adding derivations for central potential and hydrogen atom

lecture_notes/3-25/central potential states

@@ -0,0 +1,131 @@
+Bound states of a central potential
+━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━
+
+    For any central potential V(r) = V(│r│) the eigenfunctions of H can be
+    separated as
+    
+        ❬r❙E,l,mₗ❭ = Rₑ﹐ₗ(r) Yₗ﹐ₘ(θ,φ)
+    
+    (pic) The radial S.E. is 
+    
+        ⎡−͟ħ͟²⎛ ∂͟²͟  +  2͟ ∂͟  ⎞ + l͟(l͟+͟1͟)ħ͟² + V(│r│) ⎤ Rₑ﹐ₗ(r) = E Rₑ﹐ₗ(r)
+        ⎣2m ⎝ ∂r²    r ∂r ⎠    2 m r²           ⎦
+
+    (pic) Developed radial schrodinger equation using U(r) replacement
+
+        Rₑ﹐ₗ(r) = Uₑ﹐ₗ(r) /r
+
+        ⎡−͟ħ͟²⎛ ∂͟²͟  +  2͟ ∂͟  ⎞⎤ U͟ = E ∂͟²͟ U
+        ⎣2m ⎝ ∂r²    r ∂r ⎠⎦ r     ∂r²
+
+            ∂͟  U͟ = 1͟ ∂͟ U - U͟ 
+            ∂r r   r ∂r    r²
+    
+            ∂͟²͟  U͟ = 1͟ ∂͟²͟ U + 2͟U͟ - 2͟ ∂͟ U
+            ∂r² r   r ∂r²    r³   r²∂r
+
+        ⎡−͟ħ͟²⎛ ∂͟²͟  U͟ +  2͟ ∂͟  U͟ ⎞⎤ = E ∂͟²͟ U
+        ⎣2m ⎝ ∂r² r    r ∂r r ⎠⎦     ∂r²
+────────────────────────────────────────────────────────────────────────────────
+        ⎡−͟ħ͟²⎛ 1͟ ∂͟²͟ U + 2͟U͟ - 2͟ ∂͟ U + 2͟ 1͟ ∂͟ U - 2͟ U͟  ⎞⎤ 
+        ⎣2m ⎝ r ∂r²    r³   r²∂r    r r ∂r    r r² ⎠⎦ 
+
+                            = E 1͟ ∂͟²͟ U + E 2͟U͟ - E 2͟ ∂͟U
+                                r ∂r²      r³     r²∂r    
+────────────────────────────────────────────────────────────────────────────────
+        ⎡−͟ħ͟²⎛ 1͟ ∂͟²͟  + 2͟ - 2͟ ∂͟  + 2͟  ∂͟  - 2͟ ⎞U⎤ 
+        ⎣2m ⎝ r ∂r²   r³  r²∂r   r² ∂r   r³⎠ ⎦ 
+
+                            = E ⎛1͟ ∂͟²͟  + 2͟ - 2͟ ∂ ⎞U
+                                ⎝r ∂r²   r³  r²∂r⎠
+────────────────────────────────────────────────────────────────────────────────
+        ⎡⎛−͟ħ͟² 1͟ ∂͟²͟  + −͟ħ͟² 2͟ - −͟ħ͟² 2͟ ∂͟  + −͟ħ͟² 2͟  ∂͟  - −͟ħ͟² 2͟ ⎞U⎤ 
+        ⎣⎝2m  r ∂r²   2m  r³  2m  r²∂r   2m  r² ∂r   2m  r³⎠ ⎦ 
+
+                            = E ⎛1͟ ∂͟²͟  + 2͟ - 2͟ ∂ ⎞U
+                                ⎝r ∂r²   r³  r²∂r⎠
+────────────────────────────────────────────────────────────────────────────────
+        ⎡⎛−͟ħ͟² 1͟ ∂͟²͟  - −͟ħ͟² 2͟ ∂͟  + −͟ħ͟² 2͟  ∂͟  + −͟ħ͟² 2͟  - −͟ħ͟² 2͟ ⎞U⎤ 
+        ⎣⎝2m  r ∂r²   2m  r²∂r   2m  r² ∂r   2m  r³   2m  r³⎠ ⎦ 
+
+                            = E ⎛1͟ ∂͟²͟  + 2͟ - 2͟ ∂ ⎞U
