Formalism of Quantum Mechanics 2006 Quantum MechanicsProf. Y. F. Chen Formalism of Quantum...

Formalism of Quantum Mechanics

2006 Quantum Mechanics Prof. Y. F. Chen


linear DE. ： the main foundation of QM consists in the Schrödinger eq.

The formalism of QM deals with linear operators & wave functions that f

orm a Hilbert space

ch4 will focus on the Hermitian operators & the superposition properties

of linear DE. in Hilbert space




inhomogeneous linear differential ：

(1) = linear differential operator acting upon

(2) = eigenvalue & = eigenfunction

(3) is a weight function

any function in this vector space can be expanded as

　　　　　　 , =a set of linearly indep. basis functions

inner product ：


Definition of Inner Product & Hilbert Space


)(xyL )(xy

)(xw

)(xy

( ) ( ) ( )i i iy x w x y xL

)(xf

0

)()(n

nn xycxf

b

adxxwxgxfgf )()()(

)(xyn

orthogonal ： if , then & are orthogonal.

the norm of ：

a basis of orthnormal, linearly independent basis functions satisfie

s


Definition of Inner Product & Hilbert Space


0gf )(xf )(xg

)(xf b

adxxwxffff )()( 22/1

)(xn

ji

b

a jiji dxxwxx )()()(

Gram-Schmidt orthogonalization ：

= linearly independent, not orthonormal basis

= orthonormal basis produced by the Gram-Schmidt orthogonalizat

ion, in which is to be normalized

→


Gram-Schmidt Orthogonalization


)(xn

)(xn

2/1)()(

nnnn xx

)(xn

1

0

21120022

10011

00

)()()(

)()()()(

)()()(

)()(

n

iiiinn yxxyx

yxyxxyx

yxxyx

xyx

although the Gram-Schmidt procedure constructed an orthonormal set,

are not unique. There is an infinite number of possible orthonor

mal sets.

construct the first three orthonormal functions over the range ：




)(xn

11 x

1/ 211/ 2

0 0 0 01

1 1 0 0 1 1 1

1/ 2

1 1 1

22 2 0 0 2 1 1 2

22

( ) 1 , 2 , ( ) 1/ 2

( ) ( ) ( ) , ( ) ( )

( ) 3 / 2

1( ) ( ) ( ) ( )

3

1 5 ( ) (3 1)

2 2

x dx x

x y x x y x y x x

x x

x y x x y x y x

x x

→

it can be shown that 　　　　　　　　

where is the nth-order Legendre polynomials

the eq. for Gram-Schmidt orthogonalization tend to be ill-conditioned be

cause of the subtractions. A method for avoiding this difficulty is to use t

he polynomial recurrence relation 　




)35(2

7

2

1)( 3

3 xxx

)(2

1)( n xP

nx n

)(xPn

the adjoint/Hermitian conjugate of a matrix A：

from inner product space, the definition of the adjoint ：

the adjoint of an operator in inner product function ：


Definition of Self-Adjoint (Hermitian Operators)


)()( TT† AAA

),(),( 1†

221 xAxAxx

fggf †LL

self-adjoint/Hermitian operator ：

→(1)

→(2)

measurement of the physical quantity 　　：

(1) → , is real

(2) is not necessarily an eigenfunction of 　


Definition of Self-Adjoint (Hermitian Operators)


