Perturbation Theory (quantum Mechanics) - Strong Perturbation Theory

Strong Perturbation Theory

In a similar way as for small perturbations, it is possible to develop a strong perturbation theory. Let us consider as usual the Schrödinger equation

and we consider the question if a dual Dyson series exists that applies in the limit of a perturbation increasingly large. This question can be answered in an affirmative way and the series is the well-known adiabatic series. This approach is quite general and can be shown in the following way. Let us consider the perturbation problem

being . Our aim is to find a solution in the form

but a direct substitution into the above equation fails to produce useful results. This situation can be adjusted making a rescaling of the time variable as producing the following meaningful equations

that can be solved once we know the solution of the leading order equation. But we know that in this case we can use the adiabatic approximation. When does not depend on time one gets the Wigner-Kirkwood series that is often used in statistical mechanics. Indeed, in this case we introduce the unitary transformation

that defines a free picture as we are trying to eliminate the interaction term. Now, in dual way with respect to the small perturbations, we have to solve the Schrödinger equation

and we see that the expansion parameter appears only into the exponential and so, the corresponding Dyson series, a dual Dyson series, is meaningful at large s and is

After the rescaling in time we can see that this is indeed a series in justifying in this way the name of dual Dyson series. The reason is that we have obtained this series simply interchanging and and we can go from one to another applying this exchange. This is called duality principle in perturbation theory. The choice yields, as already said, a Wigner-Kirkwood series that is a gradient expansion. The Wigner-Kirkwood series is a semiclassical series with eigenvalues given exactly as for WKB approximation.