Spektrum hermitescher Operatoren ˆA =ˆA

Spektrum hermitescher Operatoren  = †
Spektrum
Â|ϕa i = a|ϕa i
, |ϕa i = Eigenzustand
,
a = Eigenwert
a∗ = hϕa |Â|ϕa i∗ = hϕa |† |ϕa i = hϕa |Â|ϕa i = a
a∈R
2
Varianz
ˆ |ϕa i = 0
hϕa |∆A
orthogonal
a 6= b
ˆ = Â − hϕa |Â|ϕa i
∆A
| {z }
=⇒
ˆ a i = Â|ϕa i − a|ϕa i = 0
∆A|ϕ
=a
vollständig
=⇒
hϕa |ϕb i = 0
Bew.: 0 = hϕb |Â|ϕa i = hϕb |† |ϕa i = hϕa |Â|ϕb i∗ = hϕa |b|ϕb i∗ = hϕb |b∗ |ϕa i , b∗ = b
=⇒
0 = hϕb |† |ϕa i − hϕb |Â|ϕa i = hϕb |b|ϕa i − hϕb |a|ϕa i = (b − a) hϕb |ϕa i
| {z } | {z }
⇒0
6=0
Z
X
|ϕa ihϕa | = 1̂
a
Funktion
diskret
Auswertung f (Â) per Spektrum:
f (Â)|ϕa i = f (a)|ϕa i
X
Â|ϕn i = an |ϕn i , hϕn |ϕm i = δnm ,
|ϕn ihϕn | = 1̂
n
r
π 2
~2 π 2 n2
Bsp.: Kastenpotential ϕn (x) =
sin
nx
, Ĥ|ϕn i =
|ϕn i
2
L
L
|2mL
{z }
{|ϕn i, an , n ∈ N} :
En
Kontinuum
{|ϕa i, a ∈ R} :
Â|ϕa i = a|ϕn i ,
Bsp.: p̂ = −i~∂x
Messung
präzise
←→
,
p̂ϕp (x) = pϕp (x) ,
2
ˆ |ψi = 0
hψ|∆A
Bew.: hΦ|Φi = 0
hϕa |ϕb i = δ(a − b) ,
←→
Z
1
ϕp (x) = √ exp
2π
∞
−∞
da |ϕaihϕa | = 1̂
i
px
~
Â|ψi = A|ψi notwendig Eigenzustand
ˆ 2 |ψi = hψ|(Â − A) (Â − A)|ψi
|Φi = 0 & hψ|∆A
| {z }
=⇒
≡|Φi
=⇒
gemischt
Phase
(Â − A)|ψi = 0
X
ˆ 2 |ψi =
hψ|∆A
6 0 =⇒ |ψi =
|ϕn ihϕn |ψi
n
2
=⇒ wn = hϕn |ψi = Wahrscheinlichkeit, Messwert an zu finden
|ψi −→ |ψ̃i = eiδ |ψi ,
δ ∈ R beliebige Phase
=⇒
iδ
hψ̃|Â|ψ̃i = hψ|e−iδ Âeiδ |ψi = e|−iδ
{ze }hψ|Â|ψi = hψ|Â|ψi = A
=1
Â|ψi = a|ψi
=⇒
Â|ψ̃i = a|ψ̃i
irrelevant für Messung