Resonances in the complex k plane

TL;DR

Resonances and poles in the scattering amplitude can be understood by analyzing the complex plane, with poles representing points of infinite scattering amplitudes and complex wave numbers having implications for bound states and anti-bound states.

Transcript

PROFESSOR: So I want to go a little further to try to put resonances in a more intriguing footing. That you can play with and if you-- at some point interested. So let's think of discovering [INAUDIBLE] that we have. We had A s-- remember the scattered wave was A s e to the ikx [INAUDIBLE] that divided 2. And what was A s? Well, A s squared-- the s... Read More

Key Insights

• 🫵 Resonances can be viewed as points in the complex plane where the scattering amplitude becomes infinite.
• 👋 Complex wave numbers in the imaginary axis can represent bound states, indicated by negative energy values.
• 💈 Poles in the complex plane can also represent anti-bound states.
• 🤑 The complex plane provides a rich and unified framework for understanding scattering solutions, resonances, bound states, and anti-bound states.
• 👻 The complex plane allows for a more precise and mathematically rigorous approach to studying resonances.
• ❓ Bound states can be identified by matching solutions to a pure decaying exponential in the asymptotic region.
• 💈 The interpretation of resonances and poles in the complex plane provides valuable insights into the behavior of scattering amplitudes.

Questions & Answers

Q: How can resonances and poles be analyzed in the context of scattering amplitudes?

Resonances and poles can be analyzed by solving the equation tan(delta) = -i, which guarantees a pole in the scattering amplitude. This allows us to identify points in the complex plane where the scattering amplitude becomes infinite.

Q: What is the significance of complex wave numbers in the imaginary axis?

Complex wave numbers in the imaginary axis, represented as k = i * kappa, can indicate bound states. The energy corresponding to these wave numbers is negative, suggesting the presence of confined states.

Q: What are anti-bound states and how do they relate to the complex plane?

Anti-bound states are solutions of the scattering amplitude that match a pure increasing exponential. In the complex plane, these states are represented by poles in the negative k region. Some nuclear states are associated with anti-bound states.

Q: How does the complex plane provide a valuable framework for understanding scattering solutions?

The complex plane accommodates scattering solutions, resonances, bound states, and anti-bound states. By analyzing the complex plane, we gain a more comprehensive understanding of the behavior of scattering amplitudes and the properties of these states.

Summary & Key Takeaways

• Resonances and poles in the scattering amplitude can be analyzed in the complex plane, with poles representing points where the scattering amplitude becomes infinite.

• Complex wave numbers in the imaginary axis can represent bound states, while poles in the complex plane can also represent anti-bound states.

• The complex plane provides a rich framework for understanding scattering solutions, resonances, bound states, and anti-bound states.