Exponential Potential Born Approximation

(Submitted by team 1)

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Using the Born approximation, find the differential cross section for the next exponential potential:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V(r)= -V_0 e^{-\frac{r}{a}} }

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If the potential Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V \!} is spherical symmetric we can use the equation:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f_{born}(\theta) = - \frac{2m}{\hbar^2} \int_0^\infin dr' V(r') \frac{\sin(qr')}{qr'} {r'}^2 }

So,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f_{born}(\theta) = - \frac{2mV_0}{\hbar^2 q} \int_0^\infin r' \sin(qr') e^{-\frac{r'}{a}} dr' }

Solving this integral by parts,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} f_{born}(\theta) &= - \frac{2mV_0}{\hbar^2 q} \frac{\partial}{\partial q} \int_0^\infin \cos(qr') e^{-\frac{r'}{a}} dr' \\ &= - \frac{2mV_0}{\hbar^2 q} \frac{\partial}{\partial q} Re\left[ \int_0^\infin e^{iqr'} e^{-\frac{r'}{a}} dr' \right] \\ &= - \frac{2mV_0}{\hbar^2 q} \frac{\partial}{\partial q} Re\left[ \frac { e^{(iq - \frac{1}{a})r'} }{iq - \frac{1}{a}} \right]_{_0}^{^\infin} \\ &= -\frac{2mV_0}{\hbar^2 q} \frac{\partial}{\partial q} Re\left[ \frac { 1 }{\frac{1}{a} + iq }\right] \end{align} }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f_{born}(\theta) = \frac{4mV_0}{\hbar^2 a} (\frac{1}{ \frac{1}{a^2} +q^2 })^2 }

So, the differential cross section,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{d\sigma}{d \theta} = \left|f_{born}(\theta) \right|^2 = \frac{16m^2V_0^2}{\hbar^4 a^2} \left(\frac{1}{ \frac{1}{a^2} +q^2 }\right)^4 }