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