Question

In the context of scattering in a central potential, phase shifts indicate

• A. The change in the direction of a wavefront upon reflection
• B. Variations in the central potential's strength
• C. The alteration of a wave's phase after passing through a potential
• D. The number of particles scattered per second

Hint:

Phase shifts in scattering theory reveal how much a wave has been delayed (or advanced) due to a potential — they’re key to calculating cross-sections and resonance behavior!

Answer 1

The Correct Option is C

In quantum scattering theory, especially for central potentials (spherically symmetric), the concept of phase shifts arises from the partial wave analysis. When a wave function is scattered by a potential, its phase changes compared to free (unscattered) wave evolution. This deviation is quantified by the phase shift \( \delta_l \) for each angular momentum quantum number \( l \).
Mathematically, the radial part of the wavefunction far from the scattering center behaves like: \[ R_l(r) \sim \sin\left(kr - \frac{l\pi}{2} + \delta_l\right) \] The term \( \delta_l \) is the phase shift due to the interaction with the potential. It captures how the potential modifies the wavefunction’s phase even though the amplitude remains unaffected far from the scattering region. Why other options are incorrect:

• (A) This describes reflection, not the subtle phase modification post-scattering.
• (B) Variations in the potential may influence phase shifts, but the phase shift itself is not a direct indicator of the potential’s variation.
• (D) This refers to scattering rate or cross-section, not the wave's phase.