MIT18_102s09_lec24 - MIT OpenCourseWare

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

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

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

Unformatted text preview: MIT OpenCourseWare 18.102 Introduction to Functional Analysis Spring 2009 For information about citing these materials or our Terms of Use, visit: . 140 LECTURE NOTES FOR 18.102, SPRING 2009 Lecture 24. Thursday, May 7: Completeness of Hermite basis Here is what I claim was done last time. Starting from the ground state for the harmonic oscillator d 2 (24.1) H = − dx 2 + x 2 , Hu = u , u = e − x 2 / 2 and using the creation and annihilation operators d d (24.2) A = + x, C = − + x, AC − CA = 2Id , H = CA + Id dx dx I examined the higher eigenfunctions: (24.3) u j = C j u = p j ( x ) u ( c ) , p ( x ) = 2 j x j + l.o.ts , Hu j = (2 j + 1) u j and showed that these are orthogonal, u j ⊥ u k , j = k, and so when normalized give an orthonormal system in L 2 ( R ) : u j (24.4) e j = 2 j/ 2 ( j !) 1 1 . 2 π 4 Now, what I want to show today, and not much more, is that the e j form an orthonormal basis of L 2 ( R ) , meaning they are complete as an orthonormal sequence. There are various proofs of this, but the only ‘simple’ ones I know involve the Fourier inversion formula and I want to use the completeness to prove the Fourier inversion formula, so that will not do. Instead I want to use a version of Mehler’s formula. I also tried to motivate this a bit last time. Namely, I suggested that to show the completeness of the e j ’s it is enough to find a compact self-adjoint operator with these as eigenfunctions and no null space. It is the last part which is tricky. The first part is easy. Remembering that all the e j are real, we can find an operator with the e j ;s as eigenfunctions with corresponding eigenvalues λ j > (say) by just defining ∞ ∞ (24.5) Au ( x ) = λ j ( u,e j ) e j ( x ) = λ j e j ( x ) e j ( y ) u ( y ) ....
View Full Document

This note was uploaded on 11/08/2011 for the course PHY 18.102 taught by Professor Staff during the Spring '09 term at MIT.

Page1 / 5

MIT18_102s09_lec24 - MIT OpenCourseWare

This preview shows document pages 1 - 3. Sign up to view the full document.

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