L7.2 Approximate WKB solutions

TL;DR

The WKB approximation is a method used to solve certain types of differential equations in quantum mechanics.

Transcript

PROFESSOR: OK. So let's try to solve this. So my classical approximation is about solving these equations. So let's see what we get. Well, the first equation is kind of simple. I think everybody has the temptation there to just take the square root, and that's what we should do, s0 prime is equal to plus minus p of x. And therefore s0 of x is equal... Read More

Key Insights

• 🍉 The WKB approximation relies on solving differential equations in terms of the function p(x).
• 🫚 The first equation involves finding the square root of p(x), while the second equation involves taking the derivative of the logarithm of p(x).
• 💁 The solutions derived using the WKB approximation are of the form e^is, where s is the integral of p(x).
• 👋 The WKB approximation provides a good approximation for the wave function in the classically allowed and classically forbidden regions.

Questions & Answers

Q: What is the WKB approximation used for?

The WKB approximation is used to solve differential equations in quantum mechanics.

Q: How do you solve the first equation in the WKB approximation?

The first equation involves finding the square root of the function p(x) and integrating it to obtain the solution.

Q: How do you solve the second equation in the WKB approximation?

The second equation involves taking the derivative of the logarithm of p(x) to find the second solution.

Q: What is the significance of the plus-minus sign in the solutions?

The plus-minus sign represents the ambiguity in the direction of the particle's motion, as it can be moving to the left or right.

Summary & Key Takeaways

• The WKB approximation is used to solve differential equations in quantum mechanics.

• The first equation in the WKB approximation involves finding the square root of the function p(x) and integrating it to obtain the solution.

• The second equation involves taking the derivative of the logarithm of p(x) to find the second solution.