QMGreensite_merged

332 CHAPTER21. QUANTUMMECHANICSASLINEARALGEBRA

wherethistime

En=n^2

̄h^2 π^2

2 mL^2

(21.177)

ThematrixrepresentationsXmnisgivenby

Xmn=<φm|X|φn> (21.178)

whichisbestevaluatedintheX-representation,whichwealreadyknowshouldbe

Xmn =

∫

dxφ∗m(x)xφn(x)

=

1

2 L

∫L

0

dxxsin

mπx

L

sin

nπx

L

(21.179)

Togofromthebra-ketexpression(21.178)toitsx-representation(21.179),weuse
againtheidentityoperator

Xmn = <φm|IXI|φn>

= <φm|

{∫

dx|x><x|

}

X

{∫

dy|y><y|

}

|φn>

=

∫

dx

∫

dy <φm|x><x|X|y><y|φn>

=

∫

dx

∫

dyφ∗m(x)xδ(x−y)φn(y) (21.180)

Carryingoutthey-integration,weobtaineq.(21.179).Carryingoutthex-integration,
wefinallyobtain

Xmn=







−π (^2) (n^8 Lmn (^2) −m (^2) ) 2 (n−m) odd
0 (n−m) even
L
2 n=m

(21.181)

Similarly,themomentumoperatorinthesquare-wellrepresentationis

Pmn = <φm|P|φn>

=

∫

dxdyφ∗m(x)

(

−i ̄h

∂

∂x

δ(x−y)

)

φn(y)

=

∫

dxφ∗n(x)

(

−i ̄h

∂

∂x

)

φn(x) (21.182)

whichisevaluatedtobe

Pmn=−i ̄h

{ 4 nm

L(n^2 −m^2 ) (n−m) odd

0 (n−m) even

(21.183)