Delta Potential Born Approximation: Difference between revisions
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In Born approximation, | In the Born approximation, the scattering amplitude is | ||
<math> | <math>f_{\text{Born}}(\theta)=-\frac{m}{2\pi\hbar^2}\int d^3\mathbf{r'}\,V(\mathbf{r'})e^{-i\mathbf{q}\cdot\mathbf{r}'},</math> | ||
where <math>\mathbf{q}=\mathbf{k}'-\mathbf{k}</math> | where <math>\mathbf{q}=\mathbf{k}'-\mathbf{k}</math> and <math>\mathbf{k}</math> and <math>\mathbf{k}'</math> are the wave vectors of the incident and scattered waves, respectively. Substituting in the delta function potential, we get | ||
<math>f_B(\theta)=-\frac{m}{2\pi\hbar^2}\int g\delta^3(\mathbf{r}')e^{-i\mathbf{q}\cdot\mathbf{r}'} | <math>f_B(\theta)=-\frac{m}{2\pi\hbar^2}\int d^3\mathbf{r'}\,g\delta^3(\mathbf{r}')e^{-i\mathbf{q}\cdot\mathbf{r}'}=-\frac{mg}{2\pi\hbar^2},</math> | ||
and the differential cross section is | and therefore the differential cross section is | ||
<math>\sigma | <math>\frac{d\sigma}{d\Omega}=|f_{\text{Born}}(\theta)|^2=\frac{m^2 g^2}{4\pi^2\hbar^4}.</math> | ||
As the distribution is isotropic, the total cross section is | As the distribution is isotropic, the total cross section is | ||
<math>\ | <math>\sigma=4\pi\sigma=\frac{m^2 g^2}{\pi\hbar^4}</math>. | ||
Back to [[ | Back to [[Differential Cross Section and the Green's Function Formulation of Scattering#Problems|Differential Cross Section and the Green's Function Formulation of Scattering]] | ||
Latest revision as of 13:48, 18 January 2014
In the Born approximation, the scattering amplitude is
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_{\text{Born}}(\theta)=-\frac{m}{2\pi\hbar^2}\int d^3\mathbf{r'}\,V(\mathbf{r'})e^{-i\mathbf{q}\cdot\mathbf{r}'},}
where 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 \mathbf{q}=\mathbf{k}'-\mathbf{k}} and 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 \mathbf{k}} and 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 \mathbf{k}'} are the wave vectors of the incident and scattered waves, respectively. Substituting in the delta function potential, we get
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_B(\theta)=-\frac{m}{2\pi\hbar^2}\int d^3\mathbf{r'}\,g\delta^3(\mathbf{r}')e^{-i\mathbf{q}\cdot\mathbf{r}'}=-\frac{mg}{2\pi\hbar^2},}
and therefore the differential cross section is
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\Omega}=|f_{\text{Born}}(\theta)|^2=\frac{m^2 g^2}{4\pi^2\hbar^4}.}
As the distribution is isotropic, the total cross section is
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 \sigma=4\pi\sigma=\frac{m^2 g^2}{\pi\hbar^4}} .
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