# The Dyson series

Several people asked to see a route from the Schrödinger equation

$i\hbar \frac{\partial |\psi(t)\rangle}{\partial t}=H(t)|\psi(t)\rangle$

to the
Dyson series for the evolution operator

$U(t,t') = 1 -\frac{i}{\hbar}\int_{t'}^{t} H(t_1)\, dt_{1} -\frac{1}{\hbar^{2}} \int_{t'}^{t}dt_{1}\int_{t’}^{t_{1}}dt_{2}\, H(t_{1})H(t_{2})+\cdots.$

Since I see that this is confined to a footnote of the Advanced Quantum course, here is the derivation. Integrate both sides of the Schrödinger equation with respect to time to give

$|\psi(t)\rangle=|\psi(t’)\rangle-\frac{i}{\hbar}\int_{t’}^{t} H(t_1)|\psi(t_1)\rangle dt_1$

and then iterate. You will see that at each stage of the iteration, the new time integral you introduce goes from the initial time $$t’$$ to the previous dummy variable. In other words, the occurrences of the $$H(t)$$ are
time ordered from later to earlier times as you read from left to right. The symbol $$\mathcal{T}$$ just tells us to do that.

Finally, what is the relation to the exponential series?

$\mathcal{T} \exp\left(-\frac{i}{\hbar}\int_{t'}^{t} H(t_{i})\, dt_{i}\right)$

The exponential series of course has a factor $$1/n!$$ in the $$n^\text{th}$$ order. Where does this come from? This is because restricting the dummy variables to an ordering $$t_1>t_2>\ldots t_n$$ restricts the n-dimensional volume of the integral by a factor $$1/n!$$ (think of the number of ways of permuting the labels).