L14.4 Landau levels (continued). Finite sample

TL;DR

The content discusses the physics of degeneracy in quantum mechanics, specifically focusing on the Landau levels and the effects of changing gauges.

Transcript

PROFESSOR: So that's part of this story. Let's try to understand it a little better still. So what does the state look like? Well the state looks like e to the ikxx times a state of the oscillator and y. OK. So what is happening here? We could re-solve this problem using a different gauge. And you will. You will solve it at least once or twice. Aga... Read More

Key Insights

• 👀 Solving the Schrodinger equation in different gauges can yield different looking solutions, but the Landau levels and infinite degeneracy remain the same.
• ❓ Comparing solutions in different gauges requires understanding gauge transformation and infinite degeneracy.
• 😰 In a finite size sample, the degeneracy of the system can be calculated based on the values of Nx consistent with states within the sample.
• ❓ The degeneracy of a system is determined by the sample area and the magnetic flux quantum.

Questions & Answers

Q: What does the state of a system look like in quantum mechanics?

The state of a system can be represented as e to the ikxx times a state of the oscillator and y.

Q: How do solutions in different gauges relate to each other?

Solutions in different gauges, such as the symmetric gauge, can be related to each other through superpositions and gauge transformations.

Q: How can the physics of degeneracy be understood in a finite size sample?

To understand degeneracy in a finite size sample, one can impose periodic boundary conditions in x and quantize kx. The degeneracy is determined by the values of Nx that are consistent with states within the sample.

Q: How can the degeneracy of a system be calculated?

The degeneracy of a system can be calculated by dividing the sample area by the magnetic flux quantum, giving the number of degenerate states of each Landau level.

Summary & Key Takeaways

• The state of a system in quantum mechanics can be represented as e to the ikxx times a state of the oscillator and y.

• Solving the Schrodinger equation in different gauges can yield different looking solutions, but the Landau levels and infinite degeneracy remain the same.

• To compare solutions in different gauges and understand the physics of degeneracy, one must consider gauge transformation and infinite degeneracy.