Born Approximation Validity Conditions
The Born approximation is a method used in quantum mechanics to calculate the scattering of particles. The first Born approximation simplifies the calculation by approximating the scattered wave using a plane wave.
First Born Approximation Requirements
This approximation is particularly effective under specific physical conditions:
- High Incident Energies: The incident particle must have a significantly large energy (E). This ensures that the particle's momentum is high relative to the interaction potential, meaning the potential causes only a minor perturbation to the particle's path. Mathematically, this often corresponds to the condition that the momentum transfer during scattering is small compared to the incident momentum.
- Weak Scattering Potentials: The interaction potential (V) responsible for scattering must be weak. A weak potential means it does not drastically alter the incident wave function. The approximation assumes the potential is a small perturbation, which holds true only when the potential's strength is low.
When both high incident energies and weak potentials are present, the replacement of the scattered wave by a plane wave is a valid simplification, leading to accurate results for the scattering amplitude.
Therefore, the first Born approximation is valid for large incident energies and weak scattering potentials.
Important Questions from Scattering Cross Section Phase Shift Method
- A particle is scattered from a potential $ V(\vec{r}) = g\delta^3(\vec{r}) $, where $ g $ is a positive constant. Using the first Born approximation, the angular $ (\theta, \phi) $ dependence of differential scattering cross section $ \frac{d\sigma}{d\Omega} $ is
Consider the potential $U(r)$ defined as $$U(r) = -U_0 \frac{e^{- \alpha r}}{r}$$ where $ \alpha$ and $U_0$ are real constants of appropriate dimensions. According to the first Born approximation, the elastic scattering amplitude calculated with $U(r)$ for a (wave-vector) momentum transfer $q$ and $ \alpha \to 0$, is proportional to
(Useful integral: $ \int_0^{ \infty} \sin(qr)e^{- \alpha r} dr = \frac{q}{ \alpha^2+q^2}$)
- Consider an elastic scattering of particles in $l = 0$ states. If the corresponding phase shift $\delta_0$ is $90^\circ$ and the magnitude of the incident wave vector is equal to $\sqrt{2}\pi$ fm$^{-1}$ then the total scattering cross section in units of fm$^2$ is ________.
- Protons and $\alpha$-particles of equal initial momenta are scattered off a gold foil in a Rutherford scattering experiment. The scattering cross sections for proton on gold and $\alpha$-particle on gold are $\sigma_p$ and $\sigma_\alpha$ respectively. The ratio $\sigma_\alpha/\sigma_p$ is _________
- Consider the scattering of neutrons by protons at very low energy due to a nuclear potential of range $r_0$. Given that,
$cot(kr_0 + \delta) \approx -\frac{\gamma}{k}$
where $\delta$ is the phase shift, $k$ the wave number and $(-\gamma)$ the logarithmic derivative of the deuteron ground state wave function, the phase shift is
