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