Leftover files added.

lecture_notes/3-30/finite spherical well

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+Finite Spherical Well and Deuterium Nucleus
+
+    Attractive potential exists between proton and neutron
+
+    (pic) step function potential
+
+    Assume l=0 then S.E. for radial function is
+
+        −͟ħ͟²͟ ⎛∂͟²͟ R + 2͟ ∂͟ R⎞V R = E R
+         2m ⎝∂r²    r ∂r ⎠
+
+    K² ≡ (E͟_͟−͟_͟V͟)2m  ⇒  ∂͟²͟ R + 2͟ ∂͟ R + K² R = E R 
+            ħ²         ∂r²    r ∂r  
+
+    R(r) = U(r)/r  ⇒  ∂͟²͟ U = - K² R
+                      ∂r²  
+
+For r<a
+
+K₀ = 2͟m͟(E+V₀) ⇒ 
+     ħ²
+

lecture_notes/3-30/infinite spherical well

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+
+
+
+
+
+
+
+    
+    ⎧ 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 
+⎩

lecture_notes/3-30/nodes

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+U(ρ) = ρˡ⁺¹ exp(-ρ/2) Fₙᵣ(ρ)
+
+n ≡ nᵣ - l - 1 ⇒ l ≥ n-1 
+
+n│  l  │mₗ
+─┼─────┼───────────
+1│0    │0
+2│0,1  │-1,0,1
+3│0,1,2│-2,-1,0,1,2
+
+
+(pic) Fₙᵣ has n roots or nodes
+