Unformatted text preview: e average value of the observable
ˆ
ˆ
corresponding to A is given by a * Adx
.
all space expectation value
ParticleinaBox 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
 Fall '13
 Physical chemistry, pH, Uncertainty Principle, orthonormal set

Click to edit the document details