QMGreensite_merged

10.2. PARITY 157

Wehavealreadynotedthattheenergiesofthefreeparticleare2-folddegenerate;
i.e.foreachenergyEtherearetwolinearlyindependentstateswiththesameenergy,
andanylinearcombinationofthosetwostates

ψ(x)=aei

√

2 mEx/ ̄h+be−i

√

2 mEx/ ̄h (10.23)

isalsoaneigenstatewithenergyE. However,sincep ̃commuteswithH ̃,itfollows
fromthecommutatortheoremthatenergyEandmomentumparesimultaneously
observable. Thismeansthatwecanchoosethecomplete setofenergyeigenstates
{φα}tobeeigenstatesofH ̃andp ̃,whichis

{

φp(x)=

1

√

2 π ̄h

eipx/ ̄h; Ep=

p^2

2 m

; p∈[−∞,∞]

}

(10.24)

Eachvalueofthemomentumpsinglesoutoneandonlyoneenergyeigenstateφp(x),
whereasanenergyeigenvalue,byitself,isnotenoughtospecifythestateuniquely.

10.2 Parity

Thefreeparticlepotential
V(x)= 0 (10.25)

theharmonicoscillatorpotential

V(x)=

1

2

kx^2 (10.26)

andthefinitesquarewellpotential

V(x)=







0 x<−a

−V 0 −a≤x≤a

0 x>a

(10.27)

areallinvariantundertheParitytransformation

x′=−x (10.28)

whichisaleft-rightreflectionofthex-axis.ThekineticenergytermoftheHamilto-
nianisalsoinvariantunderthistransformation,since

−

̄h^2

2 m

∂

∂x′

∂

∂x′

= −

̄h^2

2 m

(

−

∂

∂x

)(

−

∂

∂x

)

= −

̄h^2

2 m

∂

∂x

∂

∂x

(10.29)