QMGreensite_merged

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)