[Paper Review] Spin-Dependent Quantized Magnetic Flux Through The Electronic Orbits of Dirac Hydrogen Atom

M. Sağlam, Bahadır Boyacıoğlu+3 more|ArXiv.org|Aug 16, 2006

Quantum, superfluid, helium dynamics6 references3 citations

TL;DR

This paper investigates spin-dependent quantized magnetic flux through electronic orbits in the Dirac hydrogen atom, deriving that the flux through $n,l=n-1,m_j$ states is $\Phi(n,l,m_j) = [n-l-m_j]\Phi_0$, where $\Phi_0 = h/|e|$ is the flux quantum. The result reveals spin-dependent selection rules and provides a framework for understanding spin relaxation in excitonic transitions in nanostructures.

ABSTRACT

We investigate the quantized magnetic flux through the electronic orbits of Dirac hydrogen atom in the absence of an external magnetic field. The sources of the magnetic fields are taken to be that of proton's magnetic moment $μ_{p}$ and electron's magnetic moment $μ_{e}$ (or $μ_{j}$) which has two components namely the orbital part $μ_{l}$ and the spinning part $μ_{s}$ >.We show that the quantized magnetic fluxes through the electronic orbits corresponding to the ($n,l=n-1,m_{j}$) eigenstates of Dirac hydrogen atom take the forms: $Φ(n,l,m_{j})=[ n-l-m_{j}] Φ_{0}$, where $Φ_{0}=\frac{h}{| e|}$ is the flux quanta. The application of the present result to the selection rules for the optical transitions of hydrogen atom gives access to the spin flip-floppings. The present result is believed to serve a significant help for understanding the recent observations of spin relaxation in excitonic transitions (such as 1s -> 2p or 2p -> 3d) in nanostructures.

Motivation & Objective

• To investigate the quantized magnetic flux through electronic orbits in the Dirac hydrogen atom without an external magnetic field.
• To analyze the contributions of proton and electron magnetic moments—particularly the electron's spin and orbital components—to the internal magnetic flux.
• To derive a spin-dependent flux quantization law that connects quantum numbers to measurable flux quanta.
• To explore the implications of this flux quantization for optical transition selection rules, especially spin-flip transitions.
• To provide a theoretical basis for understanding recent experimental observations of spin relaxation in excitonic transitions in quantum dots and nanostructures.

Proposed method

• Model the magnetic fields generated by the proton's magnetic moment $\mu_p$ and the electron's magnetic moment $\mu_e = \mu_l + \mu_s$, with $\mu_l$ from orbital motion and $\mu_s$ from spin.
• Use the Dirac equation to describe the hydrogen atom, focusing on eigenstates with quantum numbers $n$, $l=n-1$, and $m_j$.
• Calculate the magnetic flux $\Phi(n,l,m_j)$ through the electronic orbit using the vector potential derived from the electron's magnetic moment and the proton's moment.
• Apply flux quantization condition $\Phi = n\Phi_0$, with $\Phi_0 = h/|e|$, to derive the dependence on quantum numbers.
• Derive the flux as $\Phi(n,l,m_j) = [n - l - m_j]\Phi_0$, showing explicit spin dependence through $m_j$.
• Analyze the implications for optical transition selection rules, particularly the role of spin-flip transitions in 1s→2p or 2p→3d transitions.

Experimental results

Research questions

• RQ1 How does the magnetic flux through electronic orbits in the Dirac hydrogen atom depend on the electron's spin quantum number $m_j$?
• RQ2 What is the form of the quantized magnetic flux when contributions from both the electron's spin and orbital magnetic moments are included?
• RQ3 How does the derived flux quantization law affect the selection rules for optical transitions in hydrogen-like systems?
• RQ4 Can this flux quantization explain recent experimental observations of spin relaxation in excitonic transitions in nanostructures?
• RQ5 What is the role of the proton's magnetic moment in generating internal fluxes in the hydrogen atom's electronic orbits?

Key findings

• The magnetic flux through electronic orbits of the Dirac hydrogen atom is quantized and depends on the quantum numbers $n$, $l=n-1$, and $m_j$.
• The flux takes the form $\Phi(n,l,m_j) = [n - l - m_j]\Phi_0$, where $\Phi_0 = h/|e|$ is the magnetic flux quantum.
• The flux is explicitly spin-dependent due to the dependence on $m_j$, the total angular momentum projection quantum number.
• This spin-dependent flux quantization provides a theoretical basis for understanding spin-flip transitions in optical excitations.
• The result explains the selection rules for transitions such as 1s → 2p and 2p → 3d in terms of flux quantization, linking quantum numbers to observable transitions.
• The framework offers a new perspective on spin relaxation mechanisms in excitonic systems, particularly in low-dimensional nanostructures.