- #1
Bapelsin
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Homework Statement
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]
Homework Equations
See above.
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!