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)