《固体物理学》课程教学资源（课外阅读材料）Dirac关于多电子系统的量子力学处理 Quantum Mechanics of Many-Electron Systems.pdf_大学文库

ownloaded from rspa. royalsocietypublishing org on March 11, 2010 714 Quantum Mechanics of Many-Electron Systems By P. A. M. DIRAC, St John,s College, Cambridge (Communicated by R. H. Fowler, F.R.S.--Received March 12, 1929) The general theory of quantum mechanics is now almost complete, the imperfections that still remain being in connection with the exact fitting in of the theory with relativity ideas. These give rise to difficulties only when high-speed particles are involved, and are therefore of no importance in the con sideration of atomic and molecular structure and ordinary chemical reactions in which it is, indeed, usually sufficiently accurate if one neglects relativity variation of mass with velocity and assumes only Coulomb forces between the various electrons and atomic nuclei. The underlying physical laws necessary for the mathematical theory of a large part of physics and the whole of chemistry are thus completely known, and the difficulty is only that the exact application of these laws leads to equations much too complicated to be soluble. It there- fore becomes desirable that approximate practical methods of applying quantum mechanics should be developed, which can lead to an explanation of the main features of complex atomic systems without too much computation Already before the arrival of quantum mechanics there existed a theory of tomic structure, based on Bohrs ideas of quantised orbits, which was fairly successful in a wide field. To get agreement with experiment it was found necessary to introduce the spin of the electron, giving a doubling in the number of orbits of an electron in an atom. With the help of this spin and Paulis exclusion principle, a satisfactory theory of multiplet terms was obtained when one made the additional assumption that the electrons in an atom all set them- selves with their spins parallel or antiparallel. If s denoted the magnitude of the resultant spin angular momentum, this s was combined vectorially with the resultant orbital angular momentum l to give a multiplet of multiplicity 2s +l The fact that one had to make this additional assumption was, however,a serious disadvantage, as no theoretical reasons to support it could be given It seemed to show that there were large forces coupling the spin vectors of the electrons in an atom, much larger forces than could be accounted for as due to the interaction of the magnetic moments of the electrons. The position was thus that there was empirical evidence in favour of these large forces, but that their theoretical nature was quite unknown The Royal Society is collaborating with JSTOR to digitize, preserve, and extend access to Proceedings of the Royal Society of London. Senes A, Containing Papers of a Mathematical and Physical Character

Downloaded from rspa. royalsocietypublishing org on March 11, 2010 Quantum Mechanics of Many-Electron Systems. 715 The old orbit theory is now replaced by Hartrees method of the self-con sistent field, based on quantum mechanics. The simplifying feature of the old theory, according to which each electron has its own individual orbit, is retained, but the"orbit"is now a quantum-mechanical state of the single electron, represented by a wave function in three dimensions. The only action of one orbit on another is assumed to be that of a static distribution of electricity causing a partial screening of the nucleus. A theoretical justification for Hartree's method, showing that its results must be in approximate agreement with those of the exact Schrodinger equation for the whole system, has been given by Gaunt. f The method, however, suffers from the same limitation as the old orbit theory. It cannot give an explanation of multiplet structure without an extraneous assumption of large forces coupling the spins The solution of this difficulty in the explanation of multiplet structure is provided by the eachange(austausch) interaction of the electrons, which arises owing to the electrons being indistinguishable one from another. Two electrons may change places without our knowing it, and the proper allowance for the possibility of quantum jumps of this nature, which can be made in a treatment of the problem by quantum mechanics, gives rise to the new kind of interaction, The energies involved, the so-called exchange energies, are quite large. In fact it is these exchange energies between electrons in different atoms that give rise to homopolar valency bonds, as shown by Heitler and The application of the new exchange ideas to the problem of multiplet structure has been made by Wigner and Hund. The new theory provides no justification for the assumption that the electrons all set themselves with their spins parallel or antiparallel. In fact it does not allow any meaning to be given to this assumption, since in quantum mechanics the component of the spin angular momentum of an electron in any direction is a g-number with the two eigen-values+yh, so that one cannot in general give a meaning to the direction of the spin of an electron in a given stationary state. What the 4 D. R. Hartree, ' Proc. Camb. Phil. Soc.,' vol. 24, p. 89(1928) tJ. A. Gaunt,'Proe. Camb. Phil. Soo,' vol. 24, p. 328( 1928). It is pointed out by aunt that there does not seem to be any theoretical justification for Hartree's method of aloulating energies and that its extremely good agreement with observation is probably accidental. The somewhat different method proposed by Gaunt is the one that should be used in connection with the present paper. i w. Heitler and F. London, ' Z. Physik, ' vol 44, p. 455(1927). SE. Wigner, ' Z. Physik, 'vol 43, P. 624(1927). I F Hund, 'Z Physik, vol 43, p. 788(1927)

