Question

The Born approximation is applied in quantum mechanics to:

• A. Estimate the scattering amplitude for weak potentials
• B. Calculate exact wave functions for bound states
• C. Determine the non-relativistic limit of particle interactions
• D. Solve the Schrödinger equation for any potential

Hint:

In quantum mechanics, the Born approximation is your starting tool for scattering problems with weak potentials. It simplifies calculations by assuming minimal disturbance to the incoming wave, enabling estimation of the scattering amplitude.

Answer 1

The Correct Option is A

Step 1: In quantum mechanics, the behavior of particles like electrons or photons is described using wave functions. When these particles encounter a potential barrier or a target (like a nucleus or another atom), they get scattered. Understanding how this scattering occurs is central to many areas of physics such as atomic, nuclear, and particle physics.
Step 2: The problem of scattering is mathematically formulated using the Schrödinger equation. However, solving the Schrödinger equation exactly for a general potential is extremely difficult and often impossible, especially when the potential is complex or has no symmetrical form.
Step 3: To handle such situations, approximation methods are used. The Born approximation is one such method, and it applies when the potential causing the scattering is weak. It assumes that the incident particle's wave function is only slightly modified due to the interaction with the potential.
Step 4: In the Born approximation, the wave function of the particle is replaced with a known incident wave in the expression for the scattered wave. This allows physicists to derive a formula for the scattering amplitude — a quantity that determines the probability of a particle being scattered at a particular angle.
Step 5: The scattering amplitude is an essential part of calculating the differential cross-section, which can be measured experimentally. Thus, the Born approximation gives a practical way to link theory with experiment for weak potentials.
Step 6: The other options are not valid:
- Option (B) involves calculating exact wave functions, which is not the goal of this approximation.
- Option (C) is unrelated to the scattering process; it deals with approximating relativistic equations.
- Option (D) is incorrect because the Born approximation does not solve the Schrödinger equation for all potentials—it only provides an approximate solution for weak ones.
Therefore, the Born approximation is used to estimate the scattering amplitude for weak potentials.