added 3-28 lecture

lecture_notes/3-28/3-28-overview.txt

@@ -1 +0,0 @@
-Developed Theory for The Hydrogen Atom

lecture_notes/3-28/hydrogen atom

@@ -1,43 +1,17 @@
-Theory of the Hydrogen Atom
-━━━━━━━━━━━━━━━━━━━━━━━━━━━━
-    V(𝐫) = -e²/r
-
-    For any hydrogenic ion with nuclear charge Z
-        V(𝐫) = -Ze²/r
-
-    Eigenfunctions in spherical coordinates:
-
-        Ψₑ﹐ₗ﹐ₘ(r,θ,φ) = Rₑ﹐ₗ(r) Yₗ﹐ₘ(θ,φ) = Uₑ﹐ₗ/r Yₗ﹐ₘ(θ,φ)
-
-    Derived from first principles the wave equation of an electron in the
-    hydrogen atom.
-
-        - Start with Dirac Notation
-        - Replace with known general functions
-        - transform to eigenvalue equation in position space
-        - replace pieces with function F and find derivatives
-            F = ∑ Cₖρᵏ
-
-        pic 3 first take had factors that needed to be fixed
-
-        - Led to the Laguerre Polynomials
-            These are infinite, though, so
-        - Force Laguerre Polynomial solutions to truncate
-
-Derivation
-━━━━━━━━━━
+Derivation of Electron States in Hydrogen Atom
+━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━
     The Hydrogen atom follows the central potential development from the
     previous lecture. 
     
-    ❬r,θ,φ❙Ψ❭ = R(r)ₑ,ₗ yₗ﹐ₘ(θ,φ)
+    ❬r,θ,φ❙Ψ❭ = R(r)ₑ,ₗ yₗₘ(θ,φ)
 
-    Rₑ﹐ₗ(r) = Uₑ﹐ₗ(r) /r
+        Rₑₗ(r) = Uₑₗ(r) /r
                     
-    ρ ≡ √⎛8͟m͟ │E│⎞r
-         ⎝ ħ²   ⎠ 
+        ρ ≡ √⎛8͟m͟ │E│⎞r
+             ⎝ ħ²   ⎠ 
     
-    λ = Z͟e͟² √⎛_͟m͟_͟ ⎞
-         ħ   ⎝2│E│⎠
+        λ = Z͟e͟² √⎛_͟m͟_͟ ⎞
+             ħ   ⎝2│E│⎠
 
     d͟²͟ U - 1͟ l(l+1) + ⎛λ͟ - 1͟⎞U = 0
     dρ²    ρ²         ⎝ρ   4⎠
@@ -56,31 +30,35 @@ Derivation
             U(ρ) = A exp(-ρ/2) + B exp(ρ/2) = A exp(-ρ/2)
                                 (ρ→∞, B=0)
 
-            U(ρ) = ρˡ⁺¹ exp(-ρ/2) Fₑ﹐ₗ(ρ)
+            U(ρ) = ρˡ⁺¹ exp(-ρ/2) Fₑₗ(ρ)
 
-            d͟ U = (l+1)ρˡ exp(-ρ/2) Fₑ﹐ₗ(ρ)
+            d͟ U =  (l+1)ρˡ exp(-ρ/2) Fₑₗ(ρ)
             dρ
-                        -½ ρˡ⁺¹ exp(-ρ/2) Fₑ﹐ₗ(ρ)
 
-                            + ρˡ⁺¹ exp(-ρ/2) d͟ Fₑ﹐ₗ(ρ)
-                                             dρ
+                   -½ ρˡ⁺¹ exp(-ρ/2) Fₑₗ(ρ)
+
+
+                   + ρˡ⁺¹ exp(-ρ/2) d͟ Fₑₗ(ρ)
+                                    dρ
 
             d͟ U = ⎛l͟+͟1͟ - 1͟ ⎞ U + ρˡ⁺¹ exp(-ρ/2) d͟ F(ρ)
             dρ    ⎝ ρ    2 ⎠                    dρ
 
