  An ensemble of spins oriented along the $z$ direction exhibits nonzero
fluctuation in the transverse ($x$- and $y$-) components of the spin angular
momentum in accordance with the uncertainty principle. When the spins obey a
spin temperature distribution, the mean square fluctuation in $S_{x}$ can be
calculated by ensemble average of the expectation value of $S_{x}^{2}$ with
respect to an equilibrium density matrix $\rho =e^{\beta S_{z}}/Z$. The
fluctuation can also be calculated from the fluctuation-dissipation theorem as
has been done in literature in the context of NMR spin noise. For spin 1/2
particles in the high temperature limit, appropriate for many NMR experiments,
the two methods are known to produce the same, temperature-independent spin
noise. We show that inclusion of the zero-point fluctuation term in the
original Nyquist relation extends this correspondence to an arbitrary spin
temperature for any spin $S$. This indicates that the uncertainty
principle-limited spin projection noise can be viewed as a result of the
zero-point fluctuation in the thermal bath coupled to the spins.
