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Question #43662

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The Question

In a theoretical model of a positronium atom (a bound system of an electron and a positron, each of mass $m_e$ and charge magnitude $e$ ), the system's interaction with a weak uniform external magnetic field $\vec{B}$ is governed by the effective one-body relative Hamiltonian $H_{rel} = \frac{(\vec{p} - e\vec{A})^2}{2\mu} + V(r)$ , where $\mu$ is the reduced mass of the system, $\vec{p}$ is the relative momentum, and $\vec{A} = \frac{1}{2}(\vec{B} \times \vec{r})$ is the vector potential. Under this model, what is the magnitude of the energy splitting $\Delta E$ between adjacent orbital magnetic sublevels (states differing by magnetic quantum number $\Delta m_l = 1$ )?

Options

A.
$\frac{e \hbar B}{2 m_e}$
B.
$\frac{e \hbar B}{m_e}$
C.
$\frac{e \hbar B}{4 m_e}$
D.
$\frac{2 e \hbar B}{m_e}$

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Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Intuition_Breaker

Train your brain to resist "obvious" answers that are designed to deceive your intuition.

Concepts Behind This Question

Reduced Mass Effect|Gyromagnetic Ratio Invariance|Force Between Parallel Conductors|Vector Cross-Products

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to adapt standard quantum mechanical formulas (like the Bohr magneton and Zeeman splitting) to systems with a significantly different reduced mass.

The lethal trap occurs when students instinctively apply the standard electron energy splitting formula ( $\Delta E = e \hbar B / 2m_e$ ) without accounting for the center-of-mass motion of the positronium system.

By relying on rote memorization rather than deriving the interaction from the effective relative Hamiltonian, they miss that the gyromagnetic ratio of the effective relative particle is inversely proportional to the reduced mass ( $\mu = m_e/2$ ).

This mass correction effectively doubles the Larmor frequency and the resulting energy split magnitude, punishing the assumption that the electron mass $m_e$ is invariant across different atomic models.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Atoms - Class 12
Concept	Reduced Mass Effect
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free