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QM: Magnus expansion, Gaussian integral

  1. Jul 21, 2009 #1
    1. The problem statement, all variables and given/known data

    The time-evolution operator [tex]\hat{U}(t,t_0)[/tex] for a time-dependent Hamiltonian can be expressed using a Magnus expansion, which can be written

    [tex]\hat{U}(t,t_0)=e^{\hat{A}(t,t_0)}=exp\left(\sum_{n=1}^{\infty}\frac{1}{n!}\left(\frac{-\imath}{\hbar}\right)^n\hat{F}_n(t,t_0)\right)[/tex]

    The first two terms are given by

    [tex]\hat{F}_1=\int_{t_0}^t\hat{H}(t_1)dt_1[/tex]
    [tex]\hat{F}_2=\int_{t_0}^t\int_{t_0}^{t_1}[\hat{H}(t_2),\hat{H}(t_1)]dt_2dt_1[/tex]

    Consider a Harmonic oscillator with [tex]\hat{H}=\hat{H}_0 + \hat{V}(t)[/tex] where
    [tex]\hat{H}_0=\frac{\hat{p}^2}{2m}+\frac{1}{2}m\omega^2\hat{x}^2[/tex]

    It interacts with an external field of the form [tex]\hat{V}=\hat{V}_0f(t)[/tex]

    Evaluate the Magnus expansion to second order for the case [tex]\hat{V}_0=V_0\hat{x}[/tex] and [tex]f(t)=e^{-t^2/\sigma^2}[/tex]
    2. Relevant equations

    See above.

    3. The attempt at a solution

    First term: [tex]\hat{F}_1=\int_{t_0}^t\left(\frac{\hat{p}^2}{2m}+\frac{1}{2}m\omega^2\hat{x}^2+V_0\hat{x}e^{-t^2/\sigma^2}\right)dt_1= \left(\frac{\hat{p}^2}{2m}+\frac{1}{2}m\omega^2\hat{x}^2\right)\cdot(t-t_0)+ ???[/tex]

    Second term (after solving the commutator specified above): [tex]\hat{F}_2=\frac{1}{m}V_0\imath\hbar\hat{p}\int_{t_0}^t\int_{t_0}^{t_1}\left(e^{-t_2^2/\sigma^2}-e^{-t_1^2/\sigma^2}\right)dt_1dt_2 = ??? + ???[/tex]

    How do I integrate the Gaussian functions above? It seems like the integrals only are easily solvable when integrated from [tex]-\infty[/tex] to [tex]\infty[/tex], is there any trick to get past this problem?

    Any kind of help appreciated!
     
  2. jcsd
  3. Jul 21, 2009 #2

    gabbagabbahey

    User Avatar
    Homework Helper
    Gold Member

    Just make use of the error function:

    [tex]\text{erf}(x)\equiv\frac{2}{\sqrt{\pi}}\int_0^x e^{-u^2}du[/tex]

    And remember ,

    [tex]\int_{t_0}^{t} f(t_1)dt_1=\int_{0}^{t} f(t_1)dt_1-\int_{0}^{t_0} f(t_1)dt_1[/tex]
     
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