# How can I go into the interaction picture when it is difficult/impossible to exponentiate the operator?

I have a Hamiltonian of the form $H = H_0(t) + V(t)$ and wish to go into the interaction picture with $H_0(t)$. Normally this would be done by defining an evolution operator $U(t) = e^{-iH_0t/\hbar}$ and calculating $H_{interaction} = U ^{\dagger}V U$

However, I am having trouble exponentiating $H_0$, since it contains harmonic oscillator raising and lowering operators, $a$ and $a^{\dagger}$. Also, $H_0$ is time-dependent, which adds difficulty.

Is there a way to find $H_{interaction}$ without exponentiating $H_0$ ? Possibly a perturbative method?

Wikipedia mentions: 'It is possible to obtain the interaction picture for a time-dependent Hamiltonian $H_{0,S}(t)$ as well, but the exponentials need to be replaced by the unitary propagator for the evolution generated by $H_{0,S}(t)$, or more explicitly with a time-ordered exponential integral' on the interaction picture page but doesn't explain how one would go about doing this. Is this the Dyson series? Thanks.

• You appear to have your notation backward. Normally, $H_0$ is the operator you exponentiate, and $V$ is the "small" operator. See: Wikipedia's interaction picture article. – Sean E. Lake Jan 23 '18 at 21:09
• Hi, thanks for spotting that. I've now changed the question so that it uses standard notation and added some more information that I found on the wikipedia page. The main point of the question still remains, though. – asph Jan 24 '18 at 12:16

## 1 Answer

You can't get around doing something equivalent to exponentiating the operator. One way to do something equivalent is to solve the equations of motion that come from the commutation relations with $$H_0(t)$$. For example, when $$H_0$$ is the simple harmonic oscillator the equations of motion for $$x$$ and $$p$$ are \begin{align} \dot{x} & = i[H_0,x] = \frac{p}{m} \\ \dot{p} & = i[H_0,p] = -\omega^2 x, \end{align} with the initial condition that $$x(0) = x_S$$ and $$p(0)=p_S$$. That system of first order ordinary differential equations is solved by \begin{align} x(t) & = x_S \cos(\omega t) + p_S \sin(\omega t) \\ p(t) & = -x_S \sin(\omega t) + p_S \cos(\omega t). \end{align} Because $$x(t) = \mathrm{e}^{-iH_0t}x_S\mathrm{e}^{iH_0t}$$ (signs may be flipped), any reasonably smooth $$V(x_S,p_S,t)$$ will obey \begin{align} \mathrm{e}^{-iH_0t}V(x_S,p_S,t)\mathrm{e}^{iH_0t} = V(x(t),p(t),t). \end{align}