Abstract
We discuss an investigation of student difficulties with the corrections to the energy spectrum of the hydrogen atom for the strong and weak field Zeeman effects using degenerate perturbation theory. This investigation was carried out in advanced quantum mechanics courses by administering written free-response and multiple-choice questions and conducting individual interviews with students. We discuss the common student difficulties related to these concepts which can be used as a guide for creating learning tools to help students develop a functional understanding of concepts involving the corrections to the energy spectrum due to the Zeeman effect.
Highlights
The Zeeman effect in the hydrogen atom is the shift in the energy spectrum due to the presence of a magnetic field, and it is proportional to the strength of the magnetic field
The weak field Zeeman effect occurs when the corrections to the energies due to the fine structure term are much greater than the corrections to the energies due to the Zeeman term
Since the fine-structure term and, in general, the Zeeman term are significantly smaller than the unperturbed Hamiltonian, perturbation theory (PT) is an excellent method for determining the approximate solutions to the Time-Independent Schrödinger Equation (TISE) and corrections to the energy spectrum of the hydrogen atom
Summary
The Zeeman effect in the hydrogen atom is the shift in the energy spectrum due to the presence of a magnetic field, and it is proportional to the strength of the magnetic field. Since the fine-structure term and, in general, the Zeeman term are significantly smaller than the unperturbed Hamiltonian, perturbation theory (PT) is an excellent method for determining the approximate solutions to the TISE and corrections to the energy spectrum of the hydrogen atom. Due to the degeneracy in the hydrogen atom energy spectrum, degenerate perturbation theory (DPT) must be used to find the corrections for the strong and weak field Zeeman effect. We investigated student difficulties with finding the first-order corrections to the energies of the hydrogen atom for the strong and weak field Zeeman effects using DPT. Once a good basis for step 2 has been identified, the first order corrections to the energies due to the weaker perturbation can be determined. In the weak field Zeeman effect, the coupled representation forms a good basis for both step 1 and 2
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