# Operators

# Operators

This chapter develops a basic working knowledge of operators — objects central to quantum mechanics. The fact that quantum operators are linear determines how they may be manipulated, so linearity forms the chapter's first topic. The adjoint of an operator is then introduced. This provides the basis for Hermitian or self-adjoint operators, which are discussed for both discrete eigenvalue systems and wavefunctions. Projection operators, which ‘project out’ a certain part of a quantum state, are then introduced. These form the basis for the identity operator — a sum over projection operators — which changes the representation in which a state is expressed without changing the state itself. Finally, unitary operators are briefly discussed, including the defining unitarity condition, the fact that such operators preserve inner products, and their physical meaning as effecting transformations in space and time. Unitary operators are discussed in greater detail in Appendix D.

*Keywords:*
linear operator, adjoint operator, Hermitian operator, projection operator, identity operator, unitary operator

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