Formelblatt - thphys.uni-heidelberg.dewolschin/ed-formelblatt.pdf · Die assozierten...
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Transcript of Formelblatt - thphys.uni-heidelberg.dewolschin/ed-formelblatt.pdf · Die assozierten...
Formelblatt
Vektor-Identitaten
~A · (~B × ~C) = ~B · ( ~C × ~A) = ~C · (~A × ~B)~A × (~B × ~C) = (~A · ~C)~B − (~A · ~B) ~C
(~A × ~B) · ( ~C × ~D) = (~A · ~C)(~B · ~D) − (~A · ~D)(~B · ~C)
∇ × ∇φ = 0
∇ · (∇ × ~A) = 0
∇ × (∇ × ~A) = ∇(∇ · ~A) − ∇2 ~A
∇ · (ψ~A) = ~A · ∇ψ + ψ∇ · ~A∇ × (ψ~A) = ∇ψ × ~A + ψ∇ × ~A∇(~A · ~B) = (~A · ∇)~B + (~B · ∇)~A + ~A × (∇ × ~B) + ~B × (∇ × ~A)
∇ · (~A × ~B) = ~B · (∇ × A) − ~A · (∇ × B)
∇ × (~A × ~B) = ~A(∇ · ~B) − ~B(∇ · ~A) + (~B · ∇)~A − (~A · ∇)~B
Differential-Operatoren
Kartesische Koordinaten (x1, x2, x3):
∇ψ = ~e1∂ψ
∂x1+ ~e2
∂ψ
∂x2+ ~e3
∂ψ
∂x3
∇ · ~A =∂A1
∂x1+∂A2
∂x2+∂A3
∂x3
∇ × ~A = ~e1
(
∂A3
∂x2− ∂A2
∂x3
)
+ ~e2
(
∂A1
∂x3− ∂A3
∂x1
)
+ ~e3
(
∂A2
∂x1− ∂A1
∂x2
)
∇2ψ =∂2ψ
∂x21
+∂2ψ
∂x22
+∂2ψ
∂x23
Zylinderkoordinaten (r, φ, z):
∇ψ = ~er∂ψ
∂r+ ~eφ
1r∂ψ
∂φ+ ~ez
∂ψ
∂z
∇ · ~A =1r∂(r Ar)∂r
+1r
∂Aφ
∂φ+∂Az
∂z
∇ × ~A = ~er
(
1r∂Az
∂φ−∂Aφ
∂z
)
+ ~eφ
(
∂Ar
∂z− ∂Az
∂r
)
+ ~ez1r
(
∂ (rAφ)
∂r− ∂Ar
∂φ
)
∇2ψ =1r∂
∂r
(
r∂ψ
∂r
)
+1r2
∂2ψ
∂φ2+∂2ψ
∂z2
Kugelkoordinaten (r, θ, φ):
∇ψ = ~er∂ψ
∂r+ ~eθ
1r∂ψ
∂θ+ ~eφ
1r sin θ
∂ψ
∂φ
∇ · ~A =1r2
∂(r2Ar)∂r
+1
r sin θ∂(sin θAθ)
∂θ+
1r sin θ
∂Aφ
∂φ
∇ × ~A = ~er1
r sin θ
[
∂(sin θAφ)
∂θ− ∂Aθ
∂φ
]
+ ~eθ
[
1r sin θ
∂Ar
∂φ− 1
r
∂(r Aφ)
∂r
]
+ ~eφ1r
[
∂(r Aθ)∂r
− ∂Ar
∂θ
]
∇2ψ =1r2
∂
∂r
(
r2∂ψ
∂r
)
+1
r2 sin θ∂
∂θ
(
sin θ∂ψ
∂θ
)
+1
r2 sin2 θ
∂2ψ
∂φ2
≡ 1r∂2(rψ)∂r2
+1
r2 sin θ∂
∂θ
(
sin θ∂ψ
∂θ
)
+1
r2 sin2 θ
∂2ψ
∂φ2
Kugelflachenfunktionen
Das Koordinatensystem:
x = r sin θ cos φ
y = r sin θ sin φ
z = r cos θ
Die Yl,m fur l = 0, 1, 2 sind:
Y0,0 =1√
4π
Y1,0 =
√
34π
cos θ
Y1,1 = −√
38π
sin θeiφ
Y1,−1 =
√
38π
sin θe−iφ
Y2,0 =
√
54π
12
(3 cos2 θ − 1)
Y2,1 = −√
158π
sin θ cos θeiφ
Y2,−1 =
√
158π
sin θ cos θe−iφ
Y2,2 =
√
152π
14
sin2 θe2iφ
Y2,−2 =
√
152π
14
sin2 θe−2iφ
Die assozierten Legendre-Polynome fur l = 0, 1, 2 sind:
P00(cos θ) = 1
P01(cos θ) = cos θ
P11(cos θ) = −
√1 − cos2 θ
P−11 (cos θ) =
12
√1 − cos2 θ
P02(cos θ) =
12
(3 cos2 θ − 1)
P12(cos θ) = −3
√1 − cos2 θ cos θ
P−12 (cos θ) =
12
√1 − cos2 θ cos θ
P22(cos θ) = 3(1 − cos2 θ)
P−22 (cos θ) =
18
(1 − cos2 θ)
