mirror of
https://asciireactor.com/otho/phy-4600.git
synced 2025-10-31 15:58:05 +00:00
41 lines
823 B
Plaintext
41 lines
823 B
Plaintext
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
⎧ 0 r<a
|
|||
|
|
V = ⎨
|
|||
|
|
⎩ ∞ r>a
|
|||
|
|
|
|||
|
|
for r<a
|
|||
|
|
───────
|
|||
|
|
|
|||
|
|
∂͟²͟ R + 2͟ ∂͟ R - 1͟ l(l+1) K²R = 0
|
|||
|
|
∂ρ² r ∂r 2r²
|
|||
|
|
|
|||
|
|
K ≡ √⎛2͟m͟E͟⎞ ρ ≡ Kr
|
|||
|
|
⎝ ħ²⎠
|
|||
|
|
|
|||
|
|
⇒ ∂͟²͟ R + 2͟ ∂͟ R - ⎛ 1͟ l(l+1) - 1 ⎞ R = 0
|
|||
|
|
∂ρ² ρ ∂r ⎝ ρ² ⎠
|
|||
|
|
|
|||
|
|
The solutions are Bessel functions
|
|||
|
|
|
|||
|
|
|
|||
|
|
⎧ jₗ(ρ) = (-ρ)ˡ ⎛1͟ d͟ ⎞ˡ⎛s͟i͟n͟ρ͟⎞
|
|||
|
|
⎪ ⎝ρ dρ⎠ ⎝ ρ ⎠ (regular)
|
|||
|
|
R(ρ) = ⎨
|
|||
|
|
⎪ nₗ(ρ) = -(-ρ)ˡ ⎛1͟ d͟ ⎞ˡ⎛c͟o͟s͟ρ͟⎞ ⎛ irregular or ⎞
|
|||
|
|
⎩ ⎝ρ dρ⎠ ⎝ ρ ⎠ ⎝ "Neuman function" ⎠
|
|||
|
|
|
|||
|
|
For continuity
|
|||
|
|
|
|||
|
|
|
|||
|
|
⎧
|
|||
|
|
⎪ A sin(K₀ a) = C e⁻ᵃ
|
|||
|
|
⎨
|
|||
|
|
⎪ K₀Acos(K₀ a) = -ᵩ
|
|||
|
|
⎩
|