1. Linear Vector Space and Matrix Mechanics
Chapter 1
Lecture 1.2
Dr. Arvind Kumar
Physics Department
NIT Jalandhar
e.mail:

2. The theory of quantum mechanics is based on the linear
operators and the wave functions belonging to abstract
Hilbert space.
Schrodinger Eq.:
Before going into the discussion of Hilbert space let us
review some properties of 3-dimenional Euclidian space.
A vector V can be written in Cartesian so-ordinate
system as,
Vx, Vy and Vz are components of vector V. ex, ey and ez
are the unit vectors along three orthogonal directions x, y and
z and span the vector space.

3. The inner product (dot product) of two vector is defined as
The length of vector is given by

4. Hilbert space and wave functions:
Linear vector space:
A linear vector space consist of two set of elements
(i) Set of vectors (ii) set of scalar
And two algebraic rules
(i) Rule for vector addition (ii) rule for scalar multiplication
Known as field

5. The addition rule has the properties and structure of
abelian group:
if and are vectors of a space then the sum
is also a vector of same space.
Commutativity:
Associativity:
Existence of zero vector:
Existence of symmetric or inverse vector:

6. (II) Multiplication rule: The multiplication of scalars by
vectors has following properties (scalars can be real or
complex)
The product of a scalar with a vector gives another vector
If and are two vectors of space then any linear
combination is also a vector of the space.
Here a and b are two scalars.

7. Distributive w.r.t addition
Associativity
For unitary scalar I and zero scalar O,

8. Examples:
The set of n-components of a vector form a vector
Space
The set of M by N matrices with complex entries form
a complex vector space

9. n by n Hermitian matrices with complex entries form a
real vector space.
They cannot form a complex vector space because the
multiplication of hermitian matrix by a complex
number will destroy the hermicity and will yield
anti-hermitian matrix and therefore will not be closed
under complex scalar multiplication.
An example of hermitian matrix is Pauli matrices given by,

10. Hilbert Space: The Hilbert space H consist of set of
Vectors and scalars which satisfy
following properties:
Hilbert space is a linear space satisfying the properties
discussed on last slides
(b) Hilbert space H has a well defined scalar product
and is strictly positive. The scalar product of two elements is in general complex and thus order of elements matter.

11. The scalar product of element with other element
is equal to the complex conjugate of with
Proof:
The scalar product of with is linear w.r.t 2nd factor .
If ψ =
Then we have

12. The scalar product of with is anti-linear w.r.t 1st
factor . If
Then,
Proof:

13. The scalar product of vector with itself is a positive real number
where equality hold of

14. The Hilbert space is complete.
To understand the meaning of this statement we first need
to know the meaning of sequence, convergence of sequence
and Cauchy sequence.
A sequence say an is some
ordered list and it converges
if it has some limit L.

15. A Cauchy sequence is a sequence whose terms become
arbitrarily close together as n gets very large.
An infinite sequence of vectors in a normed linear
space V is called a Cauchy sequence if

16. In the scalar product the 2nd factor belongs to the
Hilbert space whereas the 1st factor belongs to the dual
Hilbert space. This is because the scalar product is not
commutative and order matters.

17. Dimension and Basis of vector space:
A set of N nonzero vectors is said
to be linearly independent if and only if the solution of Eq is
It means any vector of the space cannot be expressed as
linear combination of other vectors of same space.

18. If there exist a set scalar which are not all zero such that one
of the vector say can be written as linear combination of
others i.e.

19. Dimensions: It is the maximum number of the linear
independent vectors of the space.
For example if we have N linearly independent vectors
then we say it is N dimensional space and any
other vector can be expanded as linear combination.
Basis: The set of maximum number of possible linearly
independent vectors of the space form the basis of that
vector space.

20. The set denoted by is called the basis and
is known as base vector.
We choose above set of linear independent vectors as
orthonormal. It means their scalar product satisfy the
condition
The basis is called complete if it span the entire space. It
means we need not to introduce any other base vector.
Any vector of the space can be written as linear combination
of base vectors.

21. The coefficients are known as the components of the
vector .
The components can be obtained by the scalar product of the
vector by base vector
Now go back to first slide of present lecture and compare the
above discussed properties with the vectors in Euclidean
space.

22. Examples:
The orthogonal unit vectors we use in the 3-Dim Euclidian
space are linearly independent.
Two parallel vectors in a plane are linearly dependent as
one can be expressed as multiple of other.
The 2 by 2 matrices form a four-dim space. To understand
we consider following four linearly independent matrices

23. Now any two dimension matrix can be written as linear
combination of above 4 matrices.
If a, b, c and d are real then we have real 4-Dim space and
if these are complex then we have complex 4-Dim space.
****From QM by R Shankar:

24. .
Theorem 2: Expansion of a vector in a given
basis is unique.
See QM by R Shankar.

25. Square integrable function: Wave functions
The scalar product of is given by
If above integral diverge scalar product does not exist.
A function is called the square integrable if
is finite.
Scalar product is also known as inner product.

26. According to Born’s probabilistic interpretation, the total
probability of finding the particle somewhere in the space is
1 i.e.
Thus wave functions of quantum mechanics are square
integrable functions.
The space of square integrable functions satisfy the
properties of Hilbert space. A wave function which is
not square integrable have no physical significance in
quantum mechanics.
Mathematician calls the Hilbert space as L2.