 
 
-            d͟²͟ U = -(l͟+͟1͟) U + ⎛l͟+͟1͟ - 1͟⎞² U 
-            dρ²       ρ²      ⎝ ρ    2⎠
-                        + 2⎛l͟+͟1͟ - 1͟⎞ρˡ⁺¹ exp(-ρ/2) d͟F͟
-                           ⎝ ρ    2⎠               dρ
-                               + ρˡ⁺¹ exp(-ρ/2) d͟²͟ F(ρ)
-                                                dρ²
+            d͟²͟ U =  -(l͟+͟1͟) U + ⎛l͟+͟1͟ - 1͟⎞² U 
+            dρ²        ρ²      ⎝ ρ    2⎠
+
+                    + 2⎛l͟+͟1͟ - 1͟⎞ρˡ⁺¹ exp(-ρ/2) d͟F͟
+                       ⎝ ρ    2⎠               dρ
+
+                    + ρˡ⁺¹ exp(-ρ/2) d͟²͟ F(ρ)
+                                     dρ²
 
 
 
 
-    ⎡−͟ħ͟² d² +  l͟  (l+1)ħ² - Z͟e͟²⎤Uₑ﹐ₗ(r) = E Uₑ﹐ₗ(r)
+    ⎡−͟ħ͟² d² +  l͟  (l+1)ħ² - Z͟e͟²⎤Uₑₗ(r) = E Uₑₗ(r)
     ⎣2m  dr²  2mr²           r ⎦
 
     ρ = √⎛8͟m͟ │E│⎞r

lecture_notes/3-28/overview

@@ -1 +1,32 @@
 Developed Theory for The Hydrogen Atom
+
+                      Theory of the Hydrogen Atom
+━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━
+    For any hydrogenic ion with nuclear charge Z
+        V(𝐫) = -Ze²/r
+
+    V(𝐫) = -e²/r
+
+    Eigenfunctions in spherical coordinates:
+
+        Ψₑₗₘ(r,θ,φ) = Rₑₗ(r) Yₗₘ(θ,φ) = Uₑₗ/r Yₗₘ(θ,φ)
+
+    Derived from first principles the wave equation of an electron
+    in the hydrogen atom.
+
+        - Start with Dirac Notation
+
+        - transform to eigenvalue equation in position space
+            Replace with known special quantities
+
+        - replace pieces with function F and find derivatives
+            F = ∑ Cₖρᵏ
+
+        pic 3 first take had factors that needed to be fixed
+
+        - Led to the Laguerre Polynomials
+            These are infinite, though, so
+            
+        - Force Laguerre Polynomial solutions to truncate
+
+

lecture_notes/3-28/overview.txt

@@ -0,0 +1,27 @@
+Developed Theory for The Hydrogen Atom
+
+    Theory of the Hydrogen Atom
+    ━━━━━━━━━━━━━━━━━━━━━━━━━━━━
+        For any hydrogenic ion with nuclear charge Z
+            V(𝐫) = -Ze²/r
+    
+        V(𝐫) = -e²/r
+    
+        Eigenfunctions in spherical coordinates:
+    
+            Ψₑₗₘ(r,θ,φ) = Rₑₗ(r) Yₗₘ(θ,φ) = Uₑₗ/r Yₗₘ(θ,φ)
+    
+        Derived from first principles the wave equation of an electron in the
+        hydrogen atom.
+    
+            - Start with Dirac Notation
+            - Replace with known general functions
+            - transform to eigenvalue equation in position space
+            - replace pieces with function F and find derivatives
+                F = ∑ Cₖρᵏ
+    
+            pic 3 first take had factors that needed to be fixed
+    
+            - Led to the Laguerre Polynomials
+                These are infinite, though, so
+            - Force Laguerre Polynomial solutions to truncate