# 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 ...

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