1. Linear Vector Space and Matrix Mechanics
Chapter 1
Lecture 1.8
Dr. Arvind Kumar
Physics Department
NIT Jalandhar
e.mail:

2. Representation in Discrete Basis:
We know that any vector in the Euclidean space can be
represented in terms of basis vectors.
The state vector of Hilbert space can be written
in terms of a complete set of mutually orthonormal base
kets.

3. Matrix representation of Ket, Bra and Operators:
Consider a discrete, complete and orthonormal basis made
of infinite set of kets
This can be denoted by
(1)
Where is the Kronecker delta symbol defined by
(2)
Completeness or closure relation is defined by

4. We can use the completeness relation to expand the vector
in terms of base kets.
We write
-----(3)
Coefficient = , represent the projection of On
is represented by set of its components a1, a2, a3.....
along respectively.

5. is represented by column vector
(4)
is represented by row vector
(5)

6. Using (4) and (5), we can write bra-ket as
----(6)
Where,

7. Matrix representation of operators:
For each linear operator we can write
Where Anm is the nm matrix element of operator

8. Operator within basis is represent by a square
matrix
e.g. Unit operator us represented by unit matrix
Kets are represented by column vectors, bra by row vectors
And operators by square matrices.

9. Hermitian adjoint operation:
The transpose of matrix A is AT, given by
Transpose of column matrix is row matrix,

10. Square matrix A is symmetric if
It is skew-symmetric if,
Hermitian adjoint operator is the complex conjugate of the
matrix transpose of A
i.e.

12. Matrix representation of
Trace of an operator: