Ⅸ SYSTEMS CONTAINING SEVERAL SIMILAR PARTICLES
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54. Symmetrical and antisymmetrical states

Ir a system in atomic physics contains a number of particles of the same kind, e.g. a number of electrons, the particles are absolutely indistinguishable one from another. No observable change is made when two of them are interchanged. This circumstance gives rise to some curious phenomena in quantum mechanics having no analogue in the classical theory, which arise from the fact that in quantum mechanics a transition may occur resulting in merely the interchange of two similar particles, which transition then could not be detected by any observational means. A satisfactory theory ought, of course, to count two observationally indistinguishable states as the same state and to deny that any transition does occur when two similar particles exchange places. We shail find that it is possible to reformu- late the theory so that this is so. Suppose we have a system containing x similar particles. We may take as our dynamical variables a set of variables ¢, describing the first particle, the corresponding set €, describing the second particle, and so on up to the set £, describing the nth particle. We shall then have the &,’s commuting with the €,’3 for r 4s. (We may require certain extra variables, describing what the system consists of in addition to the » similar particles, but it is not necessary to mention these explicitly in the present chapter.) The Hamiltonian describing the motion of the system will now be expressible as a function of the £,,£5,-5€, The fact that the particles are similar requires that the Hamiltonian shall be a symmetrical function of the &,, &o,...,€); Le. it shall remain unchanged when the sets of variables é, are interchanged or permuted in any way. This condition must hold, no matter what perturbations are applied to the system. In fact, any quantity of physical significance must be a symmetrical function of the é’s. Let |a,>, |b,>,... be kets for the first particle considered as a dynami- cal system by itself. There will be corresponding kets |a,), |b,>,... for the second particle by itself, and so on. We can get a ket for the assembly by taking the product of kets for each particle by itself, for example |@y>[bg>[€3>-+|Fn> = 1812 C3.+-In> (1) 208 SYSTEMS CONTAINING SEVERAL SIMILAR PARTICLES § 54 say, according to the notation of (65) of § 20. The ket (1) corresponds to a special kind of state for the assembly, which may be described by saying that each particle is in its own state, corresponding to its own factor on the left-hand side of (1). The general ket for the assembly is of the form of a sum or integral of kets like (1), and corresponds to a state for the assembly for which one cannot say that each particle is in its own state, but only that each particle is partly in several states, in a way which is correlated with the other particles being partly in several states. If the kets |a,), |b,>,... are a set of basic kets for the first particle by itself, the kets |a,>, |b,>,... will be a set of basic kets for the second particle by itself, and so on, and the kets (1) will be a set of basic kets for the assembly. We call the repre- sentation provided by such basic kets for the assembly a symmetrical representation, as it treats all the particles on the same footing. In (1) we may interchange the kets for the first two particles and get another ket for the assembly, namely 1by>|!4q>{63>.-19n> = [by dy g.--Gn>- More generally, we may interchange the role of the first two particles in any ket for the assembly and get another ket for the assembly. The process of interchanging the first two particles is an operator which can be applied to kets for the assembly, and is evidently a linear operator, of the type dealt with in § 7. Similarly, the process of interchanging any pair of particles is a linear operator, and by repeated applications of such interchanges we get any permutation of the particles appearing as a linear operator which can be applied to kets for the assembly. A permutation is called an even permutation or an odd permutation according to whether it can be built up from an even or an odd number of interchanges. A ket for the assembly |X is called symmetrical if it is unchanged by any permutation, i.e. if P\|X> = |X) (2) for any permutation P. It is called antisymmetrical if it is unchanged by any even permutation and has its sign changed by any odd permutation, i.e. if P\|X) = +|X), (3) the + or — sign being taken according to whether P is even or odd. The state corresponding te a symmetrical ket is called a symmetrical state, and the state corresponding to an antisymmetrical ket is called an antisymmetrical state. In a symmetrical representation, the repre- § 54 SYMMETRICAL AND ANTISYMMETRICAL STATES 209 sentative of a symmetrical ket is a symmetrical function of the variables referring to the various particles and the representative of an antisymmetrical ket is an antisymmetrical function. In the Schrédinger picture, the ket corresponding to a state of the assembly will vary with time according to Schrédinger’s equation of motion. If it is initially symmetrical it must always remain sym- metrical, since, owing to the Hamiltonian being symmetrical, there is nothing to disturb the symmetry. Similarly if the ket is initially antisymmetrical it must always remain antisymmetrical. Thus a state which is initially symmetrical always remains symmetrical and a state which is initially antisymmetrical always remains antisym- metrical. In consequence, it may be that for a particular kind of particle only symmetrical states occur in nature, or only anti- symmetrical states occur in nature. If either of these possibilities held, it would lead to certain special phenomena for the particles in question. ; Let us suppose first that only antisymmetrical states occur in nature. The ket (1) is not antisymmetrical and so does not corre- spond to a state occurring in nature. From (1) we can in general form an antisymmetrical ket by applying all possible permutations to it and adding the results, with the coefficient —1 inserted before those terms arising from an odd permutation, so as to get » +P\a, bg C3---In>> (4) the + or — sign being taken according to whether P is even or odd. The ket (4) may be written as a determinant lay> |@_> |@g> . . - (Gn? [> [62> |e . - - 10. I9> I92> IWsr> - - + IGn> and its representative in a symmetrical representation is a determi- nant. The ket (4) or (5) is not the general antisymmetrical ket, but is a specially simple one. It corresponds to a state for the assembly for which one can say that certain particle-states, namely the states a,b,c,...,g, are occupied, but one cannot say which particle is in which state, each particle being equally likely to be in any state. If 210 SYSTEMS CONTAINING SEVERAL SIMILAR PARTICLES § 54 two of the particle-states a,b,c,...,g are the same, the ket (4) or (5) vanishes and does not correspond to any state for the assembly. Thus two particles cannot occupy the same state. More generally, the occupied states must be all independent, otherwise (4) or (5) vanishes. This is an important characteristic of particles for which only anti- symmetrical states occur in nature. It leads to a special statistics, which was first studied by Fermi, so we shall call particles for which only antisymmetrical states occur in nature fermions. Let us suppose now that only symmetrical states occur in nature. The ket (1) is not symmetrical, except in the special case when all the particle-states a,b,c,...,g are the same, but we can always obtain a symmetrical ket from it by applying all possible permutations to it and adding the results, so as to get p Pla, by C3...9,>- (6) The ket (6) is not the general symmetrical ket, but is a specially simple one. It corresponds to a state for the assembly for which one can say that certain particle-states are occupied, namely the states a, 6,c,...,g, without being able to say which particle is in which state. It is now possible for two or more of the states a,b,c,...,g to be the same, so that two or more particles can be in the same state. In spite of this, the statistics of the particles is not the same as the usual statistics of the classical theory. The new statistics was first studied by Bose, so we shall call particles for which only symmetrical states occur in nature bosons. We can see the difference of Bose statistics from the usual statistics by considering a special case—that of only two particles and only two independent states a and 6 for a particle. According to classical mechanics, if the assembly of two particles is in thermodynamic equilibrium at a high temperature, each particle will be equally likely to be in either state. There is thus a probability 4 of both particles being in state a, a probability } of both particles being in state b, and a probability 4 of one particle being in each state. In the quan- tum theory there are three independent symmetrical states for the pair of particles, corresponding to the symmetrical kets |a,)|a.), |5>|b,>, and |a,>[b,>+ |ag>|6,>, and describable as both particles in state a, both particles in state 6, and one particle in each state respectively. For thermodynamic equilibrium at a high temperature these three states are equally probable, as was shown in § 33, so that § 54 SYMMETRICAL AND ANTISYMMETRICAL STATES 211 there is a probability 4 of both particles being in state a, a probability 4 of both particles being in state b, and a probability 4 of one particle being in each state. Thus with Bose statistics the probability of two particles being in the same state is greater than with classical statistics. Bose statistics differ from classical statistics in the opposite direction to Fermi statistics, for which the probability of two particles being in the same state is zero. In building up a theory of atoms on the lines mentioned at the beginning of § 38, to get agreement with experiment one must assume that two electrons are never in the same state. This rule is known as Pauli’s exclusion principle. It shows us that electrons are fermions. Planck’s law of radiation shows us that photons are bosons, as only the Bose statistics for photons will lead to Planck’s law. Similarly, for each of the other kinds of particle known in physics, there is experi- mental evidence to show either that they are fermions, or that they are bosons. Protons, neutrons, positrons are fermions, «-particles are bosons. It appears that all particles occurring in nature are either fermions or bosons, and thus only antisymmetrical or symmetrical states for an assembly of similar particles are met with in practice. Other more complicated kinds of symmetry are possible mathemati- cally, but do not apply to any known particles. With a theory which allows only antisymmetrical or only symmetrical states for a particu- lar kind of particle, one cannot make a distinction between two states which differ only through a permutation of the particles, so that the transitions mentioned at the beginning of this section disappear. 55. Permutations as dynamical variables We shall now build up a general theory for a system containing n similar particles when states with any kind of symmetry properties are allowed, i.e. when there is no restriction to only symmetrical or only antisymmetrical states. The general state now will not be sym- metrical or antisymmetrical, nor will it be expressible linearly in terms of symmetrical and antisymmetrical states when n > 2. This theory will not apply directly to any particles occurring in nature, but all the same it is useful for setting up an approximate treatment for an assembly of electrons, as will be shown in § 58. We have seen that each permutation P of the n particles is a linear operator which can be applied to any ket for the assembly. Hence we can regard P as a dynamical variable in our system of » particles. 212 SYSTEMS CONTAINING SEVERAL SIMILAR PARTICLES § 55 There are »! permutations, each of which can be regarded as a dynamical variable. One of them, F, say, is the identical permutation, which is equal to unity. The product of any two permutations is a third permutation and hence any function of the permutations is reducible to a linear function of them. Any permutation P has a reciprocal P-1 satisfying PP = PAP =P =1., A permutation P can be applied to a bra , the product must be unchanged, since it is just a number, independent of any order of the particles. Thus (PCX|)P|¥> = and is thus equal to would change it into B,.P,|X>, ie. P,P,|X> = BPX. Hence Pi = P,P, Pz, (13) which expresses the condition for P, and FP, to be similar as an algebraic equation. The existence of any P, satisfying (11) is suffi- cient to show that P, and P, are similar. 56. Permutations as constants of the motion Any symmetrical function V of the dynamical variables of all the particles is unchanged by the application of any permutation P, so P applied to the product V|X> affects only the factor |X), thus PV|X> = VP|X). Hence PV =VP, (12) showing that a symmetrical function of the dynamical variables com- mutes with every permutation. The Hamiltonian is a symmetrical fanction of the dynamical variables and thus commutes with every permutation. It follows that each permutation is a constant of the _ motion. This holds even if the Hamiltonian is not constant. If |Xt is any solution of Schrédinger’s equation of motion, P|X¢) is another. In dealing with any system in quantum mechanics, when we have found a constant of the motion a, we know that if for any state of motion, « initially has the numerical value «’, then it always has this value, so that we can assign different numbers «’ to the different states and so obtain a classification of the states. The procedure is not so straightforward, however, when we have several constants of the motion « which do not commute (as is the case with our permuta- tions P), since we cannot in general assign numerical values for all the a’s simultaneously to any state. Let us first take the case of a system whose Hamiltonian does not involve the time explicitly. The existence of constants of the motion « which do not commute is then a sign that the system is degenerate. This is because, for a 3595.57 | P 214 SYSTEMS CONTAINING SEVERAL SIMILAR PARTICLES § 56 non-degenerate system, the Hamiltonian H by itself forms a complete set of commuting observables and hence, from Theorem 2 of § 19, each of the a’s is a function of H and therefore commutes with any other a. We must now look for a function f of the «’s which has one and the same numerical value f’ for all those states belonging to one energy-level H’, so that we can use f for classifying the energy-levels of the system. We can express the condition for f by saying that it must be a function of H and must therefore commute with every dynamical variable that commutes with H, ie. with every constant of the motion. If the a’s are the only constants of the motion, or if they are a set that commute with all other independent constants of the motion, our problem reduces to finding a function B of the a’s which commutes with all the a’s. We can then assign a numerical value ’ for 8 to each energy-level of the system. If we can find several such functions 8, they must all commute with each other, so that we can give them all numerical values simultaneously. We ob- tain thus a classification of the energy-levels. When the Hamiltonian involves the time explicitly one cannot talk about energy-levels, but * the f’s will still give a useful classification of the states. We follow this method in dealing with our permutations P. We must find a function y of the P’s such that PyP-! = x for every P. It is evident that a possible y is > P,, the sum of all the permutations in a certain class c, i.e. the sum of a set of similar permutations, since > PF, P- must consist of the same permutations summed in a differ- ent order. There will be one such y for each class. Further, there can be no other independent y, since an arbitrary function of the P’s can be expressed as a linear function of them with numerical coefficients, and it will not then commute with every P unless the coefficients of similar P’s are always the same. We thus obtain all the x’s that can be used for classifying the states. It is convenient to define each x as an average instead of a sum, thus Xe = NDP, ' where 7, is the number of P’s in the class c. An alternative expression for x, is Xe = nt Y PP, P-l, (13) P the sum being extended over all the n! permutations P, it being easy to verify that this sum contains each member of the class ¢ the same number of times. For each permutation P there is one x, x(P) say, §56 PERMUTATIONS AS CONSTANTS OF THE MOTION 235 equal to the average of ail permutations similar to P. One of the xs is x(R) = 1. The constants of the motion y,, X2,---, ¥_ Obtained in this way will each have a definite numerical value for every stationary state of the system, in the case when the Hamiltonian does not involve the time explicitly, and also in the general case can be used for classifying the states, there being one set of states for every permissible set of numerical values yj, %2)---, X¥m for the y’s. Since the y’s are always constants of the motion, these sets of states will be exclusive, i.e. transitions will never take place from a state in one set to a state in another. The permissible sets of values x’ that one can give to the y’s are limited by the fact that there exist algebraic relations between the x’s. The product of any two x’s, x, X,. is of course expressible as a linear function of the P’s, and since it commutes with every P it must be expressible as a linear function of the y’s, thus { i} [ A) Xp Xq = 8% Xi 8 Xo Tn Xm: (14) where the a’s are numbers. Any numerical values x’ that one gives to the y’s must be eigenvalues of the y’s and must satisfy these same algebraic equations. For every solution x’ of these equations there is one exclusive set of states. One solution is evidently y,, = 1 for every Xp, giving the set of symmetrical states. A second obvious solution, giving the set of antisymmetrical states, is y, = --1, the + or — sign being taken according to whether the permutations in the class p are even or odd. The other solutions may be worked out in any special case by ordinary algebraic methods, as the coefficients @ in (14) may be obtained directly by a consideration of the types of permutation to which the y’s concerned refer. Any solution is, apart from a certain factor, what is called in group theory a character of tke group of permutations. The y’s are all real dynamical variables, since each P and its conjugate complex P-! are similar and will occur added together in the definition of any x, so that the x’’s must be all real numbers. The number of possible solutions of the equations (14) may easily be determined, since it must equal the number of different eigen- values of an arbitrary function B of the y’s. We can express B as a linear function of the y’s with the help of equations (14); thus B == by yy tbe xo+--- tlm Xm (i on \ i 216 SYSTEMS CONTAINING SEVERAL SIMILAR PARTICLES § 56 Similarly, we can express each of the quantities B®, B°,..., B™ as a linear function of the y’s. From ‘the m equations thus obtained, ‘together with the equation x(P,) = 1, we can eliminate the m un- knowns X14, Xo.» Xm» Obtaining as result an algebraic equation of degree m for B, Bre, Bt +-¢, Bre, == 0, The m solutions of this equation give the m possible eigenvalues for B, each of which will, according to (15), be a linear function of 6,, b,..., b, whose coefficients are a permissible set of values x1, Xos--63 Xm The sets of values x’ thus obtained must be all different, since if there were fewer than m different permissible sets of values y’ for the x’s, there would exist a linear function of the y’s every one of whose eigenvalues vanishes, which would mean that the linear function itself vanishes and the y’s are not linearly independent. Thus the number of permissible sets of numerical values for the y’s is just equal tom, which is the number of classes of permutations or the number of partitions of x. This number is therefore the number of exclusive sets of states. All dynamical variables of physical importance and all observable quantities are symmetrical between the particles and thus commute with all the P’s. Thus the only functions of the P’s of physical importance are the y’s. The states corresponding to |y’> and to S(P)\x’>, where |y’> is any eigenket of the y’s belonging to the eigen- values x’ and f(P) is any function of the P’s such that f(P)|y> < 9, are observationally indistinguishable and are thus physically equiva- lent. There is a definite number, n(x’) say, of independent kets which can be formed by multiplying |y’> by functions of the P’s, which number depends only on the y”s. It is the number of rows and columns in a matrix representation of the P’s in which each x is equal to x’. If |x’> corresponds to a stationary state, n(x‘) will be its degree of degeneracy (so far as concerns degeneracy caused by the symmetry between the particles). This degeneracy cannot beremoved by any perturbation that is symmetrical between the particles. 57. Determination of the energy-levels Let us apply the perturbation method of § 43 and make a first-order calculation of the energy-levels in the case when the Hamiltonian does not involve the time explicitly. We suppose that for our unper- turbed stationary states of the assembly each of the similar particles has its own individual state. With n particles, we shall have n of § 57 DETERMINATION OF THE ENERGY-LEVELS 217 these states, corresponding to kets Jat, |a*>,..., |a”> say, which we assume for the present to be all orthogonal. The ket for the assembly is then [XD = foed> odd... fo®, (16) like (1) with a, «?,... instead of a, b,.... If we apply any permutation P to it we get another ket P|X> = [ap> 05>... lay (17) say, 7, 8... 2 being some permutation of the numbers 1, 2,..., ”, corresponding to another stationary state of the assembly with the same energy. There are thus altogether m! unperturbed states with this energy, if we assume there are no other causes of degeneracy. According to the method of § 43 when the unperturbed system is degenerate, we must consider those elements of the matrix represent- ing the perturbing energy V that refer to two states with the same energy, i.e. those of the type (X|P, VB,|X>. These will form a matrix with 2! rows and columns, whose eigenvalues are the first-order corrections in the energy-levels. We must now introduce another kind of permutation operator which can be applied to kets of the form (17), namely a permutation which acts on the indices of the «’s. We denote such a permutation operator by P*. The essential difference between the P’s and the P’s may be seen in the following way. Let us consider a permutation in the general sense, say that consisting of the interchange of 2 and 3. This may be interpreted either as the interchange of the objects 2 and 3 or as the interchange of the objects in the places 2 and 3, these two operations producing in general quite different results. The first of these interpretations is the one that gives the operators P, the objects concerned being the similar particles. A permutation P can be applied to an arbitrary ket for the assembly. A permutation with the second interpretation has a meaning, however, only when applied to a ket of the form (17), for which each of the particles is in a ‘place’ specified by an «, or to a sum of kets of the form (17). A permutation P may be considered as an ordinary dynamical variable. A permuta- tion P* may be considered as a dynamical variable in a restricted sense, valid when one is dealing only with states obtainable by super- position of the various states (17). This is the case for our present perturbation problem. We can form algebraic functions of the P* which will be other operators applicable to kets of the form (17). In particular we can 218 SYSTEMS CONTAINING SEVERAL SIMILAR PARTICLES § 57 form x(P2%), the average of all P*’s in a certain class c. This must equal y(P,), the average of the permutation operators P in the same class, since the total set of all permutations in a given class must evidently be the same whether the permutations are applied to the particles or to the places the particles arein. Any P commutes with any P%, ie. P, Px = P2P,, (18) By labelling the «’s by the same numbers 1, 2, 3,..., » which label the particles, we set up a one-one correspondence between the «’s and the particles, so that given any permutation PF, applying to the par- ticles, we can give a meaning to the same permutation P% applying to the a’s. This meaning is such that, for the ket |X given by (16), PGL|X> = |X). (19) Since the various kets |a1>, |o*),... are orthogonal, |X> and P|X) are orthogonal unless P = 1. It follows that, for any coefficients cp, D2 6p = Cp, (20) provided |X> is normalized, the summation being over all the n! permutations P or P*, with P, fixed. Now define VY by . Vp = (X|VP|X). (21) We then have, for any two permutations P, and P,, (AP, VEX = = Ve. p, = 2 Vp (X| Ph, Py|X> with the help of (20). From (18) this gives (XP, VE,|X> = & Ye KE, P*B,|X>. (22) We may write this result as VS Pe, (23) B where the sign means an equation in a restricted sense, the operators on the two sides being equal so long as they are used only with kets of the form P|X> and their conjugate imaginary bras. The formula (23) shows that the perturbing energy V is equal, in the restricted sense, to a linear function of the permutation operators P* with coefficients Vp given by (21). The restricted sense is adequate for the calculation of the first-order correction in the energy-levels, as this calculation involves only those matrix elements of V given by § 57 DETERMINATION OF THE ENERGY-LEVELS 219 (22). The formula (23) is a very convenient one because the expression on its right-hand side is easily handled. As an example of an application of (23) we shall determine the average energy of all those states, arising from the unperturbed state (16), that belong to one exclusive set. This requires us to calculate the average eigenvalue of V for those states (17) for which the y’s have specified numerical values x’. Now the average eigenvalue of P* for any of these states equals that of P*P%(P%)-1 for arbitrary P” and thus equals that of ni > P¢p% P%)-1, which is y‘(P%) or x'(P,). Hence the average eigenvalue of V is } Vpx'(P). A similar P method could be used for calculating the average eigenvalue of any function of V, it being necessary only to replace each P* by x’(P) to perform the averaging. The number of energy-levels in an exclusive set x = x’ that arise from a given state of the unperturbed system is equal to the number of eigenvalues of the right-hand side of (23) that are consistent with the equations y = x’. This number is the number n(x’) introduced at the end of the preceding section, and is thus just the degree of degeneracy of the states in this set. We have assumed that the individual kets |o!), {a”),... which deter- mine the unperturbed state according to (16) are all orthogonal. The theory can easily be extended to the case when some of these kets are equal, any two that are not equal being still restricted to be orthogonal. We now have some permutations P* such that P*|X> = |X), namely those permutations which involve only interchanges of equal «’s. Equation (20) will now hold if the summation is extended only over those P’s which make-P*|X> different. With this change in the meaning of 2 all the previous equations still hold, including the result (23). For the present |X) there will be restrictions on the possible numerical values of the y’s, e.g. they cannot have those values corresponding to |X) being antisymmetrical. 58. Application to electrons Let us consider the case when the similar particles are electrons. This requires, according to Pauli’s exclusion principle discussed in § 54, that we take into account only the antisymmetrical states. It is now necessary to make explicit reference to the fact that electrons have spins, which show themselves through an angular momentum 220 SYSTEMS CONTAINING SEVERAL SIMILAR PARTICLES § 58 and a magnetic moment. The effect of the spin on the motion of an electron in an electromagnetic field is not very great. There are additional forces on the electron due to its magnetic moment, requiring additional terms in the Hamiltonian. The spin angular momentum does not have any direct action on the motion, butit comes into play when there are forces tending to rotate the magnetic moment, since the magnetic moment and angular momentum are constrained to be always in the same direction. In the absence of a strong magnetic field these effects are all small, of the same order of magni- tude as the corrections required by relativistic mechanics, and there would be no point in taking them into account in a