+                                ⎝r ∂r²   r³  r²∂r⎠
+────────────────────────────────────────────────────────────────────────────────
+                                ↓
+            Not sure this makes sense, but the final result is
+                                ↓
+
+        ⎡−͟ħ͟²∂͟²͟  -  ħ͟² l(l+1) + V(│r│)⎤ Uₑ﹐ₗ(r) = E Uₑ﹐ₗ(r)
+        ⎣2m ∂r²   2mr²               ⎦           
+
+    
+    Normalization Condition
+
+        ∞                 ∞
+        ∫ │Rₑ﹐ₗ│² r² dr = ∫ │Uₑ﹐ₗ│² dr
+        0                 0
+
+
+    If V(r) is not more singular at the origin than 1/r^2 then the SE has power
+    series solutions.
+
+    Thus for small r we take U(r) → rˢ
+
+    (pic) substitute U(r) = rˢ into S.E.
+
+        Uₑ﹐ₗ(r) ≈ rˢ
+
+        ⎡−͟ħ͟² ∂͟²͟  +  ħ͟² l(l+1) + V(│r│)⎤ rˢ = E rˢ
+        ⎣2m  ∂r²   2mr²               ⎦           
+
+        −͟ħ͟² s(s-1)∂͟ rˢ⁻² + l͟ (l+1)ħ²rˢ⁻² + V(│r│)rˢ = E rˢ
+        2m        ∂r       2m                              
+
+        −͟ħ͟² ⎛s(s-1)∂͟  + l͟ (l+1)ħ²⎞rˢ⁻² + V(│r│)rˢ = E rˢ
+        2m  ⎝      ∂r   2m       ⎠                                   
+
+
+        −͟ħ͟² ⎛s(s-1)∂͟  + l͟ (l+1)ħ²⎞rˢ⁻² + V(│r│)rˢ = E rˢ
+        2m  ⎝      ∂r   2m       ⎠                      
+
+        −͟ħ͟² ⎛s(s-1)∂͟  + l͟ (l+1)ħ²⎞ + V(│r│)r² = E r²
+        2m  ⎝      ∂r   2m       ⎠                      
+
+        
+    For r → 0,
+        r² → 0,
+        V(r) r² → 0.
+            ⇓
+        s(s-1) + l(l+1) = 0
+            ⇓
+        s = l+1 or s = -l
+    
+    If s = -l, the normalization conditions
+                                        
+         ∞                             │∞ 
+        ∫ r⁻²ˡ dr = 1/(2l-1) 1/(r²ˡ⁻¹) │  → diverges
+         0                             │0
+    
+
+    So, for small r,     
+    
+        Uₑ﹐ₗ(r) → (r→0) → rˡ⁺¹;
+    
+        Rₑ﹐ₗ(r) → (r→0) → rˡ.
+    
+Eigenfunctions
+━━━━━━━━━━━━━━
+
+    Ψₑ﹐ₗ﹐ₘ(r,θ,φ) = Rₑ﹐ₗ(r) Yₗ﹐ₘ(θ,φ) = Uₑ﹐ₗ /r Yₗ﹐ₘ(θ,φ)
+
+        ⇒ d͟²͟ U - 1͟ l(l+1) + ⎛λ͟ - 1͟⎞U = 0
+          dρ²    ρ²         ⎝ρ   4⎠
+
+    ⎡−͟ħ͟² d² +  l͟  (l+1)ħ² - Z͟e͟²⎤Uₑ﹐ₗ(r) = E Uₑ﹐ₗ(r)
+    ⎣2m  dr²  2mr²           r ⎦
+
+    ρ = √⎛8͟m͟ │E│⎞r
+         ⎝ ħ²   ⎠
+
+    λ = Z͟e͟² √⎛_͟m͟_͟ ⎞
+         ħ   ⎝2│E│⎠

