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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 on a given axis can never equal the magnitude of the angular momentum vector. This is based on the quantum mechanical quantization of angular momentum, where the value of the component of angular momentum on an axis is bounded by \(-j\hbar\) to \(j\hbar\), while the total angular momentum is given by \(\sqrt{j(j+1)}\hbar\), which is always more than \(j\hbar\).

Step by step solution

01

Understanding Quantum Mechanical Angular Momentum

To answer this question, you must first understand the quantum mechanical angular momentum. In quantum mechanics, the angular momentum is always quantized, and it takes only specific values. The total angular momentum \(J\), which is the magnitude of the angular momentum vector, is given by \(\sqrt{j(j+1)}\hbar\), where \(j\) is a non-negative integer or half-integer, and \(\hbar\) is the reduced Planck's constant.
02

Understanding Components of Angular Momentum

Next, you need to understand the components of angular momentum. When we measure the component of the angular momentum on a given axis (usually z-axis), we measure the angular momentum along that axis. This angular momentum \(m_J\) has discrete values given by \(m_J\hbar\), where \(m_J\) is an integer such that \(-j \leq m_J \leq j\). In other words, if we measure the angular momentum along an axis, we can get values from \(-j\hbar\) to \(j\hbar\).
03

Comparing Component and Magnitude

Now, compare the maximum magnitude of the angular momentum component on a given axis, which is \(j\hbar\), with the total angular momentum given by \(\sqrt{j(j+1)}\hbar\). As can be seen, \(\sqrt{j(j+1)}\hbar > j\hbar\). This is because \(\sqrt{j(j+1)}\) is always greater than \(j\), for all positive \(j\). So, the component of a quantized angular momentum measured on a given axis is always less than the magnitude of the angular momentum vector.

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