Question
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A particle of charge e and mass m scatters elastically on the electric potential V(r) produced by the hydrogen atom: V(r) = e2 / 4 pi 0 - 1 / r + where rho(r) = | (r)|2 is the electron probability density in the ground state, see Eq. (4.80). In the Born approximation (11.79), find a general expression for the amplitude of scattering f (theta) in terms of the atomic form-factor F (k) = z p(r)eiKf d3r, which is the Fourier transform of the electron density. Hint: The Fourier transform of a convolution of two functions is the product of their Fourier transforms. The convolution appears in the second term of the expression for V(r). How does the answer in Part (a) simplify when Ka 1, where a is the width of the electron charge distribution in the atom, i.e. the Bohr radius? Using your solution in Part (a), find a general expression fa the amplitude of scattering in the forward direction (theta = 0) in terms of rho(r). Hint: Taking into account that R rho(r)d3r = 1, take the limit k rightarrow 0 in the expression for f to the first non-vanishing term. Substituting the wave function (4.80) into the general formula obtained in Part (a), calculate the amplitude f (theta) and the di erential cross-section D (0) of scattering on the hydrogen at cm in the ground state. Check the limits Ka 1 and Ka 1 and verify that your answer agrees with the results of Parts (b) and (c). Hint: Because p(r) is spherically symmetric, you can use a formula similar to (11.88) to perform the Fourier transform of rho(r).
Explanation / Answer
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