Linear Vector Space and Matrix Mechanics Chapter 1 Lecture 1.6 Dr. Arvind Kumar Physics Department NIT Jalandhar e.mail: iitd.arvind@gmail.com https://sites.google.com/site/karvindk2013/

Projection operator: idempotent

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

Commutator algebra:

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.

Functions of operator:

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

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

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)

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)

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)

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

Properties:

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