Can be distributed among the oscillators. that is, we want the multiplicity function gn) for the n oscillators. the oscillator multiplicity function is not the same as the spin muli- plicity function found earlier. u(s)/mb gis) log g(s) 0 2.30 3.8 4.79 5.35 5.53 5.35 4.79 3.81 2.30 we begin the analysis by going back to the multiplicity function for a single oscillator, for which gi for all values of the quantunumbers, here identical ton. tosolve the 45 120 210 252 210 120 45 problem of (53) below, we need a function to represent or generate the series (51) 0 all σ run from 0 to oo. here r is just a temporary tool that will us find the result (52) < l. for the problem of n oscillators, the generating function is (53), but does not appear in the final result. the answer is provided we assume figure 1.10 energy levels of the model system of 10 magnetic moments m in a magnetic field b. the levels are labelcd by their s values, where 2s is the spin excess and in + s-5+sis the number of up spins the cnergies u() and multiplicities gis) are showa. for this problem the cnegy levels are spaced cqually, with separation ac-2mb between adjacent levels. (53) because the number of ways a tecan appcat in the n-fold product is precisely the nuanber of ordered ways in which the integer n can be formed as tac sum of n non·negative we observe that example: multiplicity function for harmonic oscillators. the problem of the binaty model system is the simplest problem for which an exact solution for the multiplicity function is known. another exactly soivable problem is the harmonic oscillator, for which the solution was originally given by max planck、the original derivation is often felt to be not cntirely simple. the beginning student nced not worry about this derwation. the modern way i。 do the problem is given in chapter 4 and is simple. (54) the quantum states of a hanmonic oscillator have the energy eigenvalues (49) thus for the system of oscillators, where the quantum number s is a positive integer or zero, and ω is the angular frequency of the oscillator. the number of states is infinite, and the multiplicity of cach is one. now consider a system of n such oscillators, all of the same frequency. we want to find the number of ways in which a given total excitation cnergy n +n 1)! this result will be nceded in solving a problem in the next chapter. (50)