Ⅶ PERTURBATION THEORY
摂動論

42. General remarks

In the preceding chapter exact treatments were given of some simple dynamical systems in the quantum theory. Most quantum problems, however, cannot be solved exactly with the present resources of mathematics, as they lead to equations whose solutions cannot be expressed in finite terms with the help of the ordinary functions of analysis. For such problems one can often use a perturbation method. This consists in splitting up the Hamiltonian into two parts, one of which must be simple and the other small. The first part may then be considered as the Hamiltonian of a simplified or unperturbed system, which can be dealt with exactly, and the addition of the second will then require small corrections, of the nature of a perturba- tion, in the solution for the unperturbed system. The requirement that the first part shall be simple requires in practice that it shall not involve the time explicitly. If the second part contains a small numerical factor e, we can obtain the solution of our equations for the perturbed system in the form of a power series in e, which, pro- vided it converges, will give the answer to our problem with any desired accuracy. Even when the series does not converge, the first approximation obtained by means of it is usually fairly accurate. There are two distinct methods in perturbation theory. In one of these the perturbation is considered as causing @ modification of the states of motion of the unperturbed system. In the other we do not consider any modification to be made in the states of the unperturbed system, but we suppose that the perturbed system, instead of remain- ing permanently in one of these states, is continually changing from one to another, or making transitions, under the influence of the perturbation. Which method is to be used in any particular case depends on the nature of the problem to be solved. The first method is useful usually only when the perturbing energy (the correction in the Hamiltonian for the undisturbed system) does not involve the time explicitly, and is then applied to the stationary states. It can be used for calculating things that do not refer to any definite time, such as the energy-levels of the stationary, states of the perturbed system, or, in the case of collision problems, the probability of scattering through 168 PERTURBATION THEORY § 42 a given angle. The second method must, on the other hand, be used for solving all problems involving a consideration of time, such as those about the transient phenomena that occur when the perturba- tion is suddenly applied, or more generally problems in which the perturbation varies with the time in any way (i.e. in which the per- turbing energy involves the time explicitly). Again, this second method must be used in collision problems, even though the per- turbing energy does not here involve the time explicitly, if one wishes to calculate absorption and emission probabilities, since these probabilities, unlike a scattering probability, cannot be defined with- out reference to a state of affairs that varies with the time. One can summarize the distinctive features of the two methods by saying that, with the first method, one compares the stationary states of the perturbed system with those of the unperturbed system; with the second method one takes a stationary state of the unperturbed system and sees how it varies with time under the influence of the perturbation. 43. The change in the energy -levels caused by a perturbation The first of the above-mentioned methods will now be applied to the calculation of the changes in the energy-levels of a system caused by a perturbation. We assume the perturbing energy, like the Hamil- tonian for the unperturbed system, not to involve the time explicitly. Our problem has a meaning, of course, only provided the energy-levels of the unperturbed system are discrete and the differences between them are large compared with the changes in them caused by the perturbation. This circumstance results in the treatment of perturba- tion problems by the first method having some different features according to whether the energy-levels of the unperturbed system are discrete or continuous. Let the Hamiltonian of the perturbed system be H=E-+Y, (1) H being the Hamiltonian of the unperturbed system and V the small perturbing energy. By hypothesis each eigenvalue H’ of H lies very close to one and only one eigenvalue H’ of H. We shall use the same number of primes to specify any eigenvalue of H and the eigenvalue of E to which it lies very close. Thus we shall have H” differing from E" by a small quantity of order V and differing from H’ by a quantity that is not small unless H’ = H”. We must now take care always to § 43 CHANGE IN THE ENERGY-LEVELS 169 use different numbers of primes to specify eigenvalues of H and & which we do not want to lie very close together. To obtain the eigenvalues of H, we have to solve the equation H\H'> = H’|H’> or (H'—H)|H> = VIA’). (2) Let |0) be an eigenket of E belonging to the eigenvalue H’ and suppose the |H’) and H’ that satisfy (2) to differ from |0> and LZ’ only by small quantities and to be expressed as |H'> = |0)+[1>+ 129+... 6) A! = E'+a,+a_+..., where |1) and a, are of the first order of smallness (i.e. the same order as V), |2> and a, are of the second order, and so on. Substituting these expressions in (2), we obtain {B'—EB+a,y+a,+..}{|0>+|I+|2+..} = V{JO>+/D+...