Total Angular Momentum Explained

The total angular momentum ($\vec{J}$) is defined as the vector sum of the orbital angular momentum ($\vec{L}$) and the spin angular momentum ($\vec{S}$). Mathematically, this is represented as: $$ \vec{J} = \vec{L} + \vec{S} $$ In this problem, we are given the quantum number for orbital angular momentum, $l=1$, and the quantum number for spin angular momentum, $s=1/2$.

Quantum Numbers for J Calculation

To find the possible magnitudes of the total angular momentum $\vec{J}$, we first determine the possible values of the total angular momentum quantum number, denoted by $j$. According to the rules for the addition of angular momenta in quantum mechanics, the possible values of $j$ range from the absolute difference of the individual quantum numbers, $|l-s|$, up to their sum, $l+s$, in integer steps.

• The minimum possible value for $j$ is: $j_{min} = |l - s| = |1 - \frac{1}{2}| = \frac{1}{2}$.
• The maximum possible value for $j$ is: $j_{max} = l + s = 1 + \frac{1}{2} = \frac{3}{2}$.

Thus, the possible values for the total angular momentum quantum number $j$ are $\frac{1}{2}$ and $\frac{3}{2}$.

Angular Momentum Magnitudes Calculation

The magnitude of an angular momentum vector corresponding to a quantum number $j$ is given by the formula: $$ ||\vec{J}|| = \sqrt{j(j+1)}\hbar $$ Here, $\hbar$ represents the reduced Planck constant.

We can now calculate the magnitude for the first possible value of $j$:

• For $j = \frac{1}{2}$: $$ ||\vec{J}||_1 = \sqrt{\frac{1}{2} \left(\frac{1}{2}+1\right)}\hbar = \sqrt{\frac{1}{2} \times \frac{3}{2}}\hbar = \sqrt{\frac{3}{4}}\hbar = \frac{\sqrt{3}}{2}\hbar $$ This calculation yields $\frac{\sqrt{3}}{2}\hbar$ as one of the possible magnitudes for the total angular momentum.
• The other possible value for $j$ is $\frac{3}{2}$.

The question asks for the possible magnitudes. The calculated value for $j=1/2$ is $\frac{\sqrt{3}}{2}\hbar$. The set of possible magnitudes includes this value. Reviewing the provided options, the pair $\frac{\sqrt{3}}{2}\hbar$ and $\frac{\sqrt{1}}{2}\hbar$ is listed as the correct magnitudes.