Group velocity and stationary phase approximation | Summary and Q&A

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July 31, 2017
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Group velocity and stationary phase approximation

TL;DR

The professor explains the concept of wave packets and group velocity, highlighting the principle of stationary phase.

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Questions & Answers

Q: How are omega and k related in waves?

Omega, the angular frequency, is proportional to k squared. This relationship is similar to the energy being proportional to p squared, where p is the momentum.

Q: How are wave packets constructed?

Wave packets are constructed by superposing waves with different values of k. Each wave is represented by an exponential function, ikx - omega kt, with the function phi of k determining the amplitude for each k value.

Q: What is the principle of stationary phase?

The principle of stationary phase states that when integrating a function multiplied by a wave, the integral only contributes significantly when the phase of the wave is stationary, meaning it doesn't change quickly with respect to the variable of integration.

Q: How does the principle of stationary phase apply to wave packets?

The principle of stationary phase applied to wave packets suggests that the integral for constructing the wave packet will only have a significant contribution if the phase, phi of k, is stationary around k0. This allows for a non-zero integral value, contributing to the overall shape of the wave packet.

Summary & Key Takeaways

  • The professor introduces the concept of omega (angular frequency) being proportional to k (wavenumber) squared, which relates to the energy and velocity of waves.

  • Wave packets are constructed by superposing waves with different values of k, each wave represented by an exponential function.

  • The function phi of k, which has a narrow peak around k0, determines the shape of the wave packet.

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