Mathematical Methods for Physics and Engineering - Matematica.NET

30.3 PERMUTATIONS AND COMBINATIONSmay imagine the n (distinguishable) objects set out on a table. Each combinationof k objects can then be made by pointing to k of the n objectsinturn(withrepetitions allowed). These k equivalent selections distributed amongst n differentbut re-choosable objects are strictly analogous to the placing of k indistinguishable‘balls’ in n different boxes with no restriction on the number of balls in each box.A particular selection in the case k =7,n = 5 may be symbolised asxxx||x|xx|x.This denotes three balls in the first box, none in the second, one in the third, twoin the fourth and one in the fifth. We therefore need only consider the number of(distinguishable) ways in which k crosses and n − 1 vertical lines can be arranged,i.e. the number of permutations of k + n − 1 objects of which k are identicalcrosses and n − 1 are identical lines. This is given by (30.33) as(k + n − 1)!= n+k−1 C k . (30.36)k!(n − 1)!We note that this expression also occurs in the binomial expansion for negativeinteger powers. If n is a positive integer, it is straightforward to show that (seechapter 1)∞∑(a + b) −n = (−1) k n+k−1 C k a −n−k b k ,k=0where a istakentobelargerthanb in magnitude.◮A system contains a number N of (non-interacting) particles, each of which can be inany of the quantum states of the system. The structure of the set of quantum states is suchthat there exist R energy levels with corresponding energies E i and degeneracies g i (i.e. theith energy level contains g i quantum states). Find the numbers of distinct ways in whichthe particles can be distributed among the quantum states of the system such that the ithenergy level contains n i particles, for i =1, 2,...,R, in the cases where the particles are(i) distinguishable with no restriction on the number in each state;(ii) indistinguishable with no restriction on the number in each state;(iii) indistinguishable with a maximum of one particle in each state;(iv) distinguishable with a maximum of one particle in each state.It is easiest to solve this problem in two stages. Let us first consider distributing the Nparticles among the R energy levels, without regard for the individual degenerate quantumstates that comprise each level. If the particles are distinguishable then the number ofdistinct arrangements with n i particles in the ith level, i =1, 2,...,R, is given by (30.35) asN!n 1 !n 2 ! ···n R ! .If, however, the particles are indistinguishable then clearly there exists only one distinctarrangement having n i particles in the ith level, i =1, 2,...,R . If we suppose that thereexist w i ways in which the n i particles in the ith energy level can be distributed amongthe g i degenerate states, then it follows that the number of distinct ways in which the N1137