adding to rotation lecture notes

lecture_notes/3-14/rotational invariance

@@ -12,6 +12,7 @@ Rotational Invariance
 
             = [1 - ι/ħ (x p𝓎 - y p𝓍)dϕ] ❙x,y,z❭
 
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         (x p𝓎 - y p𝓍) ≝ L̂𝓏 (angular momentum in z axis)
 
         L̂𝓏 is the z component of L̂ = 𝐫×𝐩̂
@@ -39,7 +40,9 @@ Rotational Invariance
 
             = [L̂𝓏,p̂𝓍²] + [L̂𝓏,p̂𝓎²] + [L̂𝓏,p̂𝓏²]
 
-            = p̂𝓍[L̂𝓏,p̂𝓍] + [L̂𝓏,p̂𝓍]p̂𝓍 + p̂𝓎[L̂𝓏,p̂𝓎] + [L̂𝓏,p̂𝓎]p̂𝓎 + p̂𝓏[L̂𝓏,p̂𝓏] + [L̂𝓏,p̂𝓏]p̂𝓏`
+            = p̂𝓍[L̂𝓏,p̂𝓍] + [L̂𝓏,p̂𝓍]p̂𝓍 
+                + p̂𝓎[L̂𝓏,p̂𝓎] + [L̂𝓏,p̂𝓎]p̂𝓎 
+                    + p̂𝓏[L̂𝓏,p̂𝓏] + [L̂𝓏,p̂𝓏]p̂𝓏  (f.n. 1)
 
             = ιħp̂𝓍p̂𝓎 + ιħp̂𝓎p̂𝓍 - ιħp̂𝓎p̂𝓍 - ιħp̂𝓍p̂𝓎 = 0
 
@@ -56,3 +59,31 @@ Rotational Invariance
 
             [L̂²,Ĥ] = 0, so L̂² is also a constant of motion.
 
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+(f.n. 1)
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+            [L̂𝓏,p̂𝓍²] = [L̂𝓏,p̂𝓍 p̂𝓍] 
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+                = L̂𝓏 p̂𝓍 p̂𝓍 - p̂𝓍 p̂𝓍 L̂𝓏
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+                    = ?
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+            = p̂𝓍[L̂𝓏,p̂𝓍] + [L̂𝓏,p̂𝓍]p̂𝓍 
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+            = (p̂𝓍 L̂𝓏 p̂𝓍 - p̂𝓍 p̂𝓍 L̂𝓏) + (L̂𝓏 p̂𝓍 p̂𝓍 - p̂𝓍 L̂𝓏 p̂𝓍) 
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+            = L̂𝓏 p̂𝓍 p̂𝓍  - p̂𝓍 p̂𝓍 L̂𝓏 
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+        ✓   = [L̂𝓏,p̂𝓍²] = [L̂𝓏,p̂𝓍 p̂𝓍] 
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