†LL

fggf LL

dxxgxfdxxfxgdxxgxfb

a

b

a

b

a

)()()()()()( LLL

L

dLL

fggf †LL LL L

L

(1) the eigenvalues of an hermitian operator are real

(2) the eigenfunctions of an hermitian operator are orthogonal

(3) the eigenfunctions of an hermitian operator form a complete set

proof (1) & (2) ：

×

× 　　　　　　　


The Properties of Hermitian Operators


)()( xyxy iii L

)()( xyxy jjj L

)(xy j

)(xyi

dxxwxyxydxxyxy i

b

a jii

b

a j )()()()()( L

dxxwxyxydxxyxy j

b

a ijj

b

a i )()()()()( L

integrating

dxxwxyxydxxyxy i

b

a jii

b

a j )()()()()(

Lcomplex

conjugate

proof (1) & (2) ：

∵

∴

→ if i=j, then → → 　 is real

if i≠j, then → & are orthogonal

∵ the eigenfunctions of an hermitian operator form a complete set

∴any function


The Properties of Hermitian Operators


dxxgxfdxxfxgdxxgxfb

a

b

a

b

a

)()()()()()( LLL

0)()()( b

a jiji dxxwxyxy

0)()()( b

a ii dxxwxyxy ii i

0)()()( b

a ji dxxwxyxy )(xyi )(xy j

0

)()(n

nn xycxy

general form of SL eq. ：

with ,where p(x), q(x), and r(x) are real functions of x

Ex. Legendre’s eq. ：　　　　　　　　　　　　　　　　

& eigenvalues l(l+1)

linear operator that are self-adjoint can be written in the form ：

linear operator=Hermitian over [a,b] satisfies BCs ：


The Sturm-Liouville Eq.


0)()()()()(

)()(

)(2

2

xyxwxyxqxd

xydxr

xd

xydxp

xd

xpdxr

)()(

2( ) 1 , ( ) 2 , ( ) 0, ( ) 1,p x x r x x q x w x

0)()()()()()( xyxwxyxqxyxp

0)()()(

bx

axji xyxpxy

BCs ： (1) → the wave with fixed ends

(2) → the wave with free ends

(3) → the periodic wave

show that subject to the BCs, the SL operator is Hermitian over [a, b] ：

putting into

→

　




0)()( byay

0)()( byay

0)()( bpap

)()()()()( xyxqxyxpxy L ( ) ( ) ( ) ( )b b

a af x g x dx g x f x dx

L L

b

a jijij

b

a i dxxyxqxyxyxpxydxxyxy )()()()()()()()( L




( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )

b bx b

i j i j i jx aa a

b

i ja

i j

y x p x y x y x p x y x y x p x y x dx

p x y x y x dx

p x y x y x

( ) ( ) ( )

( ) ( ) ( )

bx b

j ix a a

b

j ia

y x p x y x dx

y x p x y x dx

integrating by parts for the first term & using the BCs

→

→

→ the SL operators is Hermitian over the prescribed interval

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( )

b bx b

i j i j j ix aa a

b

j ia

p x y x y x dx p x y x y x y x p x y x dx

y x p x y x dx

b

a ijij

b

a jiji

dxxyxqxyxyxpxy

dxxyxqxyxyxpxy

)()()()()()(

)()()()()()(


Transforming an Eq. into SL Form


any eq. can be put into

self-adjoin form by introducing in place of

proof ： Let

→

→ to satisfy the requirement of SL eq. form for

0)()()()()()()()( xyxwxyxqxyxrxyxp

)(x )(xy

dxxp

xrxpxxy

)(2

)()(exp)()(

)()()( xxFxy

0)()()()(

)()(

)(

)()(

)()()(

)()(2)()(

xxwxqxF

xFxr

xF

xFxp

xxrxF

xFxpxxp

)(x

)()(

)()(2)( xr

xF

xFxpxp

)(2

)()(exp)(

xp

xrxpxF


Transforming an Eq. into SL Form


rewrite eq.

as the SL form for ：

with

0)()()()(

)()(

)(

)()(

)()()(

)()(2)()(

xxwxqxF

xFxr

xF

xFxp

xxrxF

xFxpxxp

)(x

0)()()()(~)()( xxwxxqxxp

)()(2

)()()(

)(2

)()(

)(2

)()()()(~

2

xqxp

xrxpxr

xp

xrxp

xp

xrxpxpxq

Formalism of Quantum Mechanics 2006 Quantum MechanicsProf. Y. F. Chen Formalism of Quantum...

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