QMGreensite_merged

12.1. THESCALEOFTHEWORLD 201

normalizationgivesN=(2a 0 )−^1 /^2 ,andtherefore

φ 200 =

1

√

8 πa^30

(

1 −

r

2 a 0

)

e−r/^2 a^0 (12.51)

Ingeneral,thewavefunctionforprincipalquantumnumberhastheform

φnlm=(polynomialinroforder≤n-1)×e−r/na^0 Ylm(θ,φ) (12.52)

Apolynomialcangothroughzero,soingeneraltheprobabilitydistributioninthe
radialdirectionhas”bumps”,whosepositionsdependonbothnandl.Inparticular,
ifweareonlyinterestedintheprobabilityoffindinganelectronbetweenradiir 1 and
r 2 ,then

prob(r 1 <r<r 2 ) =

∫r 2

r 1

drr^2

∫

dΩφ∗nlmφnlm

=

∫r 2

r 1

drr^2 Rnl^2 (r)

=

∫r 2

r 1

drP(r) (12.53)

wherewedefinetheradialprobabilitydensity

P(r) = r^2 R^2 nl(r)

= ((2n-thorderpolynomialinr)×e−^2 r/na^0 (12.54)

AsketchofP(r)vs. risshowninFig. [12.2],foranumberof low-lyingenergy
eigenstates.

12.1 The Scale of the World

Weknowthatthevolumeofagramofwaterisonecubiccentimeter. Whyisn’tthe
volumeinsteadonecubickilometer,oronecubicparsec?
Ifyouburnaloginafireplace,enoughenergyisreleasedtowarmtheroom. If
youburninsteadastickofdynamite,theenergyreleasedwilldestroytheroom,and
probablythe restofthehouse. Butneitherthelognorthe dynamitewillrelease
enoughenergytoflattenacity,orvaporizetheplanet. Whynot?
Quantummechanicsanswers thesequestions, givenasinput themasses of the
electronandproton,andtheelectroncharge. Infact,thesolutionoftheHydrogen
atomprovidesuswithrough,order-of-magnitudeestimatesforthevolumeofamole
ofsolid anything,andtheenergyreleasedbyburningamoleofanything. Thisis