QMGreensite_merged

6.3. THEENERGYOFENERGYEIGENSTATES 93

Ontheotherhand,usingtheintegrationbypartsformula(5.7),

Hmn =

∫

dx

{

−

̄h^2

2 m

φ∗m

∂^2 φn

∂x^2

+V(x)φ∗mφn

}

=

∫

dx

{

−

̄h^2

2 m

∂^2 φ∗m

∂x^2

φn+V(x)φ∗mφn

}

=

∫

dx(H ̃φm)∗φn

= Em∗ <φm|φn> (6.44)

Comparing(6.43)and(6.44),

En<φm|φn>=E∗m<φm|φn> (6.45)

Forthecasen=m,thisequationimplies

En=E∗n (6.46)

i.e.theenergyeigenvaluesarereal,whileforn+=m

(En−Em)<φm|φn>= 0 (6.47)

Whentheeigenvaluesarenon-degenerate,itmeansthat

n+=m=⇒En+=Em (6.48)

andtherefore
<φm|φn>= 0 (n+=m) (6.49)

Choosingtheeigenstatesφntosatisfythenormalizationcondition

<φn|φn>= 1 (6.50)

establishestherelation
<φm|φn>=δmn (6.51)

whichwehavealreadyseentobetrue(eq. (5.122))inthecaseoftheparticleina
tube.
Accordingtoeq. (5.38),thegeneralsolutiontothetime-dependentSchrodinger
equationcanbeexpressedintermsoftheenergyeigenstatesas

ψ(x,t)=

∑

n

anφne−iEnt/ ̄h (6.52)

Thentheenergyexpectationvalueis

=

∫

dx

[

∑

n

amφme−iEmt/ ̄h

]∗

H ̃

∑

n

anφne−iEnt/ ̄h

=

∑

m

∑

n

a∗mane−i(En−Em)t/ ̄hEn<φm|φn>

=

∑

n

Ena∗nan (6.53)