Regarding the Born approximation

In summary, the conversation discusses using the first order Born approximation to calculate the differential and total cross section for a given potential. The question is raised about the limits of this approximation and whether it is valid for a particle with a given mass. The criterion for the validity of the Born approximation is determined to be when the expression m|V0|a^2/\hbar^2 is less than 1.
  • #1
Kontilera
179
24
Hello!
In order to prepare for an exam I have started solving exercies problems and have gotten most of them right but have quetion a regarding a solution.

In this probelm I used the first order Born approximation in order to calculate the differential and total cross section for the potential,
[tex] V(r) = \lambda e^{-r^2 / 4a^2}. [/tex]
Now, how can I see the limits of this approximation? E.g. can we see if it is valid for a particle with a given mass assuming that we have [tex]ka \ll 1[/tex]?
 
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  • #2
The criterion for the validity of the Born approximation is
$$
\frac{m | V_0 | a^2}{\hbar^2} \ll 1
$$
where ##m## is the mass of the particle, ##V_0## and ##a## the height and range of the potential, respectively.
 

Related to Regarding the Born approximation

What is the Born approximation?

The Born approximation is a mathematical technique used in quantum mechanics to approximate the scattering of particles by a potential field. It assumes that the potential field is weak and the particles are non-interacting, allowing for a simpler calculation of the scattering amplitude.

When is the Born approximation valid?

The Born approximation is valid for low-energy scattering, where the particles have a small kinetic energy compared to the potential energy of the scattering field. It is also valid for weak scattering potentials, where the particles are not significantly affected by the potential.

How is the Born approximation calculated?

The Born approximation is calculated by taking the first-order term in a perturbation expansion of the full scattering amplitude. This involves solving the Schrödinger equation for the non-interacting particles in the presence of the scattering potential.

What are the limitations of the Born approximation?

The Born approximation is limited to weak scattering potentials and low-energy scattering. It also assumes that the particles are non-interacting, which may not be the case in certain systems. Additionally, it does not take into account multiple scattering events or higher-order terms in the perturbation expansion.

What are the applications of the Born approximation?

The Born approximation is commonly used in atomic and molecular physics to study the scattering of particles by a potential field. It is also used in other fields, such as solid-state physics and nuclear physics, to approximate the effects of a weak potential on the scattering of particles.

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