Linear vector space

CHAPTER 3
Quantum Mechanics. G. ARULDHAS
Generalized Formalism of
quantum mechanics
Safiya Amer.
Misurata University.
Spring 2010.

Introduction
Quantum theory is based on two constructs: wave
functions and operators.
The state of the system is represented by its waves
function, observables are represented by operators.
Mathematically, wave functions satisfy the defining
conditions for abstract vectors, and operators act
on them as linear transformation.
So the natural language of quantum mechanics is
linear algebra.

LINEAR VECTOR SPACE
The vector spaces of quantum mechanics are
like the ordinary three-dimensional spaces
of vectors from introductory physics.
Vector In a three-dimensional space
●Any vector can be expressed as
Where, are unit vectors,
and are scalars.
a
r
1 1 2 2 3 3a ae a e a e= + +
r r r r
1 2 3, ,e e e
r
1 2 3, ,a a a

The unit vectors are said to form a basis
for the set of all vectors in three dimensions.
Definition
A set of vectors is said to
form a basis for a vector space if any
arbitrary vector can be represented
by a linear combination of the
Where the vectors be
linearly independent
1 2 3, ,e e e
r
{ }1 2, ,... nu u u
{ }iu
1 1 2 2 ... n nx au a u a u= + + +
r r r r
x
r
{ }iu

Inner Product
●If we have two vectors
,
Then, the scalar product or inner product
is defined by
1
2
3
a
a a
a
 
 ÷=
 ÷
 ÷
 
r
1
2
3
b
b b
b
 
 ÷=
 ÷
 ÷
 
r
3
1( , ) i i ia b a b== ∑
rr

Orthogonal and orthonormal basis
●A basis is said to be orthogonal if
Where a, b any two vectors in a basis.
●A basis is said to be orthonormal if
{ }iu
( , ) 0a b =
rr
{ }iu
( , )i j ija a δ= , 1,2,...i j =
1
0
ijδ

= 

i j
i j
= 

≠ 

Vectors in an n-dimension space
◘we want to generalize precedent
concepts to n-dimension real space
●Orthonormal basis:
A vector a can be expressed in this
Orthonormal basis as
1
n
i i ia a e== ∑
r r r

●Inner Product:
If the vectors are complex, then
We observe that for any vector
1( , ) n
i i ia b a b∗
== ∑
rr
1( , ) n
i i ia b a b== ∑
rr
2
1 1( , ) n n
i ii i ia a a a a∗
= == =∑ ∑
r r

◘The norm or length of a vector a define as
◘Then a vector whose norm is unity is said
to be normalized.
1/2
( , )N a a=
2
1 1( , ) 1n n
i ii i ia a a a a∗
= == = =∑ ∑
r r

Linear Dependence and Independence
◘The set of vectors in a
vector space V is said to be
linearly dependent if there exist
scalars , not all zero,
such that
1 2, ,..., nc c c
1 1 ... 0n nc a c a+ + =
r r
1{ ,... }na a
r r

◘ The set of vectors is linearly
independent if
can only be satisfied when
1{ ,... }na a
r r
1 1 ... 0n nc a c a+ + =
r r
1 2 ... 0nc c c= = = =

Hilbert Space
◘In quantum mechanics , very often we
deal with complex function and the
corresponding function space is called
the Hilbert Space.
◘the Hilbert Space is a complete linear
vector space with an inner product.

an example of Hilbert Space is , the
space of square-integrable functions
on the real line. Here the inner product
is define by:
2
( )L R
( )f x
( , ) ( ) ( ),f g dxf x g x∞ ∗
−∞
= ∫

Orthogonal functions
The important definitions regarding
Orthogonal functions.
◘The Inner Product of two functions
F(X) and G(X) define in the interval
denoted as (F,G) OR (F G), is
( , ) ( ) ( )b
a
F G F x G x dx∗
= ∫
a x b≤ ≤

◘These functions are Orthogonal if
◘A function is normalized if its norm is
unity:
( , ) ( ) ( )
b
aF G F x G x dx∗
= ∫
1/221/2
( , ) ( ) 1
b
a
F F F x dx = =∫ 

◘Functions that are Orthogonal and
normalized are called orthonormal
Functions.
◘A set of functions is
linearly dependent if
Where ci are not all zero. Otherwise
they are linearly independent
( , ) , 1,2,...i j ijF F i jδ= = =
1 2( ), ( ),...F X F X
( ) 0i ic F x =∑

The expansion theorem
◘Any function defined in the same
interval can be expanded in terms of
the set of linearly-independent
functions as
Then, the coefficients are given by
( ) ( )i i
i
x c F xφ = ∑
( , )i ic F φ=
( )xφ

Linear vector space

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Linear vector space