How large is the magnetic moment of the orbital motion in a...

Name: Problem 4, Chapter 12: The Physics of Atoms and Quanta
Uploaded: 2021-11-19T20:38:29-08:00
Duration: 5 min 30 s
Channel: Chai Santi
Description: How large is the magnetic moment of the orbital motion in a muonium atom, in which the electron of a ground-state hydrogen atom has been replaced by a muon? How large is the moment in positronium (an electron and a positron, i.e. particles with the mass of the electron and opposite charges, moving around the common centre of mass)?

How large is the magnetic moment of the orbital motion in a muonium atom, in which the electron of a ground-state hydrogen atom has been replaced by a muon? How large is the moment in positronium (an electron and a positron, i.e. particles with the mass of the electron and opposite charges, moving around the common centre of mass)?

Key Concepts

Orbital Magnetic Moment

This concept refers to the magnetic dipole moment produced by a charged particle moving in an orbit. In both classical and quantum physics the orbital magnetic moment is proportional to the particle’s charge-to-mass ratio and its orbital angular momentum, usually expressed as ? = (q/(2m)) L. This relation shows that, for a given angular momentum, a heavier particle will have a smaller magnetic moment compared to a lighter one, and changes in the particle’s charge affect the sign and size of the moment.

Bohr Magneton

The Bohr magneton is a physical constant defined as ?_B = e?/(2m_e) and serves as a natural unit for magnetic moments in atomic physics, particularly for electrons. It provides a benchmark against which orbital magnetic moments are measured. When a different particle such as a muon takes the role of the orbiting particle, its larger mass results in a corresponding reduction in the orbital moment compared to ?_B.

Reduced Mass in Two?Body Systems

In the analysis of two-body quantum systems, such as exotic atoms, the dynamics of the relative motion are governed by the reduced mass of the two particles. The reduced mass modifies the effective scale of the orbital magnetic moment since the moment depends inversely on the mass of the moving particle. For example, when a muon replaces an electron, the effective orbital magnetic moment is reduced in proportion to the increased mass.

Center?of?Mass Motion and Magnetic Moment Cancellation

When two particles of equal mass and opposite charge orbit their common center of mass, as in positronium, each particle contributes an orbital magnetic moment with the magnitude determined by its own charge-to-mass ratio. However, because they move symmetrically and their charges have opposite signs, their individual contributions cancel each other, leading to a net orbital magnetic moment of zero. This cancellation is a direct consequence of the symmetry inherent in such two?body systems.

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