11 More Fun with Operators
11.1 Operators in a Vector Space
11.1.1 Review of Operators
First, a little review. Recall that thesquare integrable functions form a vector space, much
like the familiar 3D vector space.
~r=a~v 1 +b~v 2
in 3D space becomes
|ψ〉=λ 1 |ψ 1 〉+λ 2 |ψ 2 〉.
Thescalar productis defined as
〈φ|ψ〉=∫∞
−∞dx φ∗(x)ψ(x)and many of its properties can be easily deduced from the integral.
〈φ|ψ〉∗=〈ψ|φ〉As in 3D space,
~a·~b≤|a||b|
the magnitude of the dot product is limited by the magnitude of the vectors.
〈ψ|φ〉≤√
〈ψ|ψ〉〈φ|φ〉This is called theSchwartz inequality.
Operators are associativebut not commutative.
AB|ψ〉=A(B|ψ〉) = (AB)|ψ〉Anoperator transforms one vector into another vector.
|φ′〉=O|φ〉Eigenfunctions ofHermitian operators
H|i〉=Ei|i〉form anorthonormal
〈i|j〉=δij
complete set
|ψ〉=
∑
i〈i|ψ〉|i〉=∑
i|i〉〈i|ψ〉.