Solving particle on a circle

TL;DR

The Schrodinger equation is solved for a particle moving in a circle, with the wave function and energy eigenstate determined.

Transcript

BARTON ZWIEBACH: --that has served, also, our first example of solving the Schrodinger equation. Last time, I showed you a particle in a circle. And we wrote the wave function. And we said, OK, let's see what is the momentum of it. But now, let's solve, completely, this problem. So we have the particle in the circle. Which means particle moving her... Read More

Key Insights

• 😥 The circle can be represented as an infinite line with the identification that any point is the same as the point at which you add a certain length.
• ⭕ The Schrodinger equation for a particle in a circle is simplified by assuming a zero potential.
• ⭕ All energy eigenstates for a particle in a circle are either zero or positive, it is impossible to find negative energy solutions.
• #️⃣ The wave function for a particle in a circle can be given by e to the ikx, where k is a real number.
• 🎃 The allowed values of k determine the periodicity of the wave function in a circle.
• 👋 The momentum associated with the wave function in a circle is equal to hk, where h is Planck's constant.
• 👨‍💼 The solutions to the Schrodinger equation for a particle in a circle can be sines or cosines of kx or e to the ikx.

Questions & Answers

Q: How is the circle represented as an infinite line?

The circle is represented as an infinite line by using the identification that any point is the same as the point at which you add a certain length. This allows for a simplified representation of the circular motion.

Q: What is the Hamiltonian for solving the wave function in a circle?

The Hamiltonian for solving the wave function in a circle is given by -h squared over 2m times the second derivative of the wave function with respect to x. This simplifies the Schrodinger equation for a particle in a circle.

Q: What are the allowed energy values for the particle in a circle?

The allowed energy values for the particle in a circle are either zero or positive. It is impossible to find solutions of the Schrodinger equation with negative energies.

Q: How do you solve the Schrodinger equation for a particle in a circle?

The Schrodinger equation for a particle in a circle is solved by assuming a wave function of the form e to the ikx, where k is a real number. This leads to the solutions of the Schrodinger equation as sines or cosines of kx or e to the ikx.

Summary & Key Takeaways

• The problem of a particle moving in a circle is solved using the Schrodinger equation, with the wave function and energy eigenstate calculated.

• The circle is represented as an infinite line with the identification that any point is the same as the point at which you add a certain length.

• The Schrodinger equation is simplified by assuming a zero potential and it is shown that all energies are either zero or positive.