finished prob 11 chap 9

solutions/chap9/prob11

@@ -41,17 +41,15 @@ The expecation values ❬x̂❭ and ❬p̂❭ are of interest.
     ❬x̂❭ = ⅕⎛exp⎛ι͟E͟₀͟t⎞❬0❙ + 2 exp⎛ι⎛E͟₁͟t - _͟π͟⎞⎞❬1❙⎞ x̂ ⎛exp⎛-ι͟E͟₀͟t⎞❙0❭ + 2 exp⎛ι⎛_͟π͟ - E͟₁͟t⎞⎞❙1❭⎞.
            ⎝   ⎝ ħ  ⎠           ⎝ ⎝ ħ     2⎠⎠   ⎠   ⎝   ⎝  ħ  ⎠           ⎝ ⎝ 2    ħ ⎠⎠   ⎠ 
    
+                             
+                             
+
 The matrix representations of the position operator x̂ and momentum operator p̂ have been developed from the definition of the increment/decrement operators. The matrix elements may be ascertained by inspection.
                      
-    x̂ ≐               p̂ ≐
-        ⎛  0 √1 ⎞         ⎛  0  -ι√1 ⎞
-        ⎝ √1  0 ⎠,        ⎝ ι√1   0  ⎠.
-
-    ❬0❙x̂❙0❭ = x₀₀ = ❬1❙x̂❙1❭ = x₁₁ = 0.
-
-    ❬0❙x̂❙1❭ = x₀₁ = ❬1❙x̂❙0❭ = x₁₀ = √1 = 1.
-
+   x̂ ≐ √⎛_͟ħ͟ ⎞⎛  0 √1 ⎞  p̂ ≐ √⎛͟ħ͟m͟ω͟⎞⎛  0  -ι√1 ⎞
+        ⎝2mω⎠⎝ √1  0 ⎠,      ⎝ 2 ⎠⎝ ι√1   0  ⎠.
 
+ω is a characteristic parameter of the system. It is related to the steepness of the parabolic potential curve. m is the particle's mass.
 
     ❬x̂❭ = ⅕⎛exp⎛ι͟E͟₀͟t⎞ exp⎛-ι͟E͟₀͟t⎞❬0❙x̂❙0❭ +                        ⎞ 
            ⎜   ⎝ ħ  ⎠    ⎝  ħ  ⎠                                 ⎟ 
@@ -65,19 +63,18 @@ The matrix representations of the position operator x̂ and momentum operator p
            ⎜         2 exp⎛ι⎛E͟₁͟t - _͟π͟⎞⎞2 exp⎛ι⎛_͟π͟ - E͟₁͟t⎞⎞❬1❙x̂❙1❭⎞⎟ 
            ⎝              ⎝ ⎝ ħ     2⎠⎠     ⎝ ⎝ 2    ħ ⎠⎠       ⎠⎠.
 
+    ❬0❙x̂❙0❭ = x₀₀ = ❬1❙x̂❙1❭ = x₁₁ = 0.
+
+    ❬0❙x̂❙1❭ = x₀₁ = ❬1❙x̂❙0❭ = x₁₀ = √⎛_͟ħ͟ ⎞.
+                                     ⎝2mω⎠
+                           
 Substituting the matrix elements:
 
-    ❬x̂❭ = ⅖⎛exp⎛ι͟E͟₀͟t + ι⎛_͟π͟ - E͟₁͟t⎞⎞ +   ⎞ = ⅖⎛exp⎛ι⎛⎛E͟₀͟−͟E͟₁͟⎞t + _͟π͟⎞⎞ + ⎞  
-           ⎜   ⎝  ħ     ⎝ 2    ħ ⎠⎠     ⎟    ⎜   ⎝ ⎝⎝  ħ  ⎠     2⎠⎠   ⎟  
-           ⎜                            ⎟    ⎜                        ⎟  
-           ⎜    exp⎛-ι͟E͟₀͟t + ι⎛E͟₁͟t - _͟π͟⎞⎞⎟    ⎜    exp⎛ι͟⎛⎛E͟₁͟−͟E͟₀͟⎞ - _͟π͟⎞⎞⎟  
-           ⎝       ⎝   ħ     ⎝ ħ     2⎠⎠⎠    ⎝       ⎝ ⎝⎝  ħ  ⎠    2⎠⎠⎠;
+    ❬x̂❭ = ⅕ √⎛͟2͟͟ħ͟⎞⎛ exp⎛ι͟E͟₀͟t + ι⎛_͟π͟ - E͟₁͟t⎞⎞ + exp⎛-ι͟E͟₀͟t + ι⎛E͟₁͟t - _͟π͟⎞⎞ ⎞ 
+             ⎝mω⎠⎝    ⎝  ħ     ⎝ 2    ħ ⎠⎠      ⎝   ħ     ⎝ ħ     2⎠⎠ ⎠; 
            
