working on chap 5 homework

notes/fourier

@@ -0,0 +1,10 @@
+general series
+
+       ∞
+f(x) ~ ∑ cₙ φₙ(x)
+      ⁿ⁼⁰
+
+cₙ = ❬͟f͟,φ͟ₙ͟❭͟ᵥ where v is some modulo weighting
+      ‖φₙ‖ᵥ²
+
+

notes/particles in a box

@@ -47,21 +47,23 @@ Energy eigenstates obey the following properties:
 
     Bra-ket Notation       Wavefunction Notation
 
-    Normalization
+        Normalization
 
                            ⌠∞       
     ❬Eₙ❙Eₙ❭ = 1            ⎮ │φₙ(x)│² dx = 1
                            ⌡-∞           
 
-    Orthogonality
+        Orthogonality
 
                            ⌠∞                    
     ❬Eₙ❙Eₘ❭ = δₙₘ          ⎮ φₙ⃰(x) φₘ(x) dx = δₙₘ
                            ⌡-∞                    
 
-    Completeness
+        Completeness
-                           ⌠∞                    
+
-    ❬Eₙ❙Eₘ❭ = δₙₘ          ⎮ φₙ⃰(x) φₘ(x) dx = δₙₘ
+    ❙Ψ❭ = ∑ cₙ ❙Eₙ❭         Ψ(x) = ∑ cₙ φₙ(x)
-                           ⌡-∞                    
+          ⁿ                        ⁿ
+
+
 
 

solutions/chap5/prob2

@@ -1,17 +1,12 @@
-A particle in an infinite square well has an initial state vector
+A particle in an infinite square well has an initial state vector, with A a real number and ι the imaginary unit,
 
     ❙Ψ(t=0)❭ = A(❙φ₁❭ - ❙φ₂❭ + ι❙φ₃❭).
 
 where ❙φₙ❭ are the energy eigenstates. This also means
 
-    ❬Ψ(t=0)❙ = A⃰(❬φ₁❙ - ❬φ₂❙ + ι❬φ₃❙)
+    ❬Ψ(t=0)❙ = A(❬φ₁❙ - ❬φ₂❙ - ι❬φ₃❙)
 
 
-
-
-    ❙Ψ(t=0)❭ =  _͟A͟  (αβ❙φ₁❭ - βγ❙φ₂❭ + αγι❙φ₃❭)
-               αβγ
-
 In the energy basis,
 
     ❙φ₁❭ ≐ ⎛1⎞  ❙φ₂❭ ≐ ⎛0⎞  and ❙φ₃❭ ≐ ⎛0⎞
@@ -23,11 +18,94 @@ So,
                ⎜-A ⎟
                ⎝ιA ⎠.
 
