A mathematics manual specifically thought for undergraduate and graduate Physics students. Covers topics in Algebra, Calculus and Functional Analysis among others.
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Preliminaries:
definition convolution and definition of the Four ier transform, transform,
∫ f (r − r′)g (r ′)d r ′ = f * g (r) = h(r) → φ(r) =ɺ −ε ∫ G(r − r ′) ρ ( r′)(2π ) d r ′; G( r − r ′) ≡ G(r ′ − r); (0.1) F[ f ( x)] ≡ e f (x )(2π ) d x ≡ fɶ (k ) ↔ ∃fɶ = fɶ ( k) : f ( x) = fɶ ( k)(2 π ) d k ≡ F [ f ɶ ( k)] ɺ ∫e ∫ −1
3
f * g ≡
3
3
0
− i ⋅k • x
+3
+ i⋅ k • x
3
−3
−1
3
As in electrodynamics, we want to solve the inhomogeneous Poisson equation with the Green-ansatz, −1 −1 −1 2 ∇ φ (r ) = ( G * ρ e )( )(r) G(r − r ′) ρ e ( r ′) dr ′ ≡ ρe (r ); ∃G = G(r − r′) : φ ( r) =
ε0
ε0
∫
(1.1)
ε 0
Let’s put this ansatz (1.1) into the indicated Poisson equation ∇ 2φ (r ) = − ρ e (r ) / ε 0 , and get, ∇
2
φ (r ) =
∫
2
∇ G (r − r ′)
− ρ e (r ′)
ε0
(2π )3 d 3 r ′ =
− ρ e (r)
ε 0
2
→ ∇ G(r − r ′) = (2π )
3
2
δ (r − r ′) →
∇ G( R) = (2π )
3
δ ( R) (1.2)
Now, let’s represent G (R) in Fourier space. This sends the Laplacian of (1.2) to being an algebraic object, 2
∇ G (R) = ɺ∇ =
2
∫e
∫e
+ i ⋅k • R
+ i ⋅k • R
∫
Gɶ (k )(2π )−3 d 3k = ( ik ) 2 e+ i⋅k• RGɶ (k )(2π ) −3 d 3 k
(2π ) δ (R ) (2π ) d k = e −3
3
3
+ i⋅k • 0
0
= e =1 ↔
(1.3)
ɶ (k ) = 1 −k G 2
Inverse-Fourier-transforming (1.3), we immediately get the familiar result,
∫
G ( R ) = F −1[Gɶ (k )] = e+ i⋅k •RGɶ (k )(2π ) −3 d 3k = =
−1
2π 2
∫
∞
0
e
i ⋅k • R
dk =
−1
π
2π 2 2 R
Note: I am off by a factor of (2π )
−3
=
−1
4π
R
∫
∞
0
e + i⋅k• R ( −k −2 ) (2π ) −3 4π k 2 dk
→ φ (r ) = −ε 0
−1
−1
∫ 4π R ρ (r′)(2π ) d r ′ = 3
3
1 4πε 0
∫
ρ (r′) 3 d r′ ′ r −r
here and there, but I do not have time to go back and fix that.
