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Question #43807

Difficulty: 8/10 | 🟢 Free

The Question

A hydrogen atom is placed in a weak uniform magnetic field $\vec{B} = B_0 \hat{k}$ (where $B_0$ is the magnetic field strength in Tesla and $\hat{k}$ is the unit vector along the $z$ -axis). The atom undergoes a radiative transition from an initial state with orbital quantum numbers $l = 1, m_l = +1$ to the ground state $l = 0, m_l = 0$ . The emitted photon is detected by an observer located on the $+x$ -axis. Let $\langle L_x \rangle$ be the expectation value of the $x$ -component of the electron's orbital angular momentum in the initial state. Which of the following correctly describes the polarization state of the detected photon and the value of $\langle L_x \rangle$ ?

Options

A.
\text{Linearly polarized along the } y\text{-axis}, \ 0
B.
$\text{Circularly polarized}, \ \frac{\hbar}{\sqrt{2}}$
C.
\text{Linearly polarized along the } z\text{-axis}, \ 0
D.
\text{Circularly polarized}, \ 0

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Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Variable_Constraint

Develop pure symbolic logic skills by mastering algebraic constraints without relying on numerical data.

Concepts Behind This Question

Photon Polarization and Selection|Heisenberg's Limitation|Vector Decomposition Ambiguity

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to distinguish between classical vector models and quantum mechanical expectation values, specifically targeting the Heisenberg limitation and vector decomposition ambiguity.

Students often incorrectly assume that a $\Delta m_l = 1$ transition always produces a circularly polarized photon, neglecting that polarization depends on the transverse projection of the transition dipole moment relative to the observer's reference frame.

When viewed transversely (along the $x$ -axis), the projection of the $\hat{i} + i\hat{j}$ dipole moment leaves only the $y$ -component, resulting in linear polarization.

Furthermore, students frequently misapply classical rigid-body kinematics to the electron's angular momentum, incorrectly calculating a non-zero projection for $L_x$ (such as $\hbar/\sqrt{2}$ derived from $L^2 - L_z^2$ ) instead of recognizing that exact knowledge of $L_z$ forces $\langle L_x \rangle = 0$ due to the Heisenberg uncertainty principle.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Atoms - Class 12
Concept	Photon Polarization and Selection
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free