L3.2 Degeneracy resolved to first order; state and energy corrections | Summary and Q&A

TL;DR

This content explains the process of degenerate perturbation theory and its application in finding the first-order correction to a degenerate state.

Key Insights

• 🦾 Degenerate perturbation theory is a method used to solve for energy corrections and wavefunctions of degenerate states in quantum mechanics.
• 👖 The first step in degenerate perturbation theory is to diagonalize the perturbation matrix in the subspace spanned by the degenerate states.
• 🕰️ The second step involves finding the piece of the degenerate state within the non-degenerate subspace using the first-order equation.
• 🪈 The third step of degenerate perturbation theory is to solve the second-order equation to find the missing component of the degenerate state along the degenerate subspace.
• 🪈 The second-order correction to the degenerate state can be calculated even if the first-order correction does not split any of the energy levels.

Transcript

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Questions & Answers

Q: What is degenerate perturbation theory?

Degenerate perturbation theory is a mathematical method used to calculate the energy corrections and wavefunctions of degenerate states in quantum mechanics.

Q: How is the piece of the degenerate state within the non-degenerate subspace found?

In step two of degenerate perturbation theory, the first-order equation is used to find the piece of the degenerate state that lies within the non-degenerate subspace.

Q: What does step three of degenerate perturbation theory involve?

In step three, the second-order equation is solved to find the missing component of the degenerate state along the degenerate subspace.

Q: What happens if the first-order correction does not split any of the energy levels?

In this case, the second-order correction to the degenerate state can be calculated using degenerate perturbation theory.

Summary & Key Takeaways

• Degenerate perturbation theory is a method used to calculate the corrections to energy levels and wavefunctions of degenerate states.

• In step two of the process, the piece of the degenerate state that lies within the non-degenerate subspace is found using the first-order equation.

• Step three involves solving the second-order equation to find the missing component of the degenerate state along the degenerate subspace.