Linear Vector Space and Matrix Mechanics Chapter 1 Lecture 1.11 Dr. Arvind Kumar Physics Department NIT Jalandhar e.mail: iitd.arvind@gmail.com https://sites.google.com/site/karvindk2013/

Representation in continuous bases: Ket, bra and operators are represented in the continuous basis by continuous matrices by noncountable infinite matrices, Orthonormality condition of base kets of contiuous basis is expressed Dirac’s continuous delta function Dirac delta function

Completeness relation for continuous basis is defined by Integral State vector is expanded as where b(k) is projection.

Recall that the norm of discrete base ket was finite but norm of continuous base ket is infinite

Properties of Dirac Delta function

Ket is represented in continuous basis by a single column matrix which has continuous and infinite number of Comopnets b(k), Bra is represented by row matrix

Operators are represented by square continuous matrices whose rows and columns have continuous and infinite number of components,

Position representation: Here basis consist of infinite set of vector We have Position operator, Eigenvalue Orthonormality and completeness conditions are,

State vector can be expanded as, Above eq. Give us the wave function for state vector. give us the probability of finding The particle in volume d3r.

The scalar product between two state vector is defined as Operator is Hermitian,

Momentum Representation: Basis of momentum representation is We have Momentum operator momentum vector Orthonormality and completeness conditions in Momentum representation are,

Momentum space wave function Probability of finding system’s momentum in volume element d3p is . Scalar product between two states is given in momentum Space by

Connection between momentum and position representation: We try to establish connection between position and momentum of state vector . We write -----(1)

Similarly we write ------(2) Eq (1) and (2) implies that are Fourier transform of each other. Fourier tranform of function f(r) is defined by ----------(3)

Thus we have, comparing (1) and (3) Inverse transform is,

Position wave function is Fourier transform is Show that if is normalized then also

Find momentum operator in position representation: Find position operator in momentum representation