-(𝐚) Multiplying the state vector by its magnitude normalizes it.
+(𝐚) The state vector is normalized by taking the quotient of the magnitude.
+
+    ❙Ψ′(t=0)❭ ≐  __͟A͟__ ⎛ 1 ⎞ =  _͟1͟ ⎛ 1 ⎞     
+                √(3A²) ⎜-1 ⎟    √3 ⎜-1 ⎟     
+                       ⎝ ι ⎠       ⎝ ι ⎠.    
+
+When the Hamiltonian operates on ❙φₙ❭ with results according to the general eigenvalue equation, with Eₙ the measured energy of state n, 
+
+    Ĥ❙φₙ❭ = Eₙ❙φₙ❭.
+
+The measured energies for a point particle in an infinite square well are given by, with L the x-width of the well, and m the particle mass,
+
+    Eₙ = n͟²͟π͟²͟ħ͟².
+          2mL²
+
+So,
+
+    Ĥ❙Ψ′(t=0)❭ = _͟1͟ (E₁❙φ₁❭ - E₂❙φ₂❭ + ιE₃❙φ₃❭).
+                 √3
+
+(𝐛) There is an equal chance of measuring any of the three values, so 𝓟ₙ=1/3. The measured enemies are given by the previous expression.
+
+    Energy          Probability     
+
+     π͟²͟ħ͟².          𝓟₁=¹/₃
+     2mL²    
+
+    4͟π͟²͟ħ͟².          𝓟₂=¹/₃
+     2mL²                        
+
+    9͟π͟²͟ħ͟².          𝓟₃=¹/₃
+     2mL²                         
+
+
+The average value of the energy, or the expectation value, is given by
+
+
+    │❬Ψ′❙Ĥ❙Ψ′❭│² = │_͟1͟ (❬φ₁❙ - ❬φ₂❙ - ι❬φ₃❙)_͟1͟ (E₁❙φ₁❭ - E₂❙φ₂❭ + ιE₃❙φ₃❭)│²
+                   │√3                      √3                            │. 
+
+    ❬Ψ′❙Ĥ❙Ψ′❭ = _͟1͟ (❬φ₁❙ - ❬φ₂❙ - ι❬φ₃❙)_͟1͟ (E₁❙φ₁❭ - E₂❙φ₂❭ + ιE₃❙φ₃❭)
+                √3                      √3                                           
+
+    = ¹/₃ (❬φ₁❙ - ❬φ₂❙ - ι❬φ₃❙)(E₁❙φ₁❭ - E₂❙φ₂❭ + ιE₃❙φ₃❭)
+    
+    = ¹/₃ (❬φ₁❙E₁❙φ₁❭ - ❬φ₁❙E₂❙φ₂❭ + ❬φ₁❙ιE₃❙φ₃❭) (-❬φ₂❙E₁❙φ₁❭ + ❬φ₂❙E₂❙φ₂❭ - ❬φ₂❙ιE₃❙φ₃❭) 
+        (-ι❬φ₃❙E₁❙φ₁❭ + ι❬φ₃❙E₂❙φ₂❭ - ι❬φ₃❙ιE₃❙φ₃❭).
+
+Orthogonality is a property of the energy eigenstates: 
+
+                               ⎧ 0 if n≠m   
+    ❬Eₙ❙Eₘ❭ = δₙₘ, where δₙₘ = ⎨          , so
+                               ⎩ 1 if n=m   
+
+❬Ψ′❙Ĥ❙Ψ′❭ = ¹/₃ (❬φ₁❙E₁❙φ₁❭) (❬φ₂❙E₂❙φ₂❭) (-ι❬φ₃❙ιE₃❙φ₃❭)
+
+= ¹/₃ E₁ E₂ E₃ = ¹/₃ π͟²͟ħ͟² 4͟π͟²͟ħ͟² 9͟π͟²͟ħ͟² = _͟3͟ ⎛π͟²͟ħ͟²⎞³
+                     2mL²  2mL²  2mL²    2 ⎝ mL²⎠.
+
+(𝐜) Therefore, the energy expectation value 
+
+    │❬Ψ′❙Ĥ❙Ψ′❭│² = _͟9͟ ⎛π͟²͟ħ͟²⎞⁶
+                    4 ⎝ mL²⎠.
+
+Because the hamiltonian is time-independent, the state vector progresses timewise according to,
+
+    ❙Ψ′(t)❭ = exp(-ι͟Ĥ͟t͟ )❙Ψ′(t=0)❭ = _͟1͟ exp(-ι͟Ĥ͟t͟ ) (❙φ₁❭ - ❙φ₂❭ + ι ❙φ₃❭).
+                    ħ               √3       ħ                       
+
+    _͟1͟ ⎛exp(-ι͟E͟₁͟t͟ )❙φ₁❭ - exp(-ι͟E͟₂͟t͟ )❙φ₂❭ + ι exp(-ι͟E͟₃͟t͟ )❙φ₃❭⎞.
+    √3 ⎝       ħ                 ħ                   ħ       ⎠
+
+
+
+The coefficients for the states have the same magnitudes, so the probability of measuring a particular state at time t=0 is 1/3.
+
+
+    ❙Ψ(t=0)❭ =  _͟A͟  (αβ❙φ₁❭ - βγ❙φ₂❭ + αγι❙φ₃❭)
+               αβγ
+
+
+
+
+
+
+
+
 
-    ❙Ψ′(t=0)❭ ≐  __͟A͟__ ⎛ 1 ⎞ =  _͟1͟ ⎛ 1 ⎞    
-                √(3A²) ⎜-1 ⎟    √3 ⎜-1 ⎟.  
-                       ⎝ ι ⎠       ⎝ ι ⎠