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Time Evolution operator in Interaction Picture (Harmonic Oscillator)

  1. Mar 13, 2014 #1
    1. The problem statement, all variables and given/known data
    Consider a time-dependent harmonic oscillator with Hamiltonian

    [tex]\hat{H}(t)=\hat{H}_0+\hat{V}(t)[/tex]
    [tex]\hat{H}_0=\hbar \omega \left( \hat{a}^{\dagger}\hat{a}+\frac{1}{2} \right)[/tex]
    [tex]\hat{V}(t)=\lambda \left( e^{i\Omega t}\hat{a}^{\dagger}+e^{-i\Omega t}\hat{a} \right)[/tex]

    (i) Compute ##\hat{U}_S(t,0)## using the interaction representation formula (Equation 1 in next section) to second order perturbation theory.
    (ii) Compute ##\hat{U}_S(t,0)## using (Equation 2 in next section) to second order perturbation theory

    2. Relevant equations

    EQUATION 1:
    [tex]U_I(t,0)=1-\frac{i}{\hbar}\int_0^t dt' V_I(t')+\left( \frac{-i}{\hbar} \right)^2 \int_0^t dt' \int_0^{t'} V_I(t')V_I(t'') + \dots[/tex]

    EQUATION 2:

    [tex]U(t,0)=1+\sum_{n=1}^{∞}\left( \frac{-i}{\hbar} \right)^n\int_0^t dt_1 \int_0^{t_1} dt_2 \dots \int_0^{t_{n-1}}dt_n H(t_1)H(t_2)\dots H(t_n)[/tex]



    3. The attempt at a solution


    So I know that for the interaction picture the transformation of the operator ##\hat{V}_I## is..

    [tex]\hat{V}_I=e^{\frac{i}{\hbar}\hat{H}_0 t} \hat{V} e^{\frac{-i}{\hbar}\hat{H}_0 t}[/tex]

    I also know that both operators and kets evolve in time. So I use the interaction picture equation of motion on the ladder operators so I can obtain an expression for them as a function of time.

    [tex]\frac{d\hat{a}}{dt}=\frac{1}{i\hbar}\left[ \hat{a},\hbar \omega \left( \hat{a}^{\dagger}\hat{a} + \frac{1}{2} \right) \right][/tex]

    [tex]\frac{d\hat{a}^{\dagger}}{dt}=\frac{1}{i\hbar}\left[ \hat{a}^{\dagger},\hbar \omega \left( \hat{a}^{\dagger}\hat{a} + \frac{1}{2} \right) \right][/tex]

    I then got..

    [tex]\hat{a}(t)=\hat{a}(0)e^{-i\omega t}[/tex]
    [tex]\hat{a}^{\dagger}(t)=\hat{a}^{\dagger}(0)e^{i\omega t}[/tex]

    I plugged these into the expression for V to get..

    [tex]\hat{V}=\lambda \left[ \hat{a}^{\dagger}(0)e^{i(\Omega + \omega)t} + \hat{a}(0)e^{-i(\Omega + \omega)t} \right][/tex]

    So now what needs to be done, is to transform this into the interaction picture and then plug it into Equation 1 from above and integrate. But this seems very messy and I am having doubts if this is the correct way to go about this problem.

    If anyone can shed some light onto this I would really appreciate it!
     
  2. jcsd
  3. Mar 13, 2014 #2
    I'm pretty sure your very last line is V_I; you don't need to transform it any more. Just plug it into Equation 1.
     
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