Introduction to Quantum Mechanics 2

2Hermitian Operator, Dirac’s Notations

General objective

The general objective is to know the properties of Hermitian operators and the usefulness of Dirac’s notations.

Specific objectives

On completing this chapter, the reader should be able to:

• – define the space of square-summable wave functions;
• – know the properties of the scalar product of two functions;
• – define a discrete orthonormal basis;
• – define the Kronecker symbol;
• – define the components of a wave function;
• – define the norm of a wave function;
• – know the orthonormalization relation;
• – write the expansion of a wave function;
• – know the closing relation;
• – define the space of states;
• – know Dirac’s notations;
• – write the expansion of a state vector;
• – distinguish between a ket vector and a bra vector;
• – distinguish between a linear operator and a linear functional;
• – define the components of a ket and a bra;
• – define a matrix element;
• – define the projection operator on a ket and on a sub-space;
• – define a self-adjoint operator;
• – define a Hermitian operator;
• – give examples of Hermitian operators;
• – know the rules of Hermitian conjugation;
• – define a function of operators;
• – know the commutation rules;
• – define the Poisson brackets;
• – know the properties of commutators;
• – define the trace of an operator;
• – define a unitary operator;
• – define the density operator;
• – define the evolution operator;
• – define an observable;
• – know the properties of observables associated with spin;
• – know the properties of Pauli ...