# HW1 - Quantum Mechanics I Physics 3143 Assignment 1 Fall...

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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, ω > 0, and wavenumber, k , of light propagating in the vacuum are related by ω = c | k | , where c is the speed of light. Show that an electromagnetic wave of the form E ( x, t ) = E sin ( kx − ωt ) 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 Λ = h/p ; setting Λ = 2 π/k , this gives p = planckover2pi1 k . The energy E = planckover2pi1 ω is related to the momentum of a non-relativistic free particle by E = p 2 / (2 m ), or in terms of ω and k , ω = planckover2pi1 2 m k 2 . For a relativistic particle, the formula E 2 = p 2 c 2 + m 2 c 4 gives instead ( planckover2pi1 ω ) 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 ψ (...
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HW1 - Quantum Mechanics I Physics 3143 Assignment 1 Fall...

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