Operator methods in quantum mechanics
Chapter 3Operator methods inquantum mechanicsWhile the wave mechanical formulation has proved successful in describingthe quantum mechanics of bound and unbound particles, some properties cannot be represented through a wave-like description. For example, the electronspin degree of freedom does not translate to the action of a gradient is therefore useful to reformulate quantum mechanics in a framework thatinvolves only discussing properties of operators, it is helpful to introduce a furthersimplification of notation. One advantage of the Operator algebra is that itdoes not rely upon a particular basis. For example, when one writes H= p22m,where the hat denotes an Operator , we can equally represent the momentumoperator in the spatial coordinate basis, when it is described by the differentialoperator, p= i!
an eigenstate of the momentum operator,ˆp = −i!∂x, with eigenvalue p. For a free particle, the plane wave is also an eigenstate of the Hamiltonian, Hˆ = pˆ2 2m with eigenvalue p2 2m. In quantum mechanics, for any observable A, there is an operator Aˆ which acts on the wavefunction so that, if a system is in a state described by |ψ",
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