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Scattering_III - Relating Scattering Amplitudes to Bound...

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Relating Scattering Amplitudes to Bound States Michael Fowler, UVa. 1/17/08 Low Energy Approximations for the S Matrix In this section, we examine the properties of the partial-wave scattering matrix ( ) ( ) 1 2 l l S k ikf k = + for complex values of the momentum variable k . Of course, general complex values of k do 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 k plane. We begin with the complex k connection 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) ( ) 0 2 ikr ikr S k e i e k r r + . An l = 0 bound state has asymptotic wave function r Ce r κ where C is a normalization constant. Notice that this resembles an “outgoing wave” with imaginary momentum . If we analytically continue the scattering wave function from real k into the complex k -plane, we get both exponentially increasing and decreasing wave functions, making no physical sense. But there is an exception to this general observation: if the scattering matrix becomes infinite at 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 k pure imaginary , k i κ = ( ) 0 S k 2 2 / 2 k = m k i κ = . Now, the existence of a low energy bound state means that the S matrix 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 ( ) ( ) ( ) 0 0 1 , cot f k k k δ = i
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