Downloaded from rspa. royalsocietypublishing org on March 11, 2010 718 P.A. M. Dirac. of the permutations is reducible to a linear function of them. Any permutation A permutation P, like any other dynamical variable, can be represented by a matrix. If we take the representation in which the g's are diagonal, P will be represented by a matrix, whose general element may be written (q1 9n)=(glIa) for brevity. This matrix must satisfy (q'Pq)dq"ψ(q")=P(q)=ψ(Pq) and hence (q|P|q")=8(Pq-q) (1) We are using the notation d(e), where m is short for a set of variables w1, wgy 3,…, to denote δ(x)=8(x1)8(2)8(3) which vanishes except when each of the as vanishes. With this notation we 8(Pq-g)=8(q-P-1y") since the condition that the left- hand side shall not vanish, which is that the q"s shall be given by applying the permutation P to the s, is the same as the condition that the right-hand side shall not vanish, which is that the 's shall be given by applying the permutation P-l to the s. Thus we have an alternative expression for the matrix representing (q|P|q")=8(-P-q") The conjugate complex of any dynamical variable is given when one writes -i for i in the matrix representing that variable and also interchanges the rows with the columns. Thus we find for the conjugate complex of a permuta- tion P, with the help of (2)and(1) (gp|q")=("P|g)=8(g"-P-g) =(q|P-1q") P=P-1 Thus a permutation is not in general a real variable, its conjugate complex ual to its Any permutation of the numbers 1, 2, 3, ., n may be expressed in the cyclic otation, e.g., for n=8 Pa=(143)(27)(58)(6)
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Downloaded from rspa. royalsocietypublishing org on March 11, 2010 Quantum Mechanics of Many-Electron Systems in which each number is to be replaced by the succeeding number in a bracket unless it is the last in a bracket, when it is to be replaced by the first in that bracket. Thus p. changes the numbers 12345678 into 47138625 The type of any permutation is specified by the partition of the number n which is provided by the number of numbers in each of the brackets. Thus the type of Pa is specified by the partition=3+2+2+1. Permutations of the same type, i. e, corresponding to the same partition, we shall call simila (The usual language of group theory is to call them conjugate. Thus, for cample, Pa in(3)is similar 871)(35)(46)(2 The whole of the n! possible permutations may be divided into sets of similar permutations, each such set being called a class. The permutation P,=1 forms a class by itself. Any permutation is similar to its reciprocal When two permutations Pa and Pb are similar, either of them Pb may be obtained by making a certain permutation P in the other Pa. Thus, in our example (8),(4)we can take P to be the permutation that changes 14327586 into 87135462, i.e., the permutation (18623)(475) We then have the algebraic relation between P, and P P= PPaP To verify this, we observe that the product Pay of Pa with any wave function yis changed into Pay if one applies the permutation P to the Pa in the product but not to the 4. If we multiply the product by P on the left we are applying this permutation to both the Pa and the y, so that we must insert another factor P- between the Pa and theψ giving us PPaP-lψ to equate to Pbψ Equation(5)is the general formula showing when two permutations Pa and Po are similar. Of course P is not uniquely determined when Pa and Pb are given, but the existence of any P satisfying (5)is sufficient to show that Pa are similar 83. Permutations as Constants of the Mo We now introduce a Hamiltonian H to describe the motion of the system so that any stationary state of energy H'is represented by a wave function y satisfying H=Hψ a which H is regarded as an operator. This Hamiltonian can be an arbitrary function of the dy variables provided it is symmetrical between all the
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