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2013-08-13T15:20:21Z

141.30.193.8: /* Calculation */ Better english..

{{Unreferenced|date=December 2009}}
In [[quantum mechanics]], and in particular in [[scattering theory]], the '''Feshbach–Fano method''', named after [[Herman Feshbach]] and [[Ugo Fano]], separates (partitions) the resonant and the background components of the [[wave function]] and therefore of the associated quantities like [[cross section (physics)|cross sections]] or [[phase (waves)|phase shift]]. This approach allows us to define rigorously the concept of [[resonance]] in quantum mechanics.

In general, the partitioning formalism is based on the definition of two complementary [[projection operator|projectors]] ''P'' and ''Q'' such that

:''P'' + ''Q'' = 1.

The subspaces onto which ''P'' and ''Q'' project are sets of states obeying the continuum and the bound state [[boundary condition]]s respectively. ''P'' and ''Q'' are interpreted as the projectors on the background and the resonant subspaces respectively.

The projectors ''P'' and ''Q'' are not defined within the Feshbach–Fano method. This is its major power as well as its major weakness. On the one hand, this makes the method very general and, on the other hand, it introduces some arbitrariness which is difficult to control. Some authors define first the P space as an [[approximation]] to the background scattering but most authors define first the ''Q'' space as an approximation to the resonance. This step relies always on some physical intuition which is not easy to quantify. In practice ''P'' or ''Q'' should be chosen such that the resulting background scattering phase or cross-section is slowly depending on the scattering energy in the neighbourhood of the resonances (this is the so-called flat continuum hypothesis). If one succeeds in translating the flat continuum hypothesis in a mathematical form, it is possible to generate a set of equations defining ''P'' and ''Q'' on a less arbitrary ground.

The aim of the Feshbach–Fano method is to solve the [[Schrödinger equation]] governing a scattering process (defined by the [[Hamiltonian (quantum theory)|Hamiltonian]] ''H'') in two steps: First by solving the scattering problem ruled by the background Hamiltonian ''PHP''. It is often supposed that the solution of this problem is trivial or at least fulfilling some standard hypotheses which allow to skip its full resolution. Second by solving the resonant scattering problem corresponding to the effective complex (energy dependent) Hamiltonian

:<math>H_\mathrm{eff} (E) = QHQ + \lim_{\varepsilon \to 0} QHP{1 \over E + i \varepsilon -PHP}PHQ = QHQ + \Delta(E) - i \Gamma(E)/2, \, </math>

whose dimension is equal to the number of interacting resonances and depends parametrically on the scattering energy ''E''. The resonance [[parameter]]s <math>E_\mathrm{res}</math> and <math>\Gamma_\mathrm{res}</math> are obtained by solving the so-called implicit equation

:<math>\det[H_\mathrm{eff}(z)-z]=0 \, </math>

for ''z'' in the lower [[complex plane]]. The solution

:<math>z_\mathrm{res} = E_\mathrm{res}-i\Gamma_\mathrm{res} \, </math>

is the resonance pole. If <math>z_\mathrm{res}</math> is close to the real axis it gives rise to a [[Breit–Wigner]] or a [[Fano resonance|Fano]] profile in the corresponding cross section. Both resulting ''T'' [[matrix (mathematics)|matrices]] have to be added in order to obtain the ''T'' matrix corresponding to the full scattering problem :

:<math>T_\mathrm{tot}=T_\mathrm{background}+T_\mathrm{resonances}. \, </math>

{{DEFAULTSORT:Feshbach-Fano Partitioning}}
[[Category:Scattering theory]]

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