QMGreensite_merged

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=∞,suchaswavefunctionswhicharenon-zero
intheregionsx< 0 orx>L.