new lecture notes

lecture_notes/4-11/overview

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+                            Review of Exam 2
+━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━
+
+V = ⎧ 1/2 k r²  (x>0)
+    ⎨
+    ⎩ ∞         (x<0)
+
+    Gives the set L𝓍,L²,H due to x-symmetry
+
+(pic) schrodinger equations for radial and angular components
+
+(pic x2) working through asymptotic behaviours of diffEQ
+
+followed asymptotic approach and then set up a series polynomial solution
+for the final F(r)
+
+(pic) still working through derivatives of the polynomial solution
+
+(pic) Use the "series shift" to combine terms. 
+
+(pic) basically just continuing to prepare terms (2/r dR/dr in this case)
+to plug back into the main differential equation
+
+the final answer gives a relationship between the coeffiecients cᵢ.
+

lecture_notes/4-6/lectures

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+reestablish fundamental brakets
+
+    spin Z operators
+
+    raising/lowering operators (lowering erased before pic)
+
+    commutations relations (erased before pic)
+
+Now setup a two-spin system
+
+    (pic) Look at general operator expressions ?? in different spaces ??
+
+    Take direct product of A and B
+
+(pic) Developed 𝐒 using raising lowering operators
+
+    (pic: last y should be an x) Proved this.
+
+    

lecture_notes/4-6/overview

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+reestablish fundamental brakets
+
+    spin Z operators
+
+    raising/lowering operators (lowering erased before pic)
+
+    commutations relations (erased before pic)
+
+Now setup a two-spin system
+
+    (pic) Look at general operator expressions ?? in different spaces ??
+
+    Take direct product of A and B
+
+(pic) Developed 𝐒 using raising lowering operators
+
+    (pic: last y should be an x) Proved this.
+
+
+(pic x2) Still missing a 1/2 somewhere! But moving on to see proper solutions of direct product.
+
+    Eigenvalues are
+        
+        𝐒² = ħ² S(S+1) = ⎧ 2ħ²
+                         ⎨ 
+                         ⎩ 0ħ²
+    
+        𝐒 = ⎧ 1  (3 eigenstates)
+            ⎨
+            ⎩ 0  (1 eigenstate)
+    
+    (pic) Plugging back to direct product matrix
+
+(pic) FOUND PROBLEM of -1/2 from the up/down raising/lowering operator interactions
+
+(pic) finished diagonalizing operator product
+
+
+