-    ❬x̂❭ = ⅖⎛ exp⎛ι͟E͟₀͟t + ι⎛_͟π͟ - E͟₁͟t⎞⎞ + exp⎛-ι͟E͟₀͟t + ι⎛E͟₁͟t - _͟π͟⎞⎞ ⎞ 
-           ⎝    ⎝  ħ     ⎝ 2    ħ ⎠⎠      ⎝   ħ     ⎝ ħ     2⎠⎠ ⎠; 
-           
-    ❬x̂❭ = ⅖⎛ exp⎛ι⎛⎛E͟₀͟−͟E͟₁͟⎞t + _͟π͟⎞⎞ + exp⎛ι⎛⎛E͟₁͟−͟E͟₀͟⎞t - _͟π͟⎞⎞ ⎞
-           ⎝    ⎝ ⎝⎝  ħ  ⎠     2⎠⎠      ⎝ ⎝⎝  ħ  ⎠     2⎠⎠ ⎠.
+    ❬x̂❭ = ⅕ √⎛͟2͟͟ħ͟⎞⎛ exp⎛ι⎛⎛E͟₀͟−͟E͟₁͟⎞t + _͟π͟⎞⎞ + exp⎛ι⎛⎛E͟₁͟−͟E͟₀͟⎞t - _͟π͟⎞⎞ ⎞
+             ⎝mω⎠⎝    ⎝ ⎝⎝  ħ  ⎠     2⎠⎠      ⎝ ⎝⎝  ħ  ⎠     2⎠⎠ ⎠.
 
 For the Harmonic Oscillator, Eₙ = ħω(n + ½).
 
@@ -85,15 +82,75 @@ For the Harmonic Oscillator, Eₙ = ħω(n + ½).
         E₀−E₁ = ħω(0 - 1) = -ħω;
         E₁−E₀ = ħω.             
 
-    ❬x̂❭ = ⅖⎛ exp⎛ι⎛⎛−͟ħ͟ω͟⎞t + _͟π͟⎞⎞ + exp⎛ι⎛⎛ħ͟ω͟⎞t - _͟π͟⎞⎞ ⎞  
-           ⎝    ⎝ ⎝⎝ ħ ⎠     2⎠⎠      ⎝ ⎝⎝ ħ⎠     2⎠⎠ ⎠. 
+    ❬x̂❭ = ⅕ √⎛͟2͟͟ħ͟⎞⎛ exp⎛ι⎛⎛−͟ħ͟ω͟⎞t + _͟π͟⎞⎞ + exp⎛ι⎛⎛ħ͟ω͟⎞t - _͟π͟⎞⎞ ⎞  
+             ⎝mω⎠⎝    ⎝ ⎝⎝ ħ ⎠     2⎠⎠      ⎝ ⎝⎝ ħ⎠     2⎠⎠ ⎠. 
+
+    ❬x̂❭ = ⅕ √⎛͟2͟͟ħ͟⎞ exp(ι͟π͟)⎛ exp⎛ι⎛−͟ħ͟ω͟⎞t⎞ + exp⎛ι⎛⎛ħ͟ω͟⎞t - π⎞⎞ ⎞  
+             ⎝mω⎠      2 ⎝    ⎝ ⎝ ħ ⎠ ⎠      ⎝ ⎝⎝ ħ⎠     ⎠⎠ ⎠. 
+
+    ❬x̂❭ = ⅕ √⎛͟2͟͟ħ͟⎞ exp(ι͟π͟)⎛ exp⎛ι⎛−͟ħ͟ω͟⎞t⎞ - exp⎛ι⎛ħ͟ω͟⎞t⎞ ⎞  
+             ⎝mω⎠      2 ⎝    ⎝ ⎝ ħ ⎠ ⎠      ⎝ ⎝ ħ⎠ ⎠ ⎠. 
+
+This is a sine function.
+
+    ❬x̂❭ = ⅕ √⎛͟2͟͟ħ͟⎞ exp(ι͟π͟) 2ι sin⎛−͟ħ͟ω͟t͟⎞ = ⅕ √⎛͟8͟ħ͟⎞ ι² -sin(ωt).
+             ⎝mω⎠      2        ⎝  ħ ⎠      ⎝mω⎠          
 
 (𝐜,x̂)
-    ❬x̂❭ = ⅖⎛  ι(-ωt + _͟π͟)⎞ + exp⎛ι⎛ωt - _͟π͟⎞⎞ ⎞
-           ⎝ e         2 ⎠      ⎝ ⎝      2⎠⎠ ⎠.
+    ❬x̂❭ = ⅕ √⎛͟8͟ħ͟⎞ sin(ωt).
+             ⎝mω⎠          
 