non-relativistic theory. Theimportance of the spin lies not in these small effects on the motion of the electron, but in the fact that it gives two internal states to the electron, corresponding to the two possible values of the spin component in any assigned direction, which causes a doubling in the number of independent states of an electron. This fact has far-reaching consequences when combined with Pauli’s exclusion principle. In dealing with an assembly of electrons we have two kinds of dynamical variables. The first kind, which we may call the orbital variables, consists of the coordinates x, y, z of all the electrons and their conjugate momenta p,, p,, p,. The second kind consists of the spin variables, the variables o,, o,, o,, a8 introduced in § 37, for all the electrons. These two kinds of variables belong to different degrees of freedom. According to §§ 20 and 21, a ket fixing the state of the whole system may be of the form |.4>|B>, where |A) is a ket referring to the orbital variables alone and {B) is a ket referring to the spin variables alone, and the general ket fixing a state of the whole system is a sum or integral of kets of this form. This way of looking at things enables us to introduce two kinds of permutation operators, the first kind, P* say, applying to the orbital variables only and operating only on the factor |A> and the second kind, P* say, applying only to the spin variables and operating only on the factor |B>. The P2’s and P?’s can each be applied to any ket for the whole system, not merely to certain special kets, like the P®’s of the preceding section. The permutations P that we have had up to the present apply to all the dynamical variables of the particles concerned, so for electrons they will apply to both the orbital and the spin variables. This means that each P, equals the product P, = P= Ps, (24) § 58 APPLICATION TO ELECTRONS 221 We can now see the need for taking the spin variables into account when applying Pauli’s exclusion principle, even if we neglect the spin forces in the Hamiltonian. For any state occurring in nature each P, must have the value +1, according to whether it is an even or an odd permutation, so from (24) PtPo = -l. (25) The theory of the three preceding sections would become trivial if applied directly to electrons, for which each P, = +1. We may, however, apply it to the P* permutations of electrons. The P®’s are constants of the motion if we neglect the terms in the Hamiltonian that arise from the spin forces, since this neglect results in the Hamiltonian not involving the spin dynamical variables o at all. The P’s must then also be constants of the motion. We can now intro- duce new x’s, equal to the average of all of the P*’s in each class, and assert that for any permissible set of numerical values y’ for these y’s there will be one exclusive set of states. Thus there exist exclusive sets of states for systems containing many electrons even when we restrict ourselves to a consideration of only those states that satisfy Pauli’s principle. The exclusiveness of the sets of states is now, of course, only approximate, since the y’s are constants only so long as we neglect the spin forces. There will actually be a small probability for a transition from a state in one set to a state in another. Equation (25) gives us a simple connexion between the P*’s and P°’s, which means that instead of studying the dynamical variables P* we can get all the results we want, e.g. the characters y’, by studying the dynamical variables P’. The P°’s are much easier to study on account of there being only two independent states of spin for each electron. This fact results in there being fewer characters x’ for the group of permutations of the o-variables than for the group of general permutations, since it prevents a ket in the spin variables from being antisymmetrical in more than two of them. The study of the P?’s is made specially easy by the fact that we can express them as algebraic functions of the dynamical variables ¢. Consider the quantity Org = HI O91 Fag Oy1 Fyn tn Set = ELL + (Gy, Fa)}- With the help of equations (50) and (51) of § 37 we find readily that (6, G2)? = (Oy Fng+Fya Cyat Fn Fea)” = 3—2(Gy, Gg), (26) and hence that O48 = U1+2(6,, 03) -+(6y 9)"} = 1. (27) 222 SYSTEMS CONTAINING SEVERAL SIMILAR PARTICLES § 58 Again, we find 1 . L: O12 Ory = NOx + Op. 1024 Oyg Tt yy G9}, 1 Le . Ox2 Oy, = HOn2 + n14 WO y1 Fzg— UG zy Oyo} and hence O12 On1 = Fzq Oy. Similar relations hold for o,, and o,, so that we have O12 5, = 5, Oj, or Oy. 0, OF = a. From this we can obtain with the help of (27) O19 6, O7G' = 94. These commutation relations for O,, with o, and o, are precisely the same as those for P?,, the permutation consisting of the interchange of the spin variables of electrons 1 and 2. Thus we can put —_ Co Or = CPi, where c is