GATE 2021 PH – Question 43
Three distinguishable particles each have spin $S=1/2$, $S_z=\pm1/2$ and corresponding magnetic moments $\mu_z=\pm\mu$. In a magnetic field $H$ along $+z$, the total energy is $\mu H$. Let $x$ be the state in which the first spin has $S_z=1/2$. The probability of $x$ and the mean system magnetic moment along $+z$ in state $x$, respectively, are
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Correct answer: (A) $1/3,-\mu/3$
Explanation
With energy $E=-M_zH=\mu H$, the allowed configurations have two down moments and one up moment. There are three equally weighted configurations, according to which particle is up. The first particle is up in one of them, so its probability is $1/3$. All three configurations have total moment $-\mu$, and their mean moment per particle is $-\mu/3$. This gives the official option A. The printed phrase “mean magnetic moment of the system” is imprecise: the total moment is $-\mu$; the options use the mean per particle.