-The expectation value progresses with the time parameter t. ω is a characteristic parameter of the system. It is related to the steepness of the parabolic potential curve.
+The expectation value progresses periodically with the time parameter t. 
+
+
+A very similar argument can be made for the momentum operator.
+
+    ❬p̂❭ = ⅕⎛exp⎛ι͟E͟₀͟t⎞ exp⎛-ι͟E͟₀͟t⎞❬0❙p̂❙0❭ +                        ⎞ 
+           ⎜   ⎝ ħ  ⎠    ⎝  ħ  ⎠                                 ⎟ 
+           ⎜                                                     ⎟ 
+           ⎜   exp⎛ι͟E͟₀͟t⎞ 2 exp⎛ι⎛_͟π͟ - E͟₁͟t⎞⎞ ❬0❙p̂❙1❭ +            ⎟ 
+           ⎜      ⎝  ħ ⎠      ⎝ ⎝ 2    ħ ⎠⎠                      ⎟ 
+           ⎜                                                     ⎟ 
+           ⎜      exp⎛-ι͟E͟₀͟t⎞ 2 exp⎛ι⎛E͟₁͟t - _͟π͟⎞⎞ ❬1❙p̂❙0❭ +        ⎟ 
+           ⎜         ⎝   ħ ⎠      ⎝ ⎝ ħ     2⎠⎠                  ⎟ 
+           ⎜                                                     ⎟ 
+           ⎜         2 exp⎛ι⎛E͟₁͟t - _͟π͟⎞⎞2 exp⎛ι⎛_͟π͟ - E͟₁͟t⎞⎞❬1❙p̂❙1❭⎞⎟ 
+           ⎝              ⎝ ⎝ ħ     2⎠⎠     ⎝ ⎝ 2    ħ ⎠⎠       ⎠⎠.
 
 
 
+    ❬0❙p̂❙0❭ = p₀₀ = ❬1❙p̂❙1❭ = p₁₁ = 0.
+
+    ❬0❙p̂❙1❭ = p₀₁ = -ι√⎛͟ħ͟m͟ω͟⎞.
+                       ⎝ 2 ⎠ 
+
+    ❬1❙p̂❙0❭ = p₁₀ = ι√⎛͟ħ͟m͟ω͟⎞.
+                      ⎝ 2 ⎠ 
+
+    ❬p̂❭ = ⅖ ι √⎛͟ħ͟m͟ω͟⎞⎛-exp⎛ι͟E͟₀͟t + ι⎛_͟π͟ - E͟₁͟t⎞⎞ + exp⎛-ι͟E͟₀͟t + ι⎛E͟₁͟t - _͟π͟⎞⎞
+               ⎝ 2 ⎠⎜    ⎝  ħ     ⎝ 2    ħ ⎠⎠      ⎝   ħ     ⎝ ħ     2⎠⎠.
+    
+    ❬p̂❭ = ⅖ ι √⎛͟ħ͟m͟ω͟⎞⎛exp⎛ι⎛⎛−͟ħ͟ω͟⎞t - _͟π͟⎞⎞ + exp⎛ι⎛⎛ħ͟ω͟⎞t - _͟π͟⎞⎞⎞  
+               ⎝ 2 ⎠⎝   ⎝ ⎝⎝  ħ⎠     2⎠⎠      ⎝ ⎝⎝ ħ⎠     2⎠⎠⎠.
+    
+    ❬p̂❭ = ⅖ ι √⎛͟ħ͟m͟ω͟⎞ exp⎛−͟ι͟π͟⎞ ⎛ exp⎛ι⎛−͟ħ͟ω͟⎞t⎞ + exp⎛ι⎛ħ͟ω͟⎞t⎞ ⎞  
+               ⎝ 2 ⎠    ⎝ 2 ⎠ ⎝    ⎝ ⎝  ħ⎠ ⎠      ⎝ ⎝ ħ⎠ ⎠ ⎠.
+
+This is a cosine.
+
+(𝐜,p̂)
+
+    ❬p̂❭ = ⅘ ι √⎛͟ħ͟m͟ω͟⎞ exp⎛−͟ι͟π͟⎞ cos(ωt) = ⅕ √(8ħmω) cos(ωt).
+               ⎝ 2 ⎠    ⎝ 2 ⎠                       
+
+
+Ehrenfest's theorem states
+
+    ❬p̂❭ = m d͟❬͟x̂͟❭͟.
+            dt
+
+    m d͟❬͟x̂͟❭͟ = ⅕ mω √⎛͟8͟ħ͟⎞ cos(ωt) = ⅕ √(8mωħ) cos(ωt) = ❬p̂❭. 
+      dt           ⎝mω⎠             
+                 
+So, the theorem holds for this case.