Class_17new - PHYS809 Class 17 Notes Eigenfunction...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
1 PHYS809 Class 17 Notes Eigenfunction expansions for Green functions When we considered the Green function for the interior of a rectangle, we found that there was an asymmetry in the dependences on the coordinates. Here we briefly discuss how to develop Green functions that put the coordinates on an equal footing. Note that this can be done only at the expense of an additional sum over indices. Consider the equation 2 0, λ ∇ Φ + Φ = (17.1) subject to boundary conditions 0 Φ = on x = 0, x = a , y = 0 and y = b . By the method of separation of variables we find that non-trivial solutions are possible only if λ takes discrete eigenvalues given by 2 2 , mn m n a b π = + (17.2) where m and n are positive integers. The solutions are then the orthonormal eigenfunctions 2 sin sin . mn m x n y a b ab Φ = (17.3) The complete solution is a linear combination of terms of this form. The Green function is a solution of the Poisson equation
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

Page1 / 2

Class_17new - PHYS809 Class 17 Notes Eigenfunction...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online