Effects of resonance on phase shifts, wave amplitude and time delay

TL;DR

The professor solves a physics problem involving trigonometric and hyperbolic functions, explaining key equations and variables.

Transcript

PROFESSOR: This is the answer. Tan of 2 k a plus delta. It's a little messy, no wonder. Just plotted this functions. [INAUDIBLE] Sin k plus a cosh kappa a [INAUDIBLE] plus a prime over kappa cosh a prime a sinh kappa a. You have sin a prime a sinh. The sinh causes outer [INAUDIBLE] the same but this one changes. Centered here, k prime over kappa co... Read More

Key Insights

• 👋 The solving of the physics problem involves determining the phase shift and wave behavior in a square well potential.
• 😑 Trigonometric and hyperbolic functions are used to express the wave functions and calculate various parameters.
• 👋 The phase shift shows a linear time advance initially, followed by a rapid change and a jump in wave behavior.

Questions & Answers

Q: What functions are involved in the mathematical expression?

The mathematical expression involves trigonometric functions such as sine and cosine, as well as hyperbolic functions like hyperbolic sine and hyperbolic cosine.

Q: What variables are introduced in the problem?

The variables introduced are ka, u, v0, v1, z0, z1, k, kappa, k_prime, delta, and e. These variables represent energy, potential parameters, and wave behavior.

Q: How does the phase shift vary with energy?

The phase shift begins linearly with a negative slope, representing a time advance. Then it jumps rapidly and crosses pi/2, indicating a change in wave behavior.

Q: What is the significance of the resonance and time delay peak?

The resonance occurs at a specific energy value, leading to a sharp increase in scattering amplitude. The time delay peak represents a significant delay in the time spent traveling inside the well.

Summary & Key Takeaways

• The professor solves a mathematical expression involving sine, cosine, hyperbolic sine, and hyperbolic cosine functions.

• He introduces variables and formulas related to energy, potential, and wave functions in a square well.

• The professor discusses the behavior of the phase shift and scattering amplitude, highlighting the presence of a resonance and a peak in time delay.