7.3. EIGENSTATESASSTATESOFZEROUNCERTAINTY 111
EigenstatesofMomentum
For
p ̃=−i ̄h
∂
∂x
(7.80)
theeigenvalueequationis
−ih ̄
∂
∂x
φp 0 (x)=p 0 φp 0 (x) (7.81)
whichhassolutions
{
eigenstates φp 0 (x)=
1
√
2 π ̄h
eip^0 x/ ̄h, eigenvalues p 0 ∈[−∞,∞]
}
(7.82)
withinnerproducts
<φp 1 |φp 2 >=δ(p 1 −p 2 ) (7.83)
AsaconsequenceoftheoremH3,anyarbitraryfunctionψ(x)can bewrittenas a
superpositionoftheseeigenstates:
ψ(x)=
∫
dp 0 cp 0 φp 0 (x) (7.84)
Thecoefficientfunctioncp 0 isobtainedbymultiplyingbothsidesofthisequationby
φ∗p 1 (x),andintegratingoverx
∫
dxφ∗p 1 (x)ψ(x) =
∫
dx
∫
dp 0 cp 0 φ∗p 1 (x)φp 0 (x)
1
√
2 π ̄h
∫
dxψ(x)e−ip^1 x/ ̄h =
∫ dp
0
2 π ̄h
cp 0
∫
dxei(p^0 −p^1 )x/ ̄h
=
∫
dpcp 0 δ(p 1 −p 2 ) (7.85)
or
cp=
1
√
2 π ̄h
∫
dxψ(x)e−ip^1 x/ ̄h (7.86)
EigenstatesofEnergy: TheParticleinaTube
TheHamiltonianoperatoris
H ̃=− ̄h
2
2 m
∂^2
∂x^2
+V(x) (7.87)
whereV(x)for the particle ina tubewas givenineq. (5.100). The eigenvalue
equationisthetime-independentSchrodingerequation
[
−
h ̄^2
2 m
∂^2
∂x^2
+V(x)
]
φn(x)=Enφn(x) (7.88)