L5.1 Evaluating the Darwin correction

TL;DR

This content discusses the perturbation theory and various corrections to the Hamiltonian of the hydrogen atom, including the Darwin term, relativistic term, and spin-orbit term.

Transcript

PROFESSOR: Today we have to continue with our discussion of the hydrogen atom. We had derived or explained how you would derive the corrections to the original Hamiltonian, the Hamiltonian you've studied already in a couple of courses, this h0 Hamiltonian for the hydrogen atom that has a kinetic term and a potential term. This Hamilton we've studie... Read More

Key Insights

• 👻 Perturbation theory allows for the systematic calculation of corrections to the Hamiltonian of the hydrogen atom.
• 🍉 The corrections to the Hamiltonian, including the Darwin term, are smaller in magnitude compared to the original Hamiltonian.
• 🫀 The Darwin term affects the l=0 states of the hydrogen atom, and its correction can be calculated using the fine structure constant.

Questions & Answers

Q: What are the corrections to the original Hamiltonian of the hydrogen atom?

The corrections to the original Hamiltonian include the relativistic term, spin-orbit coupling, and the Darwin term. These corrections account for relativistic effects and interactions between the electron's spin and orbital angular momentum.

Q: How do the energies of the corrected Hamiltonian compare to the energies of the original Hamiltonian?

The energies associated with the corrections are smaller than the energies of the original Hamiltonian. These energies scale with the fine structure constant raised to the fourth power.

Q: Which states of the hydrogen atom are affected by the Darwin term?

The Darwin term only affects the l=0 states of the hydrogen atom. Other states are not affected by this correction.

Q: How is the Darwin correction calculated?

The Darwin correction, denoted as e1 Darwin for n 0 0, is calculated by taking the overlap of the wave function for the l=0 state at the origin squared. This overlap value is then substituted into the expression for the Darwin correction, which can be rewritten in terms of the fine structure constant.

Summary & Key Takeaways

• The content explores the corrections to the original Hamiltonian of the hydrogen atom, which include the relativistic term, spin-orbit coupling, and the Darwin term.

• The energies associated with these corrections are smaller than the energies of the original Hamiltonian.

• The Darwin term specifically affects the l=0 states of the hydrogen atom, and its correction can be calculated using the fine structure constant.