The GATE Grind

GATE 2021 PH – Question 64

Atomic and Molecular Physics · Spin-orbit interaction: L-S and j-j coupling schemes · 2 marks · Numerical answer

An atom’s spin $\mathbf S$ and orbital angular momentum $\mathbf L$ precess about total $\mathbf J$, which precesses about the axis of $\mathbf B_1=2B_0\hat{\mathbf z}$. The field is changed to $\mathbf B_2=B_0(\hat{\mathbf x}+\sqrt2\hat{\mathbf y}+\hat{\mathbf z})$. Given $l=2,s=1/2$, let $\theta$ be the angle between $\mathbf B_1$ and $\mathbf J$ for the largest possible $j$ and $j_z$. Find $\theta$ in degrees, rounded to the nearest integer.

Practise this question in The GATE Grind →

Show answer and explanation

Correct answer: 27 to 93

Explanation

The largest coupled angular momentum is $j=l+s=5/2$, with maximum projection $m_j=j$. The angular-momentum vector has cone angle $\alpha=\cos^{-1}[j/\sqrt{j(j+1)}]=\cos^{-1}\sqrt{5/7}=32.31^\circ$ about its quantization axis. The two field axes are $\cos^{-1}(\mathbf B_1\cdot\mathbf B_2/(|B_1||B_2|))=60^\circ$ apart. If the final precession axis is $\mathbf B_2$, the instantaneous angle to $\mathbf B_1$ varies from $60-32.31=27.69^\circ$ to $60+32.31=92.31^\circ$, depending on precession phase. The wording does not select a phase; the official key accepts 27–93 degrees. That broad range is retained.

More Spin-orbit interaction: L-S and j-j coupling schemes questions

All Spin-orbit interaction: L-S and j-j coupling schemes questions →