385lec3 - PHYS 385 Lecture 3 - Fourier transforms Lecture 3...

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PHYS 385 Lecture 3 - Fourier transforms 3 - 1 ©2003 by David Boal, Simon Fraser University. All rights reserved; further copying or resale is strictly prohibited. Lecture 3 - Fourier transforms What's important : discrete Fourier transforms continuous Fourier transforms Text : Gasiorowicz, App. A The two probability distributions P ( x ) and P ( p ) for position and momentum introduced in Lec. 2 are not independent but linked through the Uncertainty Principle. The math which underlies the relationships is the same as that of Fourier series. We introduce discrete Fourier series in this lecture, then generalize to continuous variables. Although it looks like a mathematical diversion at this point, it establishes a helpful framework for introducing the Schrödinger equation. Fourier transforms - discrete In second year calculus, series expansions are introduced as a way of representing complicated functions. For example, the sine function has the series expansion sin x = x - x 3 /3! + x 5 /5!. .. (1) This expression can be useful in evaluating sine, or some of its integrals, in the region of small x . In using a polynomial expansion such as (1), one obviously must be concerned about the convergence of the series for the range of x of interest. The de Broglie wavelength of particles provides a hint that the functions of interest in quantum mechanics are periodic. The series expansions of periodic functions can be conveniently based on other periodic functions, such as the sine and cosine functions of trigonometry. Suppose we have the simple sawtooth function f ( x ) = x , periodic over -1 x 1: What functions would be useful in representing this particular f ( x )? 1.
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385lec3 - PHYS 385 Lecture 3 - Fourier transforms Lecture 3...

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