1. Linear Vector Space and Matrix Mechanics
Chapter 1
Lecture 1.5
Dr. Arvind Kumar
Physics Department
NIT Jalandhar
e.mail:

2. Operators: An operator is an entity which is when operated
on some ket transform it into other ket and if operated
on bra then transform bra into other bra
Examples:

3. Product of operators: Product of two operators is
not commutative
However it is associative

4. Order of operation of operators matter:
Sandwiching of an operator between ket and bra in
general yield complex number.
= complex number
In evaluating it does not matter whether we
first apply operator on bra or on ket

5. Example: We shall discuss later that ket are represented
by column matrix, bra are by row matrix and operators by square matrix.
Find: (1)
(2) Verify

6. Linear operators: An operator is said to be linear if it
obey the distributive law and like all operators it
commute with constants.

7. Expectation value of operator: Expectation value of an
operator is the average of repeated measurements on an
ensemble of identically prepared systems. =
Expectation value of position and momentum are written as

8. Outer product

9. Hermitian Adjoint or Hermitian Conjugate:
Hermitian adjoint of a complex number α is the complex
Conjugate of this number and is denoted by α† .
Thus α† = α*.
The hermitian conjugate or adjoint of an operator
is such that following relation is satisfied,

10. To obtain hermitian adjoint of any expression we must
cyclically reverse the order of factor involved in the
expression

11. Properties:

13. Hermitian operators or self adjoint operators:
Using above definition of Hermitian operator in Eq.
We see that
Also we observe that expectation value of hermitian operator
are real and therefore the obervables are represented by
hermitian operator

14. Skew Hermitian operators:
The expectation values of anti-hermitian operator are
imaginary