a number. Equation (27) shows that c = +1. To deter- mine which of these values for c is the correct one, we observe that the eigenvalues of P?, are 1, 1, 1, —1, corresponding to the fact that there exist three independent symmetrical and one antisymmetrical state in the spin variables of two electrons, namely, with the notation of § 37, the states represented by the three symmetrical functions Folia) Sa(Fz2)s Selo) fp (22), Sod Oe) fp(C22) theo) fo(oz2), and the one antisymmetrical function f,(o24) fg(o22)—fe(G1) fa(ze)- Thus the mean of the eigenvalues of Pf, is 4. Now the mean of the eigenvalues of (o,, 6.) is evidently zero and hence the mean of the eigenvalues of O,, is 3. Thus we must have c = +1, and so we can put Po, = {14+ (,, 52)}. (28) In this way any permutation P? consisting simply of an interchange can be expressed as an algebraic function of the o’s. Any other per- mutation P? can be éxpressed as a product of interchanges and can therefore also be expressed as a function of the o’s. With the help of (25) we can now express the P?’s as algebraic functions of the o’s and eliminate the P°’s from the discussion. We have, since the — sign must be taken in (25) when the permutations are interchanges and since the square of an interchange is unity, Phy = —3{1+(6,, 69)}. (29) The formula (29) may conveniently be used for the evaluation of § 58 APPLICATION TO ELECTRONS 223 the characters y’ which define the exclusive sets of states. We have, for example, for the permutations consisting of interchanges, Xue = X(Pe) = 5(? ae (3,, @) a If we introduce the dynamical variable s to describe the magnitude of the total spin angular momentum, 4 2 s, in units of #, through the formula s(s-+1) = (3 9,4 > @;), r in ayreement with (39) of § 36, we have 23 (6,0) = (Yop ¥o)— S(6 = 48(s-+1)—3 1 4s(s+1)—3n n(n—4)-+4s(s+1) Xia = 5(! : n(n—1) ~~ 2n(n—1) - (9) Thus yj. is expressible as a function of the dynamical variable s and of n the number of electrons. Any of the other y’s could be evaluated on similar lines and would have to be a function of s and only, since there are no other symmetrical functions of all the o dynamical variables which could be involved. There is therefore one set of numerical values x’ for the y’s, and thus one exclusive set of states, for each eigenvalue s’ of s. The eigenvalues of s are Hence 1 1 1 gn, in—lI, én—2, see the series terminating with 0 or §. We see in this way that each of the stationary states of a system with several electrons is an eigenstate of s, the magnitude in units of hs of the total spin angular momentum } > a,, belonging to a definite r eigenvalue s’. For any given s’ there will be 2s’+1 possible values for a component of the total spin vector in any direction and these will correspond to 2s" 1 independent stationary states with the same energy. When we do not neglect the forces due to the spin magnetic moments these 2s’+-1 states will in general be split up into 2s’+1 states with slightly different energies, and will thus form a multiplet of multiplicity 2s’+-1. Transitions in which s’ changes, i.e. transitions from one multiplicity to another, cannot occur when the spin forces are neglected and will have only a small probability of occurrence when the spin forces are not neglected. 224 SYSTEMS CONTAINING SEVERAL SIMILAR PARTICES § 58 We can determine the energy-levels of a system with several electrons to the first approximation by applying the theory of the preceding section with the kets |a’> referring only to the orbital variables and using formula (23). If we consider only the Coulomb forces between the electrons, then the interaction energy V will consist of a sum of parts each referring to only two electrons, which will result in all the matrix elements Vy) vanishing except those for which P* is the identical permutation or is simply an interchange of two electrons. Thus (23) will reduce to Vaht DV. Pte (31) rcs ¥,, being the matrix element referring to the interchange of electrons ry and s. Since the P*’s have the same properties as the P*’s, any function of the P°’s will have the same eigenvalues as the corre- sponding function of the P?’s, so that the right-hand side of (31) will have the same eigenvalues as Wt 2d Veo Pres r V,{1+(e,,6,)}} . (32) r in the orbital variables for two electrons may be the same. Suppose |x and |«?> are the same. Then we must take only those eigenvalues of (31) that are consistent with P% = 1, or those eigenvalues of (32) that are consistent with P%, = 1 or P%, = —1. From (28) this condition gives (¢,,¢,) = —3, so that (o,+¢,)? = 0. Thus the resultant of the two spins «, and a, is zero, which may be interpreted as the spins o, and «, being antiparallel. Thus we may say that two electrons in the same orbital state have their spins antiparaliel. More than two electrons cannot be in the same orbital state.