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 ( ) ( ) 12 ll Sk i k fk =+ 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 Ske ie kr 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 , ki = () 0 22 /2 k = m = . 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 kk δ = i
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2 and at low energy so () 0 , kk δ =− a 0 11 1/ fk ik a a i == + −− , and () () ( ) 00 / 12 / ki a Sk i k a + =+ . Note that as , as it should, since 1 S 0 k ( ) 0 0 a = −→ and ( ) ( ) 0 2 0 ik Sk e = . Note also that this approximation correctly gives ( ) 0 1 = . This has a pole in the complex plane at 0 / kia = , and if this corresponds to a bound state having , then the binding energy In fact, though, we run into a problem here: we get the same form of a κ = 22 2 2 /2 . mm = a ( ) 0 at low energies even for a repulsive potential, which certainly doesn’t have a bound state! The pole in ( ) 0 only means that we can have an asymptotic wave function of the right form, but it does not guarantee that this asymptotic form will go smoothly to nonsingular behavior at the origin. For a repulsive potential, it’s easy to see that the zero (or negative) energy wave function on integrating out from the origin slopes more and more steeply upwards , so could never, with increasing r , go over to asymptotic decay.
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This note was uploaded on 12/07/2011 for the course PHYSICS 751 taught by Professor Michaelfowler during the Fall '07 term at UVA.

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Scattering_III - Relating Scattering Amplitudes to Bound...

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