ps03soln - 1 Physics 316 Solution for homework 3 Spring...

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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 non-relativistic 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 non-relativistic 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 non-relativistic 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 non-relativistic 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 non-relativistic, 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.

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ps03soln - 1 Physics 316 Solution for homework 3 Spring...

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