lecture_notes/3-25/hydrogen atom

@@ -0,0 +1 @@
+../3-28/hydrogen atom

lecture_notes/3-25/overview

@@ -1,65 +1,10 @@
-Bound states of a central potential
+Computed the bound states of a central potential with eigenfunctions of H
-━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━
 
+    ❬r❙E,l,mₗ❭ = Rₑ﹐ₗ(r) Yₗ﹐ₘ(θ,ϕ).
 
-For any central potential V(r) = V(│r│) the eigenfunctions of H can be separated as
+    (pic) Developed radial schrodinger equation using U(r) replacement
-
-    ❬r❙E,l,mₗ❭ = Rₑ﹐ₗ(r) Yₗ﹐ₘ(θ,ϕ)
-
-The radial S.E. is 
-
-    ⎡−͟ħ͟² ⎛ ∂͟²͟  +  2͟∂͟  ⎞ + l͟(l͟+͟1͟)ħ͟² + V(│r│) ⎤ Rₑ﹐ₗ(r) = E Rₑ﹐ₗ(r)
-    ⎣2m  ⎝ ∂r²    r∂r ⎠    2 m r²           ⎦
-
-
-    Rₑ﹐ₗ(r) = U͟ₑ͟﹐͟ₗ͟(r)
-            r
-
-            (pic) ...
-
-(pic) Developed radial schrodinger equation  using U(r) replacement
-
-    - Developed normalization condition
-
-If V(r) is not more singular at the origin than 1/r^2 then the SE has power series solutions.
-
-    Thus for small r we take U(r) → rˢ
-
-(pic) substitute U(r) = rˢ into S.E.
-
-    -ħ²/2m [(s(s-1) + l(l+1)] + V r² = E r²
-    
-    For r → 0
-        r² → 0
-        V(r) r² → 0
-    
-    ⇒ s(s-1) + l(l+1) = 0
-        ⇒ s = l+1 or s = -l
-
-    If s = -l, the normalization conditions
-                                        
-         ∞                             │∞ 
-        ∫ r⁻²ˡ dr = 1/(2l-1) 1/(r²ˡ⁻¹) │  → diverges
-         0                             │0
-
-
-    Uₑ﹐ₗ(r) → (r→0) → rˡ⁺¹
-
-    Rₑ﹐ₗ(r) → (r→0) → rˡ
-
-
-The Hydrogen Atom
-━━━━━━━━━━━━━━━━━
-
-    V(r) = -e²/r
-
-    For a hydrogenic ion with nuclear charge Z
-
-    V(r) = -Ze²/r
-
-    Eigenfunctions:
-
-        Ψₑ﹐ₗ﹐ₘ(r,θ,φ) = Rₑ﹐ₗ(r) Yₗ﹐ₘ(θ,φ) = Uₑ﹐ₗ/r Yₗ﹐ₘ(θ,φ)
 
+    If V(r) is not more singular at the origin than 1/r^2 then the SE has power series solutions.
 
+Began the Hydrogen Atom
 

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

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

lecture_notes/3-28/hydrogen atom

@@ -1,9 +1,18 @@
 Theory of the Hydrogen Atom
 ━━━━━━━━━━━━━━━━━━━━━━━━━━━━
+    V(𝐫) = -e²/r
 
-    Derived from first principles the wave equation of an electron in the hydrogen atom.
+    For any hydrogenic ion with nuclear charge Z
+        V(𝐫) = -Ze²/r
 
-        -Start with Dirac Notation
+    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
@@ -15,4 +24,68 @@ Theory of the Hydrogen Atom
             These are infinite, though, so
         - Force Laguerre Polynomial solutions to truncate
 
-        - 
+Derivation
+━━━━━━━━━━
+    The Hydrogen atom follows the central potential development from the
+    previous lecture. 
+    
+    ❬r,θ,φ❙Ψ❭ = R(r)ₑ,ₗ yₗ﹐ₘ(θ,φ)
+
+    Rₑ﹐ₗ(r) = Uₑ﹐ₗ(r) /r
+                
+    ρ ≡ √⎛8͟m͟ │E│⎞r
+         ⎝ ħ²   ⎠ 
+
+    λ = Z͟e͟² √⎛_͟m͟_͟ ⎞
+         ħ   ⎝2│E│⎠
+
+    d͟²͟ U - 1͟ l(l+1) + ⎛λ͟ - 1͟⎞U = 0
+    dρ²    ρ²         ⎝ρ   4⎠
+    
+    
+        as ρ→0, U→ρˡ⁺¹
+
+        for ρ→∞,
+
+            d͟²͟ U - 1͟U = 0
+            dρ²    4     
+
+            d͟²͟ U - 1͟U = 0
+            dρ²    4
+
+            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)
+
+            d͟ U = (l+1)ρˡ exp(-ρ/2) Fₑ﹐ₗ(ρ)
+            dρ
+                        -½ ρˡ⁺¹ exp(-ρ/2) Fₑ﹐ₗ(ρ)
+
+                            + ρˡ⁺¹ exp(-ρ/2) d͟ Fₑ﹐ₗ(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
+
+
+
+
+    ⎡−͟ħ͟² d² +  l͟  (l+1)ħ² - Z͟e͟²⎤Uₑ﹐ₗ(r) = E Uₑ﹐ₗ(r)
+    ⎣2m  dr²  2mr²           r ⎦
+
+    ρ = √⎛8͟m͟ │E│⎞r
+         ⎝ ħ²   ⎠ 
+
+    ⎛⎞
+    ⎝⎠
+