Sakurai5-41-levyb 1

5.41 Beginning with 1.7.34 b: Φ(F) = \frac{1}{(2π)^{3/2}} \int d^3x e^{-i \vec{p} \cdot \vec{x}} ψ(x) In this case ψ(x) is the ground state hydrogen atom wavefunction, given by Appendix B.6.7: ⟨ r,0,0∣1 0 0 ⟩ = R_{1 0} (r) Y^0_0 (0,d) = R_{1 0}(r) \frac{1}{√π} = \frac{1}{\sqrt{π a_o^3}} e^{-r/a_o} => Φ( \vec{p} ) = \frac{1}{ (2π)^3 \sqrt{ π a_o^3 }} \int_0^∞ \int_0^{π} \int_0^{2π} r^2 sinθ dϕ dθ dr \ e^{i \vec{p} \cdot \vec{r} } e^{-r/a_o} = \frac{2π}{(2π)^{3/2} \sqrt{ π a_o^3 }} \int_\infty^∞ \int_{0}^{π} r^2 e^{i \vec{p} \cdot \vec{r} } e^{-r/a_o} dr \, sinθ dθ choose \vec{p} = pZ (per problem + terms) => \vec{p} \cdot \vec{r} = pr(cosθ), = \frac{1}{π (2π)^{1/2}} \int_0^∞ \int_{0}^{π} r^2 e^{ic^2 pr \ cosθ - \frac {r}{a_o}} sinθ dθ = \frac{4 a_o^3}{π \sqrt{ π a_o^3 }} \frac{a_o^4}{(a_o^2 p^2 + k^2)^2 } using \ Mathematica. => |Φ( \vec{p} )|^2 d^3 p = \frac{8 a_o^5}{π (a_o^3, p^2 + k^2)^2 } d^3 \vec{p} \frac{20}{20} 77/100