v g = v , as required. B. Does the group velocity equal the particle velocity as required? Yes (see above). 2 ¢ ¡ ¡ £ ¤ ¤ ¥ ¦ § ¨ © ¦ § ¨ © § © ¨ ¨ © ¦ ! " # $ % & ' & FIG. 1: Plot of the phase and group velocities of a particle as a function of v , the velocity of the particle. The nonrelativistic, the velocity of the particle....
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This note was uploaded on 09/28/2008 for the course PHYS 316 taught by Professor Hoffstaetter during the Spring '05 term at Cornell.
 Spring '05
 HOFFSTAETTER
 Physics, Work

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