Downloaded from rspa. royalsocietypublishing org on March 11, 2010 716 P A.M. Dirac new theory shows instead is that for each stationary state of the atom there is one definite numerical walue for s, the magnitude of the total spin vector. If it were not for this theorem, a measurement of s for the atom in a given stationary state would lead to one or other of a number of possible results, according to a definite probability law. This theorem forms the basis of the theory of multiplets. It is quite sufficient to replace the previous idea of the electrons all setting themselves parallel or antiparallel, since it shows that we can take s to be a quantum number describing the states of the atom, while 8 combined vectorially with i gives a multiplet of multiplicity 28+1 Further developments of the theory of exchange have been made by Heitle London and Heisenberg, k containing applications to molecules held together by homopolar valency bonds and to ferromagnetism. The treatment given by these authors makes an extensive use of group theory and requires the reader to be well acquainted with this branch of pure mathematics. Now group theory is just a theory of certain quantities that do not satisfy the com- mutative law of multiplication, and should thus form a part of quantum mechanics, which is the general theory of all quantities that do not satisfy the commutative law of multiplication. It should therefore be possible to tran late the methods and results of group theory into the language of quantum mechanics and so obtain a treatment of the exchange phenomena which does not presuppose any knowledge of groups on the part of the reader. This is the object of the present paper The treatment of groups on the lines of quantum mechanics has the advantage that it often gives a simple physical meaning to an abstract theorem in the theory of groups, enabling one to remember the theorem more easily and perhaps suggesting a simpler way of proving it. A further advantage of the treatment of the exchange phenomena on these lines is that one can avoid doing more work in the theory of groups than is strictly necessary for the physical applications, which results in a con siderable shortening in the method In $S 2 and 3 the general theory is given of systems containing a number of similar particles, showing the existence, of exclusive sets of states (i.e,sets such that a transition can never take place from a state in one set to a state in another), and giving their main properties. In$4 an application is made to electrons, a proof being obtained of the fundamental theorem in italics above The subsequent work is concerned with an approximate calculation of the energy levels of the states, the result of this being expressible by the single See various papers in the ' Z. Physik, vols. 46-51. An excellent account of the hole theory is also contained in Weyl's book, 'Gruppentheorie und Quantummechanik

Downloaded from rspa. royalsocietypublishing org on March 11, 2010 Quantum Mechanics of Many-Electron Systems. 717 simple formula(26). This formula shows that in the first approximation the exchange interaction between the electrons may be replaced by a coupling between their spins, the energy of this coupling for each pair of electrons being multiplied by a numer coefficient given by the exchange energy. This form of coupling energy is however,just what was required in the old orbit theory. We obtain in this way a justification for the assumptions of this old theory, in so far as they can be formulated without contradicting the quantum-mechanical description of the spin. The formula(26), combined with Hartree's method for determining approximate wave functions for the different electrons, should provide a power ful way of dealing with complicated atomic systems. 82. Permutations as Dynamical variables We consider a dynamical system composed of n similar particles, the rth particle being describable by certain generalised co-ordinates denoted by the ingle symbol gr. Thus a wave function representing a state of the system will be a function of the variables g1, g2,..., gm, which may be written or brevity. Suppose now that P is any permutation of 91, %e,...,m. This P is an operator which can be applied to any wave function (g) to give as result another definite function of the a's, namely Pψ(q)=ψ(Pq) where Pq denotes the set of g's obtained by applying the permutation P to q1, qs Further P is a linear ope any dynamical variable is a linear operator which can operate on any wave function, and conversely any linear operator that can operate on every wave function may be considered as a dynamical variable. Thus any permutation P may be considered to be a dynamical variable The present paper consists in a study of these permutations P as dynamical variables. There are no classical analogues to these variables and hence they give rise to phenomena, e.g., the existence of exclusive sets of states and other exchange phenomena, which have no classical analogue. There are n! of these variables, one of them, P, say, being the identity, which must thus be qual to unity. One can add and multiply these variables and form algebr functions of them, in exactly the same way in which one can add and multiply and form algebraic functions of the ordinary co-ordinates and momenta. The productof any two permutations is a third permutation, and hence any function

