90 CHAPTER6. ENERGYANDUNCERTAINTY
inpositionandmomentumexistindependentofany”disturbance”ofthesystemby
observation, andthattheseuncertaintieshaveimportantphysical consequences, is
verywellillustratedbythestabilityoftheHydrogenatom.
- WhytheHydrogenAtomisStable
Supposethewavefunctionofanelectronisconcentrated,moreorlessuniformly,
inasphere of radiusRaroundthe nucleus, andfallsrapidlyto zerooutside this
sphere. Thepreciseformofthewavefunctionisnotsoimportant,becauseweonly
wanttomakeaveryroughestimateoftheelectronenergy,whoseexpectationvalue
isgivenby
<H> = <KineticEnergy>+<PotentialEnergy>= <p^2
2 m>+<−
e^2
r>
=
∫
d^3 xψ∗(x,y,z,t)(−̄h^2
2 m∇^2 )ψ(x,y,z,t)+
∫
d^3 xψ∗(x,y,z,t)(−e^2
r)ψ(x,y,z,t) (6.26)Firstofallwecanestimatetheexpectationvalueofthepotentialenergy, whichis
roughly
<V >∼−e^2
R(6.27)
Next,theuncertaintyintheparticlepositionis
∆x≈∆y≈∆z∼R (6.28)whichimplies,bytheUncertaintyPrinciple,
∆px≈∆py≈∆pz∼h ̄
2 R(6.29)
andtheexpectationvalueofkineticenergyisthereforeontheorder
=
1
2 m<p^2 x+p^2 y+p^2 z>=1
2 m(∆p^2 x+∆p^2 y+∆p^2 z)∼
3 ̄h^2
8 mR^2(6.30)
Theexpectationvalueofthetotalenergyisthen
=
3 ̄h^2
8 mR^2−
e^2
R