Question

To obtain the total scattering cross section ?, we first compute the scattering amplitude f(?) = -frac{2m}{hbar^2 ?} int_0^infty r V(r) sin(? r) , dr where ? = 2k sinleft(frac{?}{2} ight) Substituting Gaussian potential of the form V(r) = A e^{-? r^2}, Therefore, f(?) = -frac{2mA}{hbar^2 ?} int_0^infty r e^{-? r^2} sin(? r) , dr = -frac{2mA}{hbar^2 ?} int_0^infty frac{d}{dr}left(-frac{1}{2?} e^{-? r^2} ight) sin(? r) , dr = frac{2mA}{2? hbar^2 ?} left{left[e^{-? r^2}sin(? r) ight]_0^infty - int_0^infty e^{-? r^2} frac{d}{dr}[sin(? r)] , dr ight} = frac{mA}{? hbar^2 ?} left{0 - ? int_0^infty e^{-? r^2} cos(? r) , dr ight} = -frac{mA}{? hbar^2} left{frac{sqrt{pi}}{2sqrt{?}} e^{-frac{?^2}{4?}} ight} = -frac{mAsqrt{pi}}{2hbar^2 ?^{3/2}} e^{-frac{?^2}{4?}}

          To obtain the total scattering cross section ?, we first compute the scattering amplitude
f(?) = -frac{2m}{hbar^2 ?} int_0^infty r V(r) sin(? r) , dr
where
? = 2k sinleft(frac{?}{2}
ight)
Substituting Gaussian potential of the form
V(r) = A e^{-? r^2},
Therefore,
f(?) = -frac{2mA}{hbar^2 ?} int_0^infty r e^{-? r^2} sin(? r) , dr
= -frac{2mA}{hbar^2 ?} int_0^infty frac{d}{dr}left(-frac{1}{2?} e^{-? r^2}
ight) sin(? r) , dr
= frac{2mA}{2? hbar^2 ?} left{left[e^{-? r^2}sin(? r)
ight]_0^infty - int_0^infty e^{-? r^2} frac{d}{dr}[sin(? r)] , dr
ight}
= frac{mA}{? hbar^2 ?} left{0 - ? int_0^infty e^{-? r^2} cos(? r) , dr
ight}
= -frac{mA}{? hbar^2} left{frac{sqrt{pi}}{2sqrt{?}} e^{-frac{?^2}{4?}}
ight}
= -frac{mAsqrt{pi}}{2hbar^2 ?^{3/2}} e^{-frac{?^2}{4?}}

$To obtain the total scattering cross section ?, we first compute the scattering amplitude f(?) = -frac2mhbar^2 ? int0^infty r V(r) sin(? r) , dr where ? = 2k sinleft(frac?2 ight) Substituting Gaussian potential of the form V(r) = A e^-? r^2, Therefore, f(?) = -frac2mAhbar^2 ? int0^infty r e^-? r^2 sin(? r) , dr = -frac2mAhbar^2 ? int0^infty fracddrleft(-frac12? e^-? r^2 ight) sin(? r) , dr = frac2mA2? hbar^2 ? leftleft[e^-? r^2sin(? r) ight]0^infty - int0^infty e^-? r^2 fracddr[sin(? r)] , dr ight = fracmA? hbar^2 ? left0 - ? int0^infty e^-? r^2 cos(? r) , dr ight = -fracmA? hbar^2 leftfracsqrtpi2sqrt? e^-frac?^24? ight = -fracmAsqrtpi2hbar^2 ?^3/2 e^-frac?^24?$

Added by Robert B.

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