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

Downloaded from rspa. royalsocietypublishing org on March 11, 2010 720 P. A.M. Dirac particles. This symmetry condition requires that an element (lHI9)of the matrix representing H shall be unaltered when one applies any permutation to the 's and the same permutation to the "'s,i.e (qhi )=(pq hi Pa) (6) for arbitrary P The fact that H is symmetrical leads at once to the equation This equation may be verified by a similar argument to that used for equation (5), or alternatively by a direct application of the matrix representatives. Thus from (|PH|q")=18(P-q")d"q"Hq")=(Pg|Hlq") qHP|q")=g|Hq")d"”8("-P-1y")=(gHP-") and the two right-hand sides are now equal from(6). Equation(7)shows that each permutation variable is a constant of the motion. The P's are still constants when arbitrary perturbations are applied to the system, provided the per- urbation energy to be added to the Hamiltonian is symmetrical. Thus the constancy of the Ps is absolute. In dealing with any system in quantum mechanics, when we have found a constant of the motion o, we know that if for any state o initially has the numerical value othen it always has this value, so that we can assign different numbers a' to the different states and so obtain a classification of the state This procedure is not so straightforward, however, when we have several nstants of the motion a which do not commute(as is the case with our per mutations P), since we cannot assign numerical values for all the a's. simul- taneously to any state. The existence of constants of the motion a which do not commute is a sign that the system is degenerate. We must now look for a function B of the a's which has one and the same numerical value p for all those states belonging to one energy level H, so that we can use B for classifying the energy levels of the system. We can express the condition for p by saying that it must be a function of H (a single-valued function is implied)according to the general definition of a function of a variable in quantum mechanics or that B must commute with every variable that commutes with H, every constant of the motion. If the as 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 p of the a's which commutes

Downloaded from rspa. royalsocietypublishing org on March 11, 2010 Quantum Mechanics of Many-Electron Systems. 721 with all the as. We can then assign a numerical value B for B to each energy level of the system. If we can find several such functions B, they must all commute with each other, so that we can give them all numerical values simultaneously and obtain a complete classification of the energy levels An example of this procedure is provided by the study of the angular momen tum of an isolated system. This angular momentum has three components mx,m,,mg, each a constant of the motion, which do not commute. We look for a function of m. mm. which commutes with them all three. We can conveniently take for this function the variable k defined by k(k+h)=m2+m2+m2. For each energy level of the system there will now be one definite numerical value k' for k. This constant of the motion h is the only significant one for purposes of classifying the states, as the others merely describe the degeneracy We follow this method in dealing with our permutations P. We must find a function x of the P's such that PxP-l=x for every P. It is evident that a possible x is XPe, the sum of all the permutations Pe in a certain class i.e. the sum of a set of similar permutations, since 2PPP-I must consist of the same permutations summed in a different order. There will be one such x for each class. Further, there can be no other independent x, since an arbitrary function of the P'g 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 ∑P here ne is the number of P's in the class c. An alternative expression for lc i8 za=X,P,PP, l/n I he summation being extended over all the 9! permutations Pr. For each permutation P there is one x, x(P)say, equal to the average of all permutations similar to P. One of the x, s is %(P1)=l The dynamical variables x,, x2.xm obtained in this way will each have a definite numerical value for every stationary state of the system. Thus for every permissible set of numerical values 1, x2...m' for the x's there will be a set of states of the system. Since the xs are absolute 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

Downloaded from rspa. royalsocietypublishing org on March 11, 2010 P. A.M. Dirac The permissible sets of values x' that one can give to the x's are limited by the fact that there exist algebraic relations between the x's. The product of any two x' s, pxa, 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 x s, thus Xq=a171+a22+…+amm where the as are numbers. Any numerical values x, that one gives to the ,'s must be eigen-values of the x' s and must satisfy these same algebraic equa tions. For every solution z of these equations there is one exclusive set of or every giv of states with symmetrical wave functions. A second obvious solution is x=±1,the+or ing taken according to whether the the class p ar ld, and this gives the set of states with wave functions. The other solutions may be worked out in any special case directly cerned refer. Any solution is, apart from a certain factor, what is called in group theory a character of the group of permutations. The xs are all real variables, since each P and its conjugate complex P-l are similar and will occur added together in the definition of any x, so that the x s must be all real The number of possible solutions of the equations(10)may easily be deter- mined, since it must equal the number of different eigen-values of an arbitrar function B of the x's. We can express B as a linear function of the x,'s with the help of equations(10); thus B=b1X1+b2x2+…. Similarly we can express each of the quantities B, Bm as a linear function of the zs. From these m equations, together with the equation x(P1=l we can eliminate the m unknowns xu, x2..7m, obtaining as result an algebraic equation of degree m for B, Bm+cnBm-i+cgB-2+.+Cm The m solutions of this equation give the m possible eigen-values for B, each of which will, according to(11), be a linear function of b,b..6m whose coe are a permissible set of values %i, z2.. m. These m sets of values x thus obtained must be all different, since if there were fewer than m different permissible sets of values x' for the x,'s there would exist a linear function of the i's every one of whose eigen-values vanishes, which would mean that the