Merge branch 'master' of https://github.com/othocaes/phy-4600

lecture_notes/2-12/Time Evolution of a Free Electron

@@ -0,0 +1,10 @@
+Ĥ = p̂²
+    2mₑ
+
+                      ∞
+|Ψ(t)〉 = exp( -ι Ĥ t) ∫ dp |p〉 〈p|Ψ〉
+                 ħ   -∞
+
+〈x|Ψ(t)〉 = exp( -ι Ĥ t ) dp 〈x|p〉 〈p|Ψ〉 
+                   ħ
+

solutions/exam1/prob1

@@ -30,4 +30,10 @@ Similarly, S𝓍 (S𝓍 + ħ)(S𝓍 - ħ) ≐
 
        ⎛ 0 1 0 ⎞ ⎛ 1 1 0 ⎞ ⎛ -1  1  0 ⎞      ⎛ 1 1 1 ⎞ ⎛ -1  1  0 ⎞      ⎛ 0 1 0 ⎞
     ħ³ ⎜ 1 0 1 ⎟ ⎜ 1 1 1 ⎟ ⎜  1 -1  1 ⎟ = ħ³ ⎜ 1 2 1 ⎟ ⎜  1 -1  1 ⎟ = ħ³ ⎜ 1 0 1 ⎟.
-       ⎝ 0 1 0 ⎠ ⎝ 0 1 1 ⎠ ⎝  0  1 -1 ⎠      ⎝ 1 1 1 ⎠ ⎝  0  1 -1 ⎠      ⎝ 0 1 0 ⎠
+       ⎝ 0 1 0 ⎠ ⎝ 0 1 1 ⎠ ⎝  0  1 -1 ⎠      ⎝ 1 1 1 ⎠ ⎝  0  1 -1 ⎠      ⎝ 0 1 0 ⎠
+
+This expression results in the non-zero matrix
+
+       ⎛ 0 1 0 ⎞
+    ħ³ ⎜ 1 0 1 ⎟.
+       ⎝ 0 1 0 ⎠

solutions/exam1/prob2

@@ -1,7 +1,24 @@
 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) The eigenstates of the Hamiltonian can be determined by 
 
+🔋
+