The mathematics of quantum mechanics

Index

Index

Introduction

Ket space

Sum of kets

Commutative property

Associative property

Product for a scalar

Associative property

Distributive property with respect to scalars

Distributive property with respect to kets

Opposite of a ket

Difference of kets

Linear independence

Postulate on physical states

Bra space

Sum of bras

Commutative property

Associative property

Product for a scalar

Associative property

Distributive property with respect to scalars

Distributive property with respect to kets

Opposite of a bra

Difference of bras

Linear independence

Correspondence properties

Internal product

Fundamental properties

Linearity and antilinearity

Norm

Orthogonality

Schwarz inequality

Operators

Action on ket and bra

Null operator

Identity operator

Eigenket and eigenvalues

Sum of operators

Commutative property

Associative property

Linearity and antilinearity

Adjoint operator

Product of operators

Commutator and anticommutator

Functions of operators

Inverse operator

Adjoint of the product

Hermitian operators

Unitary operators

External product

Associativity

Projectors

Eigenkets basis

Representation of vectors and operators

Representation of a ket

Representation of an operator

Operations between matrices

Representation of a bra

Introduction

In this book we expose the mathematics for quantum mechanics. The main topics are: vectors, ket and bra space, properties and operations, product for a scalar, internal product between ket and bra, norm and Schwarz inequality, orthogonality, operators and their operations, operator acting on kets as a measure of an observable for a physical state, adjoint operator, hermitian operators, unitary operator, external product, projectors, basis of eigenkets, representation of vectors and operators, matrix algebra.

Ket space

In quantum mechanics we have to deal with physical states. A physical state is represented by a vector in an appropriate vectorial space with a certain dimension. This state vector is called ket, a term introduced by Dirac, and it contains all the information on the physical state. The dimension of the space depends on the situation, it can be finite or infinite and in the latter case it is called Hilbert space. A ket for a state a is indicated by the notation: