Linear Vector Space and Matrix Mechanics Chapter 1 Lecture 1.5 Dr. Arvind Kumar Physics Department NIT Jalandhar e.mail: iitd.arvind@gmail.com https://sites.google.com/site/karvindk2013/

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:

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

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

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

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

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

Outer product

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,

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

Properties:

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

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