phy-4600/solutions/chap9/prob11

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 ≐ √⎛_͟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  ⎠                                 ⎟ 
           ⎜                                                     ⎟ 
           ⎜   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 ⎠⎠       ⎠⎠.

    ❬0❙x̂❙0❭ = x₀₀ = ❬1❙x̂❙1❭ = x₁₁ = 0.

    ❬0❙x̂❙1❭ = x₀₁ = ❬1❙x̂❙0❭ = x₁₀ = √⎛_͟ħ͟ ⎞.
                                     ⎝2mω⎠
                           
Substituting the matrix elements:

    ❬x̂❭ = ⅕ √⎛͟2͟͟ħ͟⎞⎛ exp⎛ι͟E͟₀͟t + ι⎛_͟π͟ - E͟₁͟t⎞⎞ + exp⎛-ι͟E͟₀͟t + ι⎛E͟₁͟t - _͟π͟⎞⎞ ⎞ 
             ⎝mω⎠⎝    ⎝  ħ     ⎝ 2    ħ ⎠⎠      ⎝   ħ     ⎝ 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).

    Eₙ−Eₙ′ = ħω(n - n′):
        E₀−E₁ = ħω(0 - 1) = -ħω;
        E₁−E₀ = ħω.             

    ❬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❭ = ⅕ √⎛͟8͟ħ͟⎞ sin(ωt).
             ⎝mω⎠          

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.