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Unformatted text preview: Quantum Mechanics I, Physics 3143: Assignment 1, Fall 2010: due 9/7 in class Notation: We use t to denote /t , etc. 1. (a) The (angular) frequency, > 0, and wavenumber, k , of light propagating in the vacuum are related by = c | k | , where c is the speed of light. Show that an electromagnetic wave of the form E ( x, t ) = E sin ( kx t ) satisfies the wave equation ( 2 t c 2 2 x ) E = 0. De Broglie suggested that a particle of mass m with momentum p has a wavelength = h/p ; setting = 2 /k , this gives p = planckover2pi1 k . The energy E = planckover2pi1 is related to the momentum of a non-relativistic free particle by E = p 2 / (2 m ), or in terms of and k , = planckover2pi1 2 m k 2 . For a relativistic particle, the formula E 2 = p 2 c 2 + m 2 c 4 gives instead ( planckover2pi1 ) 2 = ( planckover2pi1 k ) 2 c 2 + m 2 c 4 . (b) Show that these non-relativistic and relativisitic dispersion relations suggest that the wave function (...
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