91影视

(a)We can represent the energy unit with (鈥�) and the partition with (mid): So, let's say we have three oscillators$N=3$and four units of energy$q=4$, the possible microstates are:

Using the general formula for Einstein solid multiplicity with $N=3$and$q=4$

$\begin{array}{c}\mathrm{惟}(N,q)=\left(\begin{array}{c}q+N鈭�1\\ q\end{array}\right)=\frac{(q+N鈭�1)!}{q!(N鈭�1)!}\\ \mathrm{惟}(3,4)=\left(\begin{array}{c}4+3鈭�1\\ 4\end{array}\right)=\frac{(4+3鈭�1)!}{4!(3鈭�1)!}=15\end{array}$

(b) For three oscillators$N=3$and five units of energy $q=5$, the possible microstates are:

Using the general formula for Einstein solid multiplicity with $N=3$and$q=5$:

$\begin{array}{c}\mathrm{惟}(N,q)=\left(\begin{array}{c}q+N鈭�1\\ q\end{array}\right)=\frac{(q+N鈭�1)!}{q!(N鈭�1)!}\\ \mathrm{惟}(3,5)=\left(\begin{array}{c}5+3鈭�1\\ 5\end{array}\right)=\frac{(5+3鈭�1)!}{5!(3鈭�1)!}=21\end{array}$

(c) For three oscillators $N=3$and five units of energy $q=6$, the possible microstates are:

Using the general formula for Einstein solid multiplicity with $N=3$and $a=6$:

$\begin{array}{c}\mathrm{惟}(N,q)=\left(\begin{array}{c}q+N鈭�1\\ q\end{array}\right)=\frac{(q+N鈭�1)!}{q!(N鈭�1)!}\\ \mathrm{惟}(3,6)=\left(\begin{array}{c}6+3鈭�1\\ 6\end{array}\right)=\frac{(6+3鈭�1)!}{6!(3鈭�1)!}=28\end{array}$

(d) For four oscillators$N=4$and two units of energy$q=2$the possible microstates are:

Using the general formula for Einstein solid multiplicity with $N=4$and $q=2$

$\begin{array}{c}\mathrm{惟}(N,q)=\left(\begin{array}{c}q+N鈭�1\\ q\end{array}\right)=\frac{(q+N鈭�1)!}{q!(N鈭�1)!}\\ \mathrm{惟}(4,2)=\left(\begin{array}{c}4+2鈭�1\\ 2\end{array}\right)=\frac{(4+2鈭�1)!}{2!(4鈭�1)!}=10\end{array}$

(e) For four oscillators $N=4$and three units of energy$q=3$the possible microstates are:

Using the general formula for Einstein solid multiplicity with$N=4$and$q=3$

$\begin{array}{c}\mathrm{惟}(N,q)=\left(\begin{array}{c}q+N鈭�1\\ q\end{array}\right)=\frac{(q+N鈭�1)!}{q!(N鈭�1)!}\\ \mathrm{惟}(4,3)=\left(\begin{array}{c}4+3鈭�1\\ 3\end{array}\right)=\frac{(4+3鈭�1)!}{3!(4鈭�1)!}=20\end{array}$

(f)Using the general multiplicity of Einstein solid formula for N oscillators$N=1$and q units of energy $q=q$:

$\mathrm{惟}(N,q)=\left(\begin{array}{c}q+N鈭�1\\ q\end{array}\right)=\frac{(q+N鈭�1)!}{q!(N鈭�1)!}$

$\mathrm{惟}(1,q)=\left(\begin{array}{c}1+q鈭�1\\ q\end{array}\right)=\frac{(1+q鈭�1)!}{q!(1鈭�1)!}=\frac{q!}{q!}$

$\mathrm{惟}(1,q)=1$

(g)Using the general multiplicity of Einstein solid formula for N oscillators)$N=N$and one units of energy $q=1$:

$\mathrm{惟}(N,q)=\left(\begin{array}{c}q+N鈭�1\\ q\end{array}\right)=\frac{(q+N鈭�1)!}{q!(N鈭�1)!}$

$\mathrm{惟}(N,1)=\left(\begin{array}{c}N+1鈭�1\\ 1\end{array}\right)=\frac{(N+1鈭�1)!}{1!(N鈭�1)!}=\frac{N(N鈭�1)!}{(N鈭�1)!}$

$\mathrm{惟}(N,1)=N$