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Chapter 36: Problem 6

Can the component of a quantized angular momentum measured on a given axis ever equal the magnitude of the angular momentum vector? Explain.

Short Answer

Expert verified
No, the component of a quantized angular momentum measured along any given axis cannot be equal to the magnitude of the angular momentum vector due to the uncertainty principle of quantum mechanics.

Step by step solution

01

Understand the Problem

The exercise is asking whether a component of a quantized angular momentum measured along a given axis can ever be equal to the magnitude of the angular momentum vector. The angular momentum is defined by \( L = \sqrt{l(l+1)}\hbar \) and the component of the angular momentum along any given axis is \( m\hbar \), where \( l \) and \( m \) are quantum numbers. \( m \) can vary from \(-l\) to \(+l\) in integer steps.
02

Apply Quantum Mechanics Principle

According to quantum mechanics, the maximum value of the component of the angular momentum \( m\hbar \) along any of the axes cannot exceed its magnitude \( L = \sqrt{l(l+1)}\hbar \). When you square both these sides, it gives \( m^2\hbar^2 \leq l(l+1)\hbar^2 \). Thus implying \( m^2 \leq l(l+1) \).
03

Provide Explanation

Looking at the inequality \( m^2 \leq l(l+1) \), it is clear that \( m^2 \) cannot be greater than \( l(l+1) \). Hence, the component of the angular momentum along any given direction can never be equal to the actual magnitude of the angular momentum vector. This is due to the principle in quantum mechanics that prohibits simultaneous exact measurements of certain pairs of physical quantities (in this case, angular momentum in different directions).

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