QMGreensite_merged

324 CHAPTER21. QUANTUMMECHANICSASLINEARALGEBRA

Theotheroperators,{P, Hho, Hsq},wedefinebytheirmatrixelementsintheX-
representation

<x|P|y> = p ̃δ(x−y) p ̃=−i ̄h

∂

∂x

<x|Hho|y> =

{

p ̃^2

2 m

+

1

2

kx^2

}

δ(x−y)

<x|Hsq|y> =

{

p ̃^2

2 m

+Vsq(x)

}

δ(x−y) (21.131)

(where Vsq(x)isthesquare-wellpotential). Wenoteonceagain thatall of these
matrixelementshavetheform

O(x,y)=O ̃δ(x−y) (21.132)

andthereforethematrixeigenvalueequation

∫

dy <x|O|y><y|ψn>=λn<x|ψn> (21.133)

becomestheoperator-acting-on-a-functionform

O ̃

∫

dyδ(x−y)ψn(y) = λnψn(x)

O ̃ψn(x) = λnψn(x) (21.134)

Inthisform,wehavealreadysolvedfortheeigenstatesofX, P, Hho, Hsq:

positioneigenstates ψx 0 (x) = <x|x 0 >=δ(x−x 0 )

momentumeigenstates ψp 0 (x) = <x|p 0 >=

1

√

2 π ̄h

eip^0 x/ ̄h

harm.osc. eigenstatesφnx = <x|φn>=

1

√

n!

(a†)ne−mωx

(^2) /2 ̄h
sq. welleigenstates φn(x) = <x|φn>=
√
1
2 L
sin
[
nπx
L
]
(21.135)
TheonlynewthingistheinsightthatthewavefunctionofastateintheX-representation
isaninnerproductψ(x)=<x|ψ>.
Giventhesematrixelements andwavefunctionsinthex-representation,wecan
findthematrixelementsandwavefunctionsinotherrepresentations. First,however,
wecheckthevalidityoftheoperatoridentity
[X,P]=i ̄hI (21.136)