Quantum numbers and angular momentum components Decide whether the statement is correct: “The magnetic quantum number determines the component of angular momentum along a prescribed (quantization) direction.”

Introduction / Context:
Atomic states are labeled by quantum numbers. Among them, the magnetic quantum number (commonly m_l for orbital angular momentum) is directly connected to the projection of angular momentum along a chosen axis, usually the z-axis. Understanding this mapping is fundamental for topics like Zeeman splitting and selection rules in spectroscopy.

Given Data / Assumptions:

• Single-electron or many-electron atoms described by orbital quantum number l and magnetic quantum number m_l.
• A fixed quantization axis (typically z) is defined by an external magnetic field or convention.
• Non-relativistic angular momentum algebra applies.

Concept / Approach:
The magnetic quantum number m_l takes integer values from −l to +l. It specifies the eigenvalue of the angular momentum operator's z-component. The total magnitude of orbital angular momentum depends only on l, while the component along the quantization axis depends on m_l. This distinction explains spectral line splitting under magnetic fields (Zeeman effect), where different m_l states shift differently.

Step-by-Step Solution:
Angular momentum magnitude L depends on l via L = sqrt(l(l+1)) * ħ (conceptual relation).The quantized component along z is L_z = m_l * ħ.Therefore, m_l determines the specific allowed component of angular momentum along the chosen axis and explains directional properties of atomic orbitals in fields.

Verification / Alternative check:
Experimental Zeeman splitting patterns confirm that energy shifts are proportional to m_l (and g-factors), matching the interpretation that m_l labels the axial component.

Why Other Options Are Wrong:
“False” ignores core quantum principles. “True only for spin” is incorrect; both orbital (m_l) and spin (m_s) have axial projections. Statements claiming it sets only the magnitude or only the energy misunderstand the roles of l and m_l.

Common Pitfalls:

• Confusing l (magnitude) with m_l (projection).
• Forgetting that external fields define the preferred axis for projection quantization.

Final Answer:
True