What are the partial wave phase shifts ( ) for hard-sphere scattering (Example 10.3)? Example 10.3

What are the partial wave phase shifts (δ_ℓ) for hard-sphere scattering (Example 10.3)?

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Example 10.3 Quantum hard-sphere scattering. Suppose V(r) = [0, The boundary condition, then, is V (a. 8) = 0, SO 00 [i (20 + 1) [je(ka) + ik ach() (ka)] Pecos 0) = 0 8=0 [∞o. (rsa). (r>a). for all 8, from which it follows (Problem 10.3) that je(ka) kh (ka) de=i- In particular, the total cross-section (Equation 10.27) is 4π -4 Σ g= je(z) h(2) JR (-1) and hence 4π je(ka) (2+1) (1) f=0)) je(z) Je(z)+ine (z) (ka) That's the exact answer, but it's not terribly illuminating, so let's consider the limiting case of low- energy scattering: ka << 1. (Since k = 27/), this amounts to saying that the wavelength is much greater than the radius of the sphere.) Referring to Table 4.4, we note that ne(z) is much larger than je(z), for small z, so -i 1 2 20+ (26)! je(z) ne(z) 2² elz/ (20 + 1)! i - (26)!z--1/20 €! 2+1 (2e)!, = (ka)+(+2 (10.30) [2001]² 2²+1. (10.31) (10.32) (10.33) (10.34) (10.35) But we're assuming ka << 1, so the higher powers are negligible in the low-energy approximation the scattering is dominated by the = 0 term. (This means that the differential cross-section is independent of 8, just as it was in the classical case.) Evidently oz4rd², (10.36) for low energy hard-sphere scattering. Surprisingly, the scattering cross-section is four times the geometrical cross-section in fact, O is the total surface area of the sphere. This "larger effective size" is characteristic of long-wavelength scattering (it would be true in optics, as well); in a sense, these waves "feel" their way around the whole sphere, whereas classical particles only see the head-on cross-section (Equation 10.8).