1. Linear Vector Space and Matrix Mechanics
Chapter 1
Lecture 1.6
Dr. Arvind Kumar
Physics Department
NIT Jalandhar
e.mail:

2. Projection operator:
idempotent

4. Example: What are conditions when is projection
Operator?
Ans: Given operator is Hermitian
Also the square of the operator is
If is normalized, then

5. Commutator algebra:

6. If two operators are Hermitian and their product is also
hermitian then these operators commute.
We write,
Also,
From above two equations, we get
Which means two operators commute.
x and px are dynamical variable but the product xpx is not
because this product does not commute.

9. Functions of operator:

11. Hermitian adjoint of function operators:
The adjoint of is given by

12. If operator is hermitian, then a function of operator
which can be expanded as
will be hermitian only if coefficients an are real numbers.
In general is not hermitian even if

13. One more important relation involving function
operator:
Consider function operator eA in terms of power series:
-----(1)
Consider function,
(2)
Where, λ is real number.
Expanding f(λ) using taylor series,
(3)

14. Note that the values of derivatives can be written as, using (2),
-----(4)
Using (2), (3) and (4) and using f(0) = B, we get
(5)

15. The rule is not valid for operators.
We use Campbell Baker Hausdorff formula, according to
Which,
(6)
F(A,B) is expressed as a infinite sum of multiple
commutators of A and B.
If A and B are two operators such that both commute
with their commutator [A,B] i.e. If [A,[A,B]] =
[B,[A,B]] = 0, then
(7)

16. Inverse an operator: The inverse of an operator
(if it exist) is defined by relation,
Where, is the unit operator.
Quotient of two operators:

17. Properties:

18. Unitary operators: A linear operator is said to be unitary
if its inverse is equal to its adjoint
Product of two unitary operators is also unitary
Product of any number of unitary operators is also unitary