Chapter 21
Quantum Mechanics as Linear
Algebra
Ihavestressedthroughoutthecoursethatphysicalstatesinquantummechanicsare
representedbyvectors(inHilbertSpace),thatobservablesareassociatedwithlinear
operators, andthat linear operators have amatrix representation. This suggests
thattheequationsofquantummechanicscan beexpressedintheterminologyand
notationoflinearalgebra,i.e. vectorsandmatrices. Thischapterwillexplorethe
linearalgebra/quantummechanicsconnectioninmoredetail. Inaddition, wewill
studyDirac’sprescriptionforquantizinganarbitrarymechanicalsystem.
Itmayhelp tohaveinmindthemaintheme. WesawinLecture 7 thatevery
linearoperatorO ̃hasamatrixrepresentationO(x,y),where
O(x,y)=O ̃δ(x−y) (21.1)Thisisactuallyonlyoneofaninfinitenumberofpossiblematrixrepresentationsof
O ̃;itisknownasthex-representation(orposition-representation). Aswewillsee,
thereexistotherusefulrepresentationsofoperators,amongthemthep-(momentum-
)representation,andtheE-(energy-)representation. Eachmatrixrepresentationis
associated withadifferentset oforthonormal basisvectors inHilbert Space, and
eachset oforthonormal basisvectorsarethe eigenstates of some completeset of
linearoperators.Attheendofthechapter,youshouldaimforagoodunderstanding
ofthelastsentence.
Webeginwithaquickreviewofsomebasicfactsaboutvectorsandmatrices.