There are a number of problems in quantum mechanics whose solution relies on time-averaging away parts of the Hamiltonian. In particular, two examples that come to mind:

  1. The rotating wave approximation for solving the two-level (Rabi) problem.
  2. Optical lattice potentials correspond to the time average of the potential induced by the electric field of a light wave.

I understand the physical basis of these approximations--that there is a scale separation between a rapidly oscillating part of the Hamiltonian and the response time of the affected degrees of freedom of the system. However, I have never seen a way of making such approximations mathematically rigorous. Always the argument boils down to something like "now, since this term is rapidly oscillating, we can take a time average", without a more formal justification.

Uncontrolled approximations like this make me uneasy, because you don't know how good they are. For example, there are known cases where the rotating wave approximation fails (for some kinds of interactions between atoms and microwaves, as I recall). So what I am looking for is a rigorous justification for time averaging a Hamiltonian.

For concreteness, I will focus on the rotating wave approximation for now. So the question is: Can someone provide a rigorous, quantitative estimate of how good an approximation it is to make the rotating wave approximation (time-average of the Hamiltonian)? "Goodness of approximation" is open to interpretation--e.g. closeness in an $L^2$ sense, closeness in some spectral sense, etc. could all be valid answers. Commentary on other time-averaging problems is also appreciated.

  • $\begingroup$ I've been under the impression that usually physically justified approximations are justified PHYSICALLY as opposed to mathematically, because providing details on the quality of the approximation either massively underestimates the quality of the approximation, or is close in difficulty to solving the unperturbed problem to much higher precision while the approximation is usually used to obtain general insights into the dynamics of the system in question and principles they follow. While I would love to see a precise answer to your question, I would be surprised to see one. $\endgroup$ – Adomas Baliuka Oct 20 '16 at 0:42

Here's a partial answer:

Suppose the Hamiltonian splits as $H=H_0 + e^{i\omega t} \Delta$, where $H_0$ and $\Delta$ are bounded at all times we care about. I will let $|H_0|$ and $|\Delta|$ refer to the supremum of the norm of the operators over all times we care about. $H_0$ is the time-averaged, or 'unperturbed', Hamiltonian, and we're interested in how close are the solutions for the perturbed and unperturbed Hamiltonians.

Consider a state $\phi$ evolving under the full Hamiltonian, and write $\phi = \phi_0 + \delta\phi$, where $\phi_0$ evolves under just the unperturbed Hamiltonian $H_0$. Explicitly, we have $$\partial_t \phi = -i\left(H_0 + e^{i\omega t}\Delta\right) \phi$$ and $$\partial_t \phi_0 = -i H_0 \phi_0.$$ Then the evolution of $\delta\phi$ is given by $$\partial_t \delta\phi = \partial_t \left(\phi - \phi_0\right) = -ie^{i\omega t} \Delta \phi_0 - i\left(H_0 + e^{i\omega t} \Delta\right) \delta\phi.$$ Suppose we start at time $t=0$ in a state of the unperturbed Hamiltonian, i.e. with $\delta\phi(0)=0$. Integrating the above equation gives $$ \begin{eqnarray} |\delta\phi(t)| & = & |\int_0^t -ie^{i\omega t} \Delta \phi_0 - i\left(H_0 + e^{i\omega t} \Delta\right) \delta\phi| \\ & \leq & |\int_0^t e^{i\omega t} \Delta \phi_0 | + \int_0^t \left(|H_0| + |\Delta|\right) |\delta\phi| \\ \end{eqnarray} $$ We estimate the first term as follows: Since the derivative $|\partial_t\phi_0|\leq |H_0| |\phi_0| = |H_0|$ (the equality uses the fact that $|\phi_0| = 1$ because it is a solution to the Schrodinger equation), in one period $T=2\pi/\omega$ of the oscillatory factor, $\phi_0$ can change by at most $T|H_0|$. So on the interval $[\tau,\tau+T]$ write $\phi_0(t) = \phi_0(\tau)+\epsilon(t)$, with $|\epsilon(t)|\leq T|H_0|$. Then we have $$ \begin{eqnarray} |\int_{\tau}^{\tau+T} e^{i\omega t} \Delta \phi_0 | & = & |\int_{\tau}^{\tau+T} e^{i\omega t} \Delta (\phi_0(\tau) + \epsilon(t)) | \\ & \leq & |\int_{\tau}^{\tau+T} e^{i\omega t} \Delta \phi_0(\tau)| + \int_{\tau}^{\tau+T} |\Delta| |\epsilon(t)| \\ & \leq & 0 + T^2 |\Delta| |H_0| \\ \end{eqnarray} $$

Breaking the integral from $0$ to $t$ into $t/T$ smaller integrals over a single period $T$ each (plus one partial interval at the end) yields the estimate $$|\int_0^t e^{i\omega t} \Delta \phi_0 | \leq t T |\Delta| |H_0| + T |\Delta| = 2 \pi |\Delta| (t|H_0| +1)/\omega.$$ Hence $$|\delta\phi(t)| \leq 2 \pi |\Delta| (t|H_0| +1)/\omega + \int_0^t \left(|H_0| + |\Delta|\right) |\delta\phi|,$$ and thus $\delta\phi$ is bounded by a solution $y(t)$ of $$y(t) = 2 \pi |\Delta| (t|H_0| +1)/\omega + \int_0^t \left(|H_0| + |\Delta|\right) y.$$ Differentiating this gives $$\partial_t y = 2 \pi |\Delta| |H_0|/\omega + \left(|H_0| + |\Delta|\right) y$$ which has solution $$y = \frac{- 2 \pi |\Delta| |H_0|}{\omega \left(|H_0| + |\Delta|\right)} + c \exp{\left[\left(|H_0| + |\Delta|\right) t\right]}$$ for some constant $c$, which from the initial condition $y(0)=0$ we find to be $\frac{2 \pi |\Delta| |H_0|}{\omega \left(|H_0| + |\Delta|\right)}$, so that $$y(t) = \frac{2 \pi |\Delta| |H_0|}{\omega \left(|H_0| + |\Delta|\right)} \left(-1 + \exp{\left[\left(|H_0| + |\Delta|\right) t\right]}\right)$$.

This is a bound on how far the perturbed solution gets from the unperturbed solution. We see that as the strength of the perturbation gets smaller ($|\Delta| \rightarrow 0$) or as the speed of the oscillation gets faster ($\omega\rightarrow \infty$) the solutions converge.

We look at this estimate a different way by asking how long it takes until the difference between the perturbed and unperturbed solutions, $\delta\phi$, becomes order 1 in magnitude. Assuming for simplicity that $|H_0| >> |\Delta|$ and that $2\pi |\Delta|/\omega << 1$, the time at which $y(t) \approx 1$ is given by $1\approx \frac{2\pi |\Delta|}{\omega} \exp{\left(|H_0|t\right)},$ equivalently $t\approx \frac{1}{|H_0|}\log{\left(\frac{\omega}{2\pi|\Delta|}\right)}.$

For the approximation to be good, we want roughly that $\delta\phi$ stay small for many oscillations of the unperturbed solution, which have a period of order $\frac{1}{|H_0|}$. So we want the above time $t$ much longer than $\frac{1}{|H_0|}$, meaning $\log{\left(\frac{\omega}{2\pi|\Delta|}\right)} >> 1$.

It must be admitted that this logarithmic estimate is pretty shoddy. But it's something.

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