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41 lines
823 B
Plaintext
41 lines
823 B
Plaintext
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⎧ 0 r<a
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V = ⎨
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⎩ ∞ r>a
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for r<a
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───────
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∂͟²͟ R + 2͟ ∂͟ R - 1͟ l(l+1) K²R = 0
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∂ρ² r ∂r 2r²
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K ≡ √⎛2͟m͟E͟⎞ ρ ≡ Kr
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⎝ ħ²⎠
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⇒ ∂͟²͟ R + 2͟ ∂͟ R - ⎛ 1͟ l(l+1) - 1 ⎞ R = 0
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∂ρ² ρ ∂r ⎝ ρ² ⎠
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The solutions are Bessel functions
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⎧ jₗ(ρ) = (-ρ)ˡ ⎛1͟ d͟ ⎞ˡ⎛s͟i͟n͟ρ͟⎞
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⎪ ⎝ρ dρ⎠ ⎝ ρ ⎠ (regular)
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R(ρ) = ⎨
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⎪ nₗ(ρ) = -(-ρ)ˡ ⎛1͟ d͟ ⎞ˡ⎛c͟o͟s͟ρ͟⎞ ⎛ irregular or ⎞
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⎩ ⎝ρ dρ⎠ ⎝ ρ ⎠ ⎝ "Neuman function" ⎠
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For continuity
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⎧
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⎪ A sin(K₀ a) = C e⁻ᵃ
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⎨
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⎪ K₀Acos(K₀ a) = -ᵩ
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⎩
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