# Solving particle on a circle | Summary and Q&A

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July 31, 2017
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MIT OpenCourseWare
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.

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### 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.