not sure

lecture_notes/1-27/Constant of Motion

@@ -0,0 +1,10 @@
+Compatible Observables occur when commutator is equal to 0.
+
+[Â,B̂] = ÂB̂ - B̂Â
+
+Constant of Motion
+------------------
+    if d/dt  = 0 and [Ĥ,Â] = 0, where Ĥ is the Hamiltonian operator.
+    then  is a constant of motion
+
+    Thm: Two operators such that [Â,B̂] always have common Eigenstates

lecture_notes/1-27/Expectation Value Time Dependence

@@ -1,13 +1,16 @@
-Computed Derivative (pic) d/dt <Â> (pic)
 
-Discovered/Introduced Commutator
+Computed Derivative d/dt <Â>:
-Cpmpatible Observables occur when commutator is equal to 0.
 
-[Ĥ,Â] = ĤÂ - ÂĤ
+	d <Â> = d 〈Ψ(t)|Â|Ψ(t)〉
+	dt      dt
+	
+	⎛d 〈Ψ(t)⎞ Â|Ψ(t)〉 + 〈Ψ(t)|Â ⎛d Ψ(t)〉⎞ + 〈Ψ(t)|d Â|Ψ(t)〉
+	⎝dt     ⎠                   ⎝dt     ⎠         dt
+						↓
+	ι 〈Ψ(t)|Ĥ Â|Ψ(t)〉 + 〈Ψ(t)|Â Ĥ -ι|Ψ(t)〉 + 〈Ψ(t)|d Â|Ψ(t)〉
+	ħ                              ħ               dt 
+						↓
+	ι 〈Ψ(t)|Ĥ Â - Â Ĥ|Ψ(t)〉 + 〈Ψ(t)|d Â|Ψ(t)〉
+	ħ                               dt 
 
-Constant of Motion
-------------------
-    if d/dt  = 0 and [Ĥ,Â] = 0,
-    then  is a constant of motion
 
-    Thm: Two operators such that [Â,B̂] always have common Eigenstates

solutions/exam1/prob3

@@ -2,5 +2,5 @@
 
     Ĥ = |a〉 δ 〈b| + |b〉 δ 〈a|, with δ a real number.
 
-a) 
+a) The eigenstates of the Hamiltonian can be determined by