330 CHAPTER21. QUANTUMMECHANICSASLINEARALGEBRA
or
X =
√
̄h
2 mω
0 1 0...
1 0
√
2...
0
√
2 0
√
3..
0 0
√
3 0
√
4.
......
P =
√
mω ̄h
2
0 −i 0...
i 0 −i√
2...
0 i√
2 0 −i√
3..
0 0 i√
3 0 −i√
4.
......
(21.167)
NoticethatthematrixrepresentationsofHho,X,andP,areallHermitian.
Exercise: Usingthematrixrepresentations(21.165)and(21.166),showthat
<φm|[X,P]|φn>=δmn (21.168)- TheSquareWellRepresentation
ThisrepresentationisonlyusefulforaHilbertSpaceinwhichtheposition-space
wavefunctionsareconstrainedto bezerooutsideafiniterangex∈[0,L]. Suchas
situationoccursiftheparticleistrappedinatubeoffinitelength.Liketheharmonic
oscillatorHamiltonian Hho, thesquarewellHamiltonianHsq hasadiscretesetof
eigenvaluesandeigenvectors,sothat”wavefunctions”arethecomponentsofinfinite-
dimensionalcolumnvectors,andoperatorsarerepresentedby∞×∞matrices.
FromtheorthonormalityofeigenstatesofaHermitianoperator,thewavefunction
ofthesquare-welleigenstate|φn>is
φn(m)=<φm|φn>=δmn m= 1 , 2 , 3 ,.... (21.169)whicharethecomponentsof∞-dimensionalcolumnvectors:
φ 1 =
1 0 0...
φ 2 =
0 1 0...
φ 3 =
0 0 1...