Incident packet and delay for reflection

TL;DR

Scattering solutions and time delay in finite range potentials are analyzed, including the construction of wave packets and the calculation of time delays.

Transcript

PROFESSOR: I'll begin by reviewing quickly what we did last time. We considered what are called finite range potentials, in which over a distance R, in the x-axis, there's a non-zero potential. So the potential is some v of x for x between capital R and 0, is equal to 0 for x larger than capital R, and it's infinity for x negative. So there's a wal... Read More

Key Insights

• ☺️ Finite range potentials have a non-zero potential over a certain distance and are infinitely high at x = 0.
• 👋 Scattering solutions for particles with potential are represented as a superposition of incoming and reflected waves.
• 📶 The scattering amplitude indicates the strength of the scattering caused by the potential.
• 🫡 Time delay in the presence of a potential is calculated as 2hbar times the derivative of the phase shift with respect to energy.
• ⌛ Time delay divided by the free transit time gives an insight into the significance of the delay compared to the time taken by a particle with a certain velocity to travel back and forth in the potential.
• 👋 Wave packets can be constructed by superposing incident and reflected waves.

Questions & Answers

Q: What are finite range potentials and how do they differ from other potentials?

Finite range potentials have a non-zero potential over a certain distance (R) and are infinite at x = 0. Unlike other potentials, nothing happens beyond distance R in a finite range potential.

Q: How are scattering solutions for particles with potential represented?

Scattering solutions are represented as a superposition of an incoming wave (e^(-ikx)/2i) and a reflected wave (e^(ikx)/2i). The incoming wave propagates from positive infinity towards 0, while the reflected wave bounces back and propagates towards more positive x.

Q: What is the scattering amplitude and its significance?

The scattering amplitude is the coefficient of the scattered wave and represents the amplitude of the wave that is scattered due to the potential. It is related to the strength of the scattering and is different from the amplitude of the free wave function.

Q: How is time delay calculated in the presence of a potential?

Time delay is calculated by taking 2hbar times the derivative of the phase shift with respect to energy. This delay represents the delay of a wave packet caused by the potential, compared to the time it takes for a particle to travel back and forth in the absence of a potential.

Summary & Key Takeaways

• The lecture begins with a review of finite range potentials, which have a non-zero potential over a certain distance and are infinitely high at x = 0.

• Scattering solutions for particles with energy h^2k^2/2m are discussed, including the representation of the solutions as incoming and reflected waves.

• The concept of the scattered wave and the scattering amplitude is explained, highlighting how the potential affects the wave function.

• The construction of wave packets and the calculation of time delay in the presence of a potential are introduced, with the time delay given by 2hbar times the derivative of the phase shift with respect to energy.