All space expectation value particle in a box example

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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...
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