ps03soln

# ps03soln - 1 Physics 316 Cornell University Solution for...

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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 = ¯ 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 0 = mc 2 / ¯ h : ξ 0 cos( ω 0 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 v 4 c 4 ¶‚ t - vx c 2 · = 1 + 1 2 v 2 c 2 t - v c 2 x + O v 3 c 3 . (1.1) Hence, we have: ω 0 t ωt - kx, with ¯ = 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 mc 2 + p 2 2 m = ω 0 + ¯ 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 = 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 and the relativistic cases yield the same group velocity, v g = v . For v p , the continuous line is given by eq. (1.4), whereas the dotted line shows the relativistic result.

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