6_postulates2-2

All space expectation value particle in a box example

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: e average value of the observable ˆ ˆ corresponding to A is given by a * Adx . all space expectation value Particle-in-a-Box Example ( x) 2 nx sin 0 x a n 1,2,... a a use in Postulate 4 a use eigenfunctions of energy operator which are not eigenfunctions of the position operator ˆ x * ( x) x ( x)dx 0 2a nx x sin 2 dx a 0 a L n n sin 2 x x cos 2 2 x2 a a x 2 n a 4 n 4 8 a a 0 2 a2 a2 a2 a a 4 8(n ) 2 8(n ) 2 2 6 9/16/2013 Wavefunction as a Vector Illustrates Adjoint (Transpose and Conjugate) Eigenfunctions of a Hermitian operator, , form a basis set. ∗ They are an orthonormal set, i.e. ,. Any function can be expressed as coefficients multiplying the basis functions. mathematical consequences of Hermitian ∗ Ket ∗ as vector | or as analytical function ∑ ⋮ ⋮ ∗ ∗ flipped and starred … ⋮ ∗ Bra ∗ ∗ Bras and Kets are adjoints ∗ … | For analytical functions, the integral form of Postulate 4 expresses the adjoint relationship Function Expressed in Terms of an Orthonormal Set orthonormal set L 1 n 1 5 ( i x) 2 L general function i x L 2 sin f( x) x 5 x 5 5 L c ( n x) f( x) dx n fit( x) c n 0 recipe for coefficients cn (n x) n 1 2.519 0.9 0.894 analytical form function as a vector 0.45 0.539 6 6 ( 1 x) c1 these add together to get ( 2 x) c2 ( 3 x) c3 4 f( x) 4 2 fit( x) 2 ( 5 x) c5 0 0 ( 4 x) c4 0 0.5 1 x 2 0 0.5 1 x Use more basis functions and do better … L n 1 50 c ( n x) f( x) dx n 50 fit( x) 0 10 times more basis functions coefficients cn (n x) n 1 analytical function 6 f( x) 4 fit( x) 2 0 0 0.5 1 x as n, fit(x)=f(x) 7 9/16/2013 Eigenfunctions of QM Operators are Orthogonal for energy to be real ˆ ˆ* E * H d H d E * If E=E’+iE”, then E*=E’-iE”, so for E=E*, then E” must be 0 and E is real true for Hermitian operators Postulate 3 with energy operator n is a quantum number for different states Consider the different states labeled by n … Use an index, k, so it can be different than n left multiply by k* and integrate ∗ complex conjugate, left multiply by n and integrate ∗ ∗ ∗ Because ∗ subtract this from that ∗ ∗ ∗ _ ∗ because * k ∗ ∗ n * d n k d ∗ is Hermitian ∗ ∗ ∗ is Hermitian, so 1st term equals 2nd 0 ∗ * if n k , then En Ek 0 and k n d 0 * if n k , then En Ek 0 and k n d 1 * k n d n , k if n k , then 0 if n k , then 1 orthogonal normalized orthonormal Kronecker delta function 8 9/16/2013 Express Function in Terms of Orthonormal Set orthonormal sets are the orthonormal basis set w...
View Full Document

This document was uploaded on 01/19/2014.

Ask a homework question - tutors are online