Relating Scattering Amplitudes to Bound States Michael Fowler, UVa. 1/17/08 Low Energy Approximations for the SMatrix In this section, we examine the properties of the partial-wave scattering matrix ()()12llSkikfk=+for complexvalues of the momentum variable k. Of course, general complex values of kdo not correspond to physical scattering, but it turns out that the scattering of physical waves can often be most simply understood in terms of dominant singularities in the complex kplane. We begin with the complex kconnection between (positive energy) scattering and (negative energy) bound states. The asymptotic form of the l= 0 partial wave function in a scattering experiment is (from the previous lecture) ()02ikrikrSk eiekrr+−⎛⎞−⎜⎟⎝⎠. An l= 0 bound state has asymptotic wave function rCerκ−where Cis a normalization constant. Notice that this resembles an “outgoing wave” with imaginary momentum . If we analytically continue the scatteringwave function from real kinto the complex k-plane, we get bothexponentially increasing and decreasing wave functions, making no physical sense. But there is an exception to this general observation: if the scattering matrix becomes infiniteat some complex value of k, the exponentially decreasing term will be infinitely larger than the exponentially increasing term. In other words, we’ll only have a decreasing wave function—a bound state. We know that the energy of a bound state has to be real and negative, and is also equal to , so this can only happen for kpure imaginary, kiκ=()0Sk22/ 2k=mkiκ=. Now, the existence of a low energy bound state means that the Smatrix has a pole (on the imaginary axis) close to the origin, so this will strongly affect low energy (near the origin, but real k) scattering. Let’s see how that works using the low-energy approximation discussed previously. Recall that the l= 0 partial wave amplitude ()()()001,cotfkkkδ=i−
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