lecture notes 3-16

lecture_notes/3-14/3d eigenstates

@@ -27,3 +27,35 @@ Understanding a System:
     Ĥ must now include angular momentum
 
     L̂²= (r̂×p̂)(r̂×p̂) = (geometric identity) = r̂²p̂ - (r̂⋅p̂) + ιħr̂⋅p̂
+
+    ❬r❙r̂²p̂²❙Ψ❭ = r² ❬r❙p̂²❙Ψ❭
+
+    r̂² p̂² = L̂² + (r̂⋅p̂)² - ιħ r̂⋅p̂
+
+    ❬r❙p̂²❙Ψ❭ = 1/r² ❬r❙L̂² + (r̂⋅p̂)² - ιħ r̂⋅p̂❙Ψ❭
+
+        ❬r❙L̂²❙Ψ❭
+
+
+        ❬r❙r̂²⋅p̂²❙Ψ❭ = r ⋅ ħ/ι ∇ ❬r❙Ψ❭ = ħ/ι r ∂/∂r ❬r❙Ψ❭
+
+
+        ❬r❙(r̂⋅p̂)²❙Ψ❭ = ❬r❙(r̂⋅p̂)(r̂⋅p̂)❙Ψ❭
+                     = r ħ/ι ∂/∂r ❬r❙r̂⋅p̂❙Ψ❭
+                     = -ħ² r ∂/∂r (r ∂/∂r) ❬r❙Ψ❭
+
+    1/2m ❬r❙p̂²❙Ψ❭ = -ħ/2m r/r² ∂/∂r r ∂/∂r ❬r❙Ψ❭ + 1/(2mr²) ❬r❙L²❙Ψ❭
+                  = -ħ²/2m (∂²/∂r² + 2/r ∂/∂r) ❬r❙Ψ❭ + 1/(2mr²) ❬r❙L̂²❙Ψ❭
+                    |               ↓               |  |        ↓       |
+                              linear energy             rotational energy
+
+    Hamiltonian can now be written
+
+        Ĥ = -ħ²/2m (∂²/∂r² + 2/r ∂/∂r) + L̂²/(2mr²) + V(│r│)
+
+        With the eigenvalue equation
+
+            ❬r❙E,l,mₗ❭ = E❬r❙E,l,mₗ❭
+
+            
+