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