QMGreensite_merged

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.