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Formalism of Quantum Mechanics
2006 Quantum Mechanics Prof. Y. F. Chen
Formalism of Quantum Mechanics
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
2006 Quantum Mechanics Prof. Y. F. Chen
Formalism of Quantum Mechanics
Formalism of Quantum Mechanics
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 ：
2006 Quantum Mechanics Prof. Y. F. Chen
Definition of Inner Product & Hilbert Space
Formalism of Quantum Mechanics
)(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
2006 Quantum Mechanics Prof. Y. F. Chen
Definition of Inner Product & Hilbert Space
Formalism of Quantum Mechanics
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
→
2006 Quantum Mechanics Prof. Y. F. Chen
Gram-Schmidt Orthogonalization
Formalism of Quantum Mechanics
)(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 ：
2006 Quantum Mechanics Prof. Y. F. Chen
Gram-Schmidt Orthogonalization
Formalism of Quantum Mechanics
)(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
2006 Quantum Mechanics Prof. Y. F. Chen
Gram-Schmidt Orthogonalization
Formalism of Quantum Mechanics
)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 ：
2006 Quantum Mechanics Prof. Y. F. Chen
Definition of Self-Adjoint (Hermitian Operators)
Formalism of Quantum Mechanics
)()( 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
2006 Quantum Mechanics Prof. Y. F. Chen
Definition of Self-Adjoint (Hermitian Operators)
Formalism of Quantum Mechanics
†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) ：
×
×
2006 Quantum Mechanics Prof. Y. F. Chen
The Properties of Hermitian Operators
Formalism of Quantum Mechanics
)()( 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
2006 Quantum Mechanics Prof. Y. F. Chen
The Properties of Hermitian Operators
Formalism of Quantum Mechanics
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 ：
2006 Quantum Mechanics Prof. Y. F. Chen
The Sturm-Liouville Eq.
Formalism of Quantum Mechanics
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
→
2006 Quantum Mechanics Prof. Y. F. Chen
The Sturm-Liouville Eq.
Formalism of Quantum Mechanics
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
2006 Quantum Mechanics Prof. Y. F. Chen
The Sturm-Liouville Eq.
Formalism of Quantum Mechanics
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( )
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
)()()()()()(
)()()()()()(
2006 Quantum Mechanics Prof. Y. F. Chen
Transforming an Eq. into SL Form
Formalism of Quantum Mechanics
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
2006 Quantum Mechanics Prof. Y. F. Chen
Transforming an Eq. into SL Form
Formalism of Quantum Mechanics
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
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