M402-iv.not.page 30.Fall.02

M402-iv.not.page 30.Fall.02 - IV-30 The Dirac Delta...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: IV-30 The Dirac Delta Function, δ(x-xo) Dirac Delta Function In one dimension, δ(x-xo) is defined to be such that: ma to b f(x) δ(x-xo)dx / + *0 if xo is not in [a,b]. *½f(xo) if xo = a or b; *f(xo) if xo ε (a,b). . Properties of δ(x-xo): (you should know those marked with *) *1. δ(x-xo) = 0 *2. m-4 to +4 δ(x)dx if x … xo =1 3. δ(ax) = δ(x)/|a| *4. δ(-x) = δ(x) 5. δ(x²-a²) = [δ(x-a) + δ(x+a)]/(2a); a $ 0 +------------------------------------------------------------------------------------------------------------------------------------------**7. δ(g(x)) = 3i δ(x-xoi)/|dg/dx|x=xoi where g(xoi) = 0 and dg/dx exists at and in a region around xoi. .------------------------------------------------------------------------------------------------------------------------------------------ 6. m-4 to +4 δ(x-a)δ(x-b)dx = δ(a-b) *8. f(x)δ(x-a) = f(a)δ(x-a) 9. δ(x) is a "symbolic" function which provides convenient notation for many mathematical expressions. Often one "uses" δ(x) in expressions which are not integrated over. However, it is understood that eventually these expressions will be integrated over so that the definition of δ (box above) applies. 10. No ordinary function having exactly the properties of δ(x) exists. However, one can approximate δ(x) by the limit of a sequence of (non-unique) functions, δn(x). Some examples of δn(x) which work are given below. In all these cases, m-4 to +4 δn(x)dx = 1 œ n and limitn --> 4 m-4 to +4 δn(x-xo)f(x)dx = f(xo). œ n. + *0 *n *0 . (a) δn(x) / for x < -1/(2n) for -1/(2n) # x # 1/(2n) for x > 1/(2n) (b) δn(x) / n//π exp[-n²x²] (c) δn(x) / (n//π)· 1/(1 + n²x²) = (d) δn(x) / sin(nx)/πx [1/(2π)]m-n to n exp(ixt)dt ...
View Full Document

This note was uploaded on 12/20/2010 for the course PHYS 402 taught by Professor J. during the Fall '09 term at Missouri S&T.

Ask a homework question - tutors are online