  Consider a quantum system $S$ weakly interacting with a very large but finite
system $B$ called the heat bath, and suppose that the composite $S\cup B$ is in
a pure state $\Psi$ with participating energies between $E$ and $E+\delta$ with
small $\delta$. Then, it is known that for most $\Psi$ the reduced density
matrix of $S$ is (approximately) equal to the canonical density matrix. That
is, the reduced density matrix is universal in the sense that it depends only
on $S$'s Hamiltonian and the temperature but not on $B$'s Hamiltonian, on the
interaction Hamiltonian, or on the details of $\Psi$. It has also been pointed
out that $S$ can also be attributed a random wave function $\psi$ whose
probability distribution is universal in the same sense. This distribution is
known as the "Scrooge measure" or "Gaussian adjusted projected (GAP) measure";
we regard it as the thermal equilibrium distribution of wave functions. The
relevant concept of the wave function of a subsystem is known as the
"conditional wave function". In this paper, we develop analogous considerations
for particles with spin. One can either use some kind of conditional wave
function or, more naturally, the "conditional density matrix", which is in
general different from the reduced density matrix. We ask what the thermal
equilibrium distribution of the conditional density matrix is, and find the
answer that for most $\Psi$ the conditional density matrix is (approximately)
deterministic, in fact (approximately) equal to the canonical density matrix.
