Widths and uncertainties

TL;DR

The content discusses the uncertainty principle and its relation to wave packets and quantum mechanics.

Transcript

PROFESSOR: So we go back to the integral. We think of k. We'll write it as k naught plus k tilde. And then we have psi of x0 equal 1 over square root of 2pi e to the ik naught x-- that part goes out-- integral dk tilde phi of k naught plus k tilde e to the ik tilde x dk. OK. So we're doing this integral. And now we're focusing on the integration ne... Read More

Key Insights

• 👋 The phase excursion of an integral in the context of wave packets affects its contribution.
• 🛩️ For small total phase excursions, there is a substantial contribution to the integral.
• 👈 The uncertainty in x is determined by the range of integration for k tilde, which affects the localization of the integral.
• 👋 Quantum mechanics is not explicitly used in deriving the relation between uncertainties but comes into play when considering waves representing states with momentum.

Questions & Answers

Q: How does the phase change affect the integral in the context of the uncertainty principle?

As the phase changes, there is a significant effect on the integral. Specifically, if the total phase excursion (delta k times x) is small, there will be a substantial contribution. However, if the total phase excursion is large, the contribution will tend towards zero.

Q: What happens to psi of x0 for x values close to zero?

For x values close to zero, psi of x0 will be sizable. This is because the phase is stationary at x equal to zero, and this leads to a substantial answer.

Q: How is the uncertainty in x determined in the context of wave packets?

The uncertainty in x is given by 2x0, where x0 represents the uncertainty in x. This is determined by the range of integration for k tilde, which is from minus delta k over 2 to delta k over 2.

Q: Where does the quantum mechanical aspect come into play?

The quantum mechanical aspect arises when considering the relation between waves representing states with momentum (e to the ikx) and the uncertainty principle. By multiplying delta k by h bar (the reduced Planck constant), one can find the relation delta p is equal to h bar delta k, leading to the uncertainty principle.

Summary & Key Takeaways

• The integral in the content focuses on the integration near k naught, where the contribution is large.

• The relevant region of integration for k tilde is from delta k over 2 to minus delta k over 2.

• The phase excursion of the integral depends on the value of delta k times x, determining the contribution to the final answer.