Question

The minimum value of angular momentum by coupling three angular momenta \(1\), \(\frac{3}{2}\) and \(\frac{5}{2}\) is:

• A. \(-5\)
• B. \(\frac{1}{2}\)
• C. \(0\)
• D. \(1\)

Hint:

When coupling three angular momenta, a total combined value of \(J=0\) is possible if and only if one of the intermediate coupled values \(J_{12}\) matches the third value \(j_3\) exactly. Here, since \(J_{12}\) can equal \(\frac{5}{2}\) and \(j_3 = \frac{5}{2}\), they can cancel each other out perfectly to give 0.

Answer 1

The Correct Option is C

Concept: In quantum mechanics, when coupling multiple angular momenta vectors, the allowed values for the total combined angular momentum quantum number \(J\) are determined by the triangle inequality rules. For two angular momenta \(j_1\) and \(j_2\), the total angular momentum \(J_{12}\) ranges in integer steps from a minimum of \(|j_1 - j_2|\) to a maximum of \(j_1 + j_2\): \[ |j_1 - j_2| \leq J_{12} \leq j_1 + j_2 \] To couple three angular momenta, we first couple any two of them together, and then couple the resulting intermediate states to the third angular momentum.

Step 1: Coupling the first two angular momenta.
Let the three given angular momenta values be: \[ j_1 = 1, \quad j_2 = \frac{3}{2}, \quad j_3 = \frac{5}{2} \] Let us first couple \(j_1\) and \(j_2\) to find the allowed intermediate total angular momentum values, \(J_{12}\): \[ |1 - \frac{3}{2}| \leq J_{12} \leq 1 + \frac{3}{2} \] \[ \frac{1}{2} \leq J_{12} \leq \frac{5}{2} \] Since the values must change in integer steps, the allowed intermediate quantum states are: \[ J_{12} = \frac{1}{2}, \, \frac{3}{2}, \, \frac{5}{2} \]

Step 2: Coupling the intermediate states \(J_{12}\) with the third angular momentum \(j_3\).
To find the absolute minimum overall combined angular momentum \(J_{\text{total}}\), we look at the lowest value produced by applying the lower bound of the triangle inequality to each intermediate state combined with \(j_3 = \frac{5}{2}\): \[ J_{\text{min}} = \min \left( |J_{12} - j_3| \right) \] Let us evaluate this absolute lower limit for each possible choice of \(J_{12}\):
• For \(J_{12} = \frac{1}{2}\): \[ |J_{12} - j_3| = \left|\frac{1}{2} - \frac{5}{2}\right| = \left|-\frac{4}{2}\right| = 2 \]
• For \(J_{12} = \frac{3}{2}\): \[ |J_{12} - j_3| = \left|\frac{3}{2} - \frac{5}{2}\right| = \left|-\frac{2}{2}\right| = 1 \]
• For \(J_{12} = \frac{5}{2}\): \[ |J_{12} - j_3| = \left|\frac{5}{2} - \frac{5}{2}\right| = 0 \]

Step 3: Determining the absolute minimum value.
Comparing the lower bounds calculated above: \[ \text{Possible lower bounds} = \{2, 1, 0\} \] The absolute minimum possible value among these outcomes is exactly 0. This means the three angular momentum vectors can align in a way that completely cancels out their net total angular momentum. This corresponds to option (3).