QMGreensite_merged

21.3. HILBERTSPACE 333

Writtenoutas∞×∞matrices,XandPare

X = L













1

2 −

16

9 π^20...

− 916 π 2 12 − 2548 π 2...

0 − 2548 π 2 12...

......
......
......













P =

ih ̄

L













0 83 0...

−^830245...

0 −^2450...

......

......

......













(21.184)

whicharebothclearlyHermitian.

Exercise: Calculatethe 11 matrixelementofthe[X,P]commutatorapproximately,
inthesquarewellrepresentation,using

[X,P]mn=

∑∞

k=1

[XmkPkn−PmkXkn] (21.185)

Carryoutthesumoverktok=16.

• TheAngularMomentumRepresentation

Asafinalexample,weconsidertheHilbertspacespannedbytheeigenstatesof
angularmomentum

J^2 |jm> = j(j+1) ̄h^2 |jm>

Jz|jm> = m ̄h|jm> (21.186)

Inthespecialcasethatjisaninteger,theinnerproductof|jm>witheigenstates
ofangles|θ,φ>givesthesphericalharmonics

Yjm(θ,φ)=<θφ|jm> (21.187)

Thesphericalharmonicsarethewavefunctionsofeigenstatesofangularmomentumin
the”anglebasis,”spannedbythetheset{|θ,φ>}ofangleeigenstates(arigidrotator
inan angleeigenstates has adefinite angular positionθ,φ However, the angular
momentumalgebra alsoallows forhalf-integer eigenvalues j = 1 / 2 , 3 / 2 , .... In
particular,thej= 1 / 2 caseisimportantfordescribingthespinangularmomentum