18.2. ADIABATICPERTURBATIONS 297
NotethatsinceEl(t)variesintime,theenergyofthesystemisnotconserved,
evenafterlongtimes. But,youmayask,did wenotprove fromtheSchrodinger
equation,backinthelastsemester,that∂t〈H〉=0?Actually,ifyoulookbackatthe
derivation,itassumedthatthepotentialV(x)wastime-independent. Ifthepotential
dependsontime,thereisnoreasonfortheenergytobeconstant. Ofcourse,ifwe
wouldtreateverythingquantum-mechanically,includingthesourceofthepotential,
thentheenergyoftheentiresystemmustbeconserved.Butthatsanotherstory.
18.2.1 Example
Consideraparticleofmassm, initially(timet =0)initsgroundstateinaone-
dimensionalboxoflengthL;i.e.
φ 1 (x) =
√
2
L
sin
(πx
L
)
E 1 =
π^2 ̄h^2
2 mL^2
(18.72)
Supposethatthewallsoftheboxmoveapartfromeachotherveryslowly,sothat
theadiabaticassumptionisjustified,andthataftersomelongtimetthewallsarea
distanceαLapart.
Wedon’tevenneedperturbationtheorytosolvethisproblem.Allthatisrequired
istosolvethetimeindependentSchrodingerequation(exactly)attimet.Theresult
is
φ 1 (x,t) =
√
2
αL
sin
(πx
αL
)
E 1 (t) =
π^2 ̄h^2
2 mα^2 L^2
(18.73)
Youcanseethattheparticlehasanenergyloss
∆E=
π^2 ̄h^2
2 mL^2
(
1 −
1
α^2
)
(18.74)
whichhasasimpleclassicalinterpretation: Theparticleintheboxexertsapressure
onthewalls. Asthewallsmove, theparticledoespositivework,andlosesenergy.
Thatsagoodthing; otherwisesteamandautomobileengineswouldn’t work! Gas
moloculesultimatelyobeythelawsofquantummechanics.Inpushingapiston,they
hadbettergiveupsomeoftheirenergytothepiston,otherwiseitwouldbehardto
understandhowquantummechanicswouldbeconsistentwiththeprinciplesofheat
enginedesign,thatwereworkedoutinthe19thcentury.