PX284 - E4 - degeneracy

degenerate states are two or more states that have the same energy
considering the spin of a particle: $S = ℏ \sqrt{s (s + 1)}$
the component of $S$ along a particular direction: $S_{z} = ℏ m_{s}$ , where $m_{s}$ is the magnetic spin quantum number
each energy level of a particle with spin, $s$ is $(2 s + 1)$ degenerate
these levels are split by a magnetic field, PX262 - G6c - zeeman effect
eg: for an electron $(s = 1 / 2)$ , there are $(2 s + 1 =) 2$ spin states

Z = \sum_{i} g (E_{i}) \exp (- β E_{i})

where, $g (E_{i})$ is the number of distinct states associated with $E_{i}$

Z = \sum_{i} (2 s + 1) \exp (- β E_{i})

\begin{matrix} E & g (E) \\ Δ & _____ & g_{1} \\ 0 & _____ & g_{2} \end{matrix}

\begin{matrix} Z = g_{1} + g_{2} \exp (- β Δ) \\ \ln Z = \ln (g_{1} + g_{2} \exp (- β Δ)) \\ U = \frac{d}{d β} (\ln Z) = g_{2} Δ \exp (- β Δ \exp (- β Δ)) \end{matrix}