working on exam 1, still

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
-Cpmpatible Observables occur when commutator is equal to 0.
+Computed Derivative d/dt <Â>:
 
-[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/prob2

@@ -1,6 +1,23 @@
 Show that ͟d͟〈͟p͟〉͟ = -〳͟d͟V͟(͟x͟)͟〵 when a particle is subjected to a potential 〈V(x)〉.
-           dt     〵  dx 〳
-
+           dt     〵dx   〳
+
+
+The time derivative of the expectation value of the momentum is a known quantity, from
+Time Dependence of Expectation Value of General Momentum Operator:
+
+d〈p〉 = 1.
+dt     ιħ
+
+The problem is therefore reduced to finding whether -/dV(x)\ reduces to 1.
+                                                     \dx   /            ιħ
+
+-/dV(x)\ = -〈Ψ| d V(x) |Ψ〉. 
+ \dx   /        dx        
+
+ Viewing the expression in this form reveals a relationship between the space derivative and the operators V(x) and |Ψ〉. The chain rule allows this derivative to be computed.
+
+-〈Ψ| d V(x) |Ψ〉 = -〈Ψ| ⎛d V(x)|Ψ> + d |Ψ> V(x)⎞. 
+     dx                ⎝dx          dx        ⎠
 
 
      

solutions/exam1/prob3

@@ -4,3 +4,5 @@
 
 a) 
 
+🔋
+