Dyson Series - The Dyson Operator

The Dyson Operator

We suppose we have a Hamiltonian H which we split into a "free" part H₀ and an "interacting" part V i.e. H = H₀ + V. We will work in the interaction picture here and assume units such that the reduced Planck constant is 1.

In the interaction picture, the evolution operator U defined by the equation:

$\Psi(t)=U(t,t_0) \Psi(t_0) \$

is called Dyson operator.

We have

$U(t,t)=I,\ U(t,t_0)=U(t,t_1)U(t_1,t_0),\ U^{-1}(t,t_0)=U(t_0,t)$

and then (Tomonaga-Schwinger equation)

$i{d \over dt} U(t,t_0)\Psi(t_0) = V(t) U(t,t_0)\Psi(t_0).$

Thus:

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Famous quotes containing the word dyson:

“The question that will decide our destiny is not whether we shall expand into space. It is: shall we be one species or a million? A million species will not exhaust the ecological niches that are awaiting the arrival of intelligence.”
—Freeman Dyson (b. 1923)