Linear Vector Space and Matrix Mechanics Chapter 1 Lecture 1.8 Dr. Arvind Kumar Physics Department NIT Jalandhar e.mail: iitd.arvind@gmail.com https://sites.google.com/site/karvindk2013/

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.

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

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.

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

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

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

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.

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

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.

Matrix representation of Trace of an operator: