Chapter 7
Operators and Observations
IsthisadaggerIseebeforeme,thehandletowardmyhand? Come,letmeclutch
thee!Ihavetheenot,andyetIseetheestill.Artthounot,fatalvision,assensibleto
feelingastosight?Orartthoubutadaggerofthemind,afalsecreation,proceeding
fromtheheat-oppressedbrain?
- Shakespeare,Macbeth
Themathematicalcoreofquantummechanicsislinearalgebrainaninfinitenum-
berofdimensions. Almosteverymanipulationinquantumtheorycanbeintepreted
asanoperationinvolvinginnerproducts and/ormatrixmultiplication. Thisisfor
threereasons. First,aswehaveseen inLecture4,physicalstatesinquantumme-
chanicsarerepresented by wavefunctions, andany functioncan beregarded as a
vectorwithacontinuous index. Second, thedynamicalequation ofquantumme-
chanics,i.e. theSchrodingerequation,isalineardifferentialequation. Aswewill
see,lineardifferentialoperatorscanbeinterpretedasinfinite-dimensionalmatrices.
Finally,thepredictionsofquantummechanics,namely,theprobabilitiesofobserving
such-and-suchanevent,areinfactdeterminedbytheinnerproductsofvectorsin
HilbertSpace. Thislecturewillbedevotedtothesemathematicalaspectsofquantum
theory.
7.1 Probabilities From Inner Products
Wehavesofarlearnedhowtousethequantumstate|ψ>tocalculatetheexpectation
valueof anyfunctionofposition, anyfunctionofmomentum,or anysumofsuch
functions,suchasenergy.Inparticular
<x> =∫
dxψ∗(x,t)x ̃ψ(x,t)<p> =∫
dxψ∗(x,t)p ̃ψ(x,t)