working on chap 9 prob 11

solutions/chap9/prob11

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+The harmonic oscillator potential is V(x) = ½kx² = ½mω²x². The hamiltonian is time independent. 
+
+The initial state vector is
+
+     Ψ(t=0) = A⎛❙0❭ + 2exp(ι͟π͟)❙1❭⎞
+               ⎝            2    ⎠
+
+Finding the normalization constant A is simple enough, since
+
+    1 = A² + (2Aexp(ι͟π͟))² = A²(1 + (2exp(ι͟π͟))²).
+                     2                    2  
+
+    A = √⎛  ______͟1͟______  ⎞ = √⎛  ________͟1͟_______  ⎞
+         ⎜ 1 + (2exp(ι͟π͟))² ⎟    ⎜ 1 + 4exp(ι͟π͟ - ι͟π͟ ) ⎟,
+         ⎝            2    ⎠    ⎝           2    2   ⎠
+
+(𝐚)
+    A = √⎛  ____͟1͟____  ⎞ = √⅕.
+         ⎝ 1 + 4exp(0) ⎠    
+                          
+
+The time evolution prescription for a time-independent hamiltonian is
+
+    Ψ(t) = exp(-ι͟E͟ₙ͟t) Ψ(t=0);
+                 ħ            
+
+    Ψ(t) = exp(-ι͟E͟ₙ͟t) √⅕⎛❙0❭ + 2exp(ι͟π͟)❙1❭⎞.
+                 ħ      ⎝            2    ⎠                    
+
+(𝐛)
+    Ψ(t) = √⅕⎛exp⎛-ι͟E͟₀͟t⎞❙0❭ + 2 exp⎛ι⎛_͟π͟ - E͟₁͟t⎞⎞❙1❭⎞.
+             ⎝   ⎝  ħ  ⎠           ⎝ ⎝ 2    ħ ⎠⎠   ⎠  
+
+The expecation values ❬x̂❭ and ❬p̂❭ are of interest.
+
+    ❬x̂❭ = ❬Ψ⃰(t)❙x̂❙Ψ(t)❭.
+
+    Ψ⃰(t) = √⅕⎛exp⎛ι͟E͟₀͟t⎞❬0❙ + 2 exp⎛ι⎛E͟₁͟t - _͟π͟⎞⎞❬1❙⎞.
+             ⎝   ⎝ ħ  ⎠           ⎝ ⎝ ħ     2⎠⎠   ⎠ 
+
+    ❬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❭ = ⅕⎛exp⎛ι͟E͟₀͟t⎞ exp⎛-ι͟E͟₀͟t⎞❬0❙x̂❙0❭ +                        ⎞ 
+           ⎜   ⎝ ħ  ⎠    ⎝  ħ  ⎠                                 ⎟ 
+           ⎜                                                     ⎟ 
+           ⎜   exp⎛ι͟E͟₀͟t⎞ 2 exp⎛ι⎛_͟π͟ - E͟₁͟t⎞⎞ ❬0❙x̂❙1❭ +            ⎟ 
+           ⎜      ⎝  ħ ⎠      ⎝ ⎝ 2    ħ ⎠⎠                      ⎟ 
+           ⎜                                                     ⎟ 
+           ⎜      exp⎛-ι͟E͟₀͟t⎞ 2 exp⎛ι⎛E͟₁͟t - _͟π͟⎞⎞ ❬1❙x̂❙0❭ +        ⎟ 
+           ⎜         ⎝   ħ ⎠      ⎝ ⎝ ħ     2⎠⎠                  ⎟ 
+           ⎜                                                     ⎟ 
+           ⎜         2 exp⎛ι⎛E͟₁͟t - _͟π͟⎞⎞2 exp⎛ι⎛_͟π͟ - E͟₁͟t⎞⎞❬1❙x̂❙1❭⎞⎟ 
+           ⎝              ⎝ ⎝ ħ     2⎠⎠     ⎝ ⎝ 2    ħ ⎠⎠       ⎠⎠.
+                           
+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❭ = 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⎠⎠ ⎠.
+
+For the Harmonic Oscillator, Eₙ = ħω(n + ½).
+
+    Eₙ−Eₙ′ = ħω(n - n′):
+        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⎠⎠ ⎠. 
+
+(𝐜,x̂)
+    ❬x̂❭ = ⅖⎛  ι(-ωt + _͟π͟)⎞ + exp⎛ι⎛ωt - _͟π͟⎞⎞ ⎞
+           ⎝ e         2 ⎠      ⎝ ⎝      2⎠⎠ ⎠.
+
+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.
+
+
+
+