112 CHAPTER7. OPERATORSANDOBSERVATIONS
whichhassolutions
eigenstates φn(x)=
√
2
L
sin[
nπx
L
], eigenvalues En=n^2
̄h^2 π^2
2 mL^2
,, n= 1 , 2 , 3 ,...
(7.89)
Theeigenstateshaveinnerproducts
<φn|φm>=δnm (7.90)
Onceagain,theoremH3insiststhatanyarbitraryfunctionψ(x)canbewrittenasa
linearcombination^1
ψ(x) =
∑∞
n=1
cnφn(x)
=
√
2
L
∑∞
n=1
cnsin[
nπx
L
] (7.91)
ThisisaFourierseries,anditiswellknownthatanyfunctionintheinterval[0,L]
canbeexpressedinthisform. Thecoefficientscnareobtainedinthesamewayas
before,i.e. bymultiplyingbothsidesbyφ∗mandintegratingoverx:
|ψ> =
∑∞
n=1
cnφn(x)
<φm|ψ> =
∑∞
n=1
cn<φm|φn>
=
∑∞
n=1
cnδnm (7.92)
andwefind
cm = <φm|ψ>
=
√
2
L
∫L
0
dxsin[
mπx
L
]ψ(x) (7.93)
7.4 The Generalized Uncertainty Principle
Supposewewanttodeviseameasurementapparatusthatwillmeasuretwoobserv-
ablesAandBsimultanously. Fromthediscussionofprevioussections,itisclear
(^1) Weexcludepathologicalcasesforwhich
intheregionsx< 0 orx>L.