6.2. THEHEISENBERGUNCERTAINTYPRINCIPLE 91
Thetotalenergyisminimizedford
at
R=
3 ̄h^2
4 me^2
(6.32)
whichisnotveryfarofftheBohrradius
r 1 =
̄h^2
me^2
(6.33)
Wecannowunderstandwhytheelectrondoesn’tfallintothenucleus. Thepo-
tentialenergyisminimized byanelectron localizedat r =0. However,themore
localizedanelectronwavefunctionis,thesmaller∆xis,andthesmaller∆xis,the
greateristheuncertaintyin∆p. Butthegreaterthevalueof∆p,thegreateristhe
expectationvalueofkineticenergy
<
p^2
2 m
>∼
̄h^2
2 m∆x^2
(6.34)
andatsomepointthis(positive)kineticenergyoverwhelmsthe(negative)Coulomb
potential,whichgoeslike−e^2 /∆x. Thisiswhytheminimumelectronenergyinthe
Hydrogenatomisobtainedbyawavefunctionofsomefiniteextent,ontheorderof
theBohrradius.
Theapplicationofthe UncertaintyprincipletotheHydrogen atomshowsthat
thereismuchmoretothisprinciplethansimplythefactthatanobservationdisturbs
anobservedobject. Thereareverymanyhydrogenatomsintheuniverse;veryfew
of themare underobservation. If∆pwere dueto a disturbanceby observation,
thentherewouldbenothingtopreventanunobservedelectronfromfallingintothe
nucleus. Nevertheless, allhydrogenatoms, observedor not,have astableground
state,andthisisduetothefactthatthereisnophysicalstateinwhichanelectron
isbothlocalized,andatrest.
Problem:Computetheexpectationvalueofthekineticandpotentialenergyfrom
eq. (6.26),usingthegaussianwavepacketinthreedimensions
φ(x,y,z)=Nexp[−(x^2 +y^2 +z^2 )/ 2 R^2 ] (6.35)
(applythenormalizationconditiontodetermineN). Incomputingtheexpectation
valueofthepotentialenergy,theformulaforsphericalcoordinates
∫
dxdydzf(r)= 4 π
∫
drr^2 f(r) (6.36)
maybeuseful.FindthevalueofRwhichminimizes
totheBohrradius. Alsocomparenumericallytheminimumvalueof
groundstateenergyE 1 oftheBohratom.