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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).
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