}. If we now separate the terms of zero order, of the first order, of the second order, and so on, we get the following set of equations, (E'— £)|0> = 0, (E’ — E)\1>+4,/0> = V{0>, (4) (£' — B)|2)+a,|1>+a,/0> = VI), The first of these equations tells us, what we have already assumed, ~ that |0) is an eigenket of # belonging to the eigenvalue £’. The others enable us to calculate the various corrections {1}, |2),..., @y,@g-0- For the further discussion of these equations it is convenient to . introduce a representation in which FH is diagonal, ie. a Heisenberg representation for the unperturbed system, and to take Z itself as one of the observables whose eigenvalues label the representatives. Let the others, in the event of others being necessary, as is the case when there is more than one eigenstate of H belonging to any eigen- value, be called §’s. A basic bra is then <#"B"|. Since |0> is an eigenket of H belonging to the eigenvalue E’, we have (ENB) = Berar f(P"), (5) where f(8”) is some function of the variables 8”. With the help of this result the second of equations (4), written in terms of representatives, becomes (B’— B’)E"B" 1) +0, bre f(B") = 2 (B"B"|V EPR HB). (8) 170 PERTURBATION THEORY § 43 Putting H” = H' here, we get a,f(8") = 2 (EVR V | E'B') f(B'). (7) Equation (7) is of the form of the standard equation in the theory of eigenvalues, so far as the variables f’ are concerned. It shows that the various possible values for a, are the eigenvalues of the matrix . This matrix is a part of the representative of the perturbing energy in the Heisenberg representation for the unper- turbed system, namely, the part consisting of those elements that refer to the same unperturbed energy-level #’ for their row and column. Each of these values for a, gives, to the first order, an energy- level of the perturbed system lying close to the energy-level E’ of the unperturbed system.} There may thus be several energy-levels of the perturbed system lying close to the one energy-level #’ of the unper- turbed system, their number being anything not exceeding the number of independent states of the unperturbed system belonging to the energy-level E’. In this way the perturbation may cause a separation or partial separation of the energy-levels that coincide at E’ for the unperturbed system. Equation (7) also determines, to the zero order, the representatives of the stationary states of the perturbed system belonging to energy-levels lying close to £’, any solution f(6’) of (7) substituted in (5) giving one such representative. Each of these stationary states of the perturbed system approximates to one of the stationary states of the unperturbed system, but the converse, that each stationary state of the unperturbed system approximates to one of the stationary states of the perturbed system, is not true, since the general stationary state of the unperturbed system belonging to the energy- level HE’ is represented by the right-hand side of (5) with an arbitrary function f(8”). The problem of finding which stationary states of the unperturbed system approximate to stationary states of the perturbed system, i.e. the problem of finding the solutions f(8') of (7), corresponds to the problem of ‘secular perturbations’ in classical mechanics. It should be noted that the above results are indepen- dent of the values of all those matrix elements of the perturbing + To distinguish these energy-levels one from another we should require some more elaborate notation, since according to the present notation they must all be specified by the same number of primes, namely by the number of primes specifying the energy-level of the unperturbed system from which they arise. For our present purposes, however, this more elaborate notation is not required. § 43 CHANGE IN THE ENERGY-LEVELS _ 171 energy which refer to two different energy-levels of the unperturbed system. Let us see what the above results become in the specially simple case when there is only one stationary state of the unperturbed system belonging to each energy-level.t In this case # alone fixes the repre- sentation, no f’s being required. The sum in (7) now reduces to a single term and we get ay = (EVE. (8) There is only one energy-level of the perturbed system lying close to any energy-level of the unperturbed system and the change wn energy ts equal, in the first order, to the corresponding diagonal element of the perturbing energy in the Heisenberg representation for the unperturbed system, or to the average value of the perturbing energy for the correspond- ing unperturbed state. The latter formulation of the result is the same as in classical mechanics when the unperturbed system is multiply periodic. We shall proceed to calculate the second-order correction a, in the energy-level for the case when the unperturbed system is non- degenerate. Equation (5) for this case reads (B'\0) = dex; with neglect of an unimportant numerical factor, and equation (6) This gives us the value of (#”|1> when £” + H’, namely ” (EVIE BY = I (Ely = Sa. (9) The third of equations (4), written in terms of representatives, becomes (E’— BB" 2)-+0,68" |) bag Sere = & CB"|V BB" 1). Putting #” = EH’ here, we get a,(H#’ \+a,= > (EB VEE"), which reduces, with the help of (8), to ag = 3S KEW EEL. EAE’ + A system with only one stationary state belonging to each energy-level is often called non-degenerate and one with two or more stationary states belonging to an energy-level is called degenerate, although these words are not very appropriate from the modern. point of view. 172 PERTURBATION THEORY § 43 Substituting for <£"}1> from (9), we obtain finally (EV | E" ><" VE") by, == EPFE’ giving for the total energy change to the second order f EK” ae E’ ay-fay = (BV |B") + S SISTINE a0) EE The method may be developed for the calculation of the higher approximations if required. General recurrence formulas giving the nth order corrections in terms of those of lower order have been obtained by Born, Heisenberg, and Jordan.t 44, The perturbation considered as causing transitions We shall now consider the second of the two perturbation methods mentioned in § 42. We suppose again that we have an unperturbed system governed by a Hamiltonian H which does not involve the time explicitly, and a perturbing energy V which can now be an arbitrary function of the time. The Hamiltonian for the perturbed system is again H = H-+V. For the present method it does not make any essential difference whether the energy-levels of the unperturbed system, i.e. the eigenvalues of H, form a discrete or continuous set. We shall, however, take the discrete case, for definiteness. We shall again work with a Heisenberg representation for the unperturbed system, but as there will now be no advantage in taking E itself as one of the observables whose eigenvalues label the representatives, we shall suppose we have a general set of «’s to label the representatives. Let us suppose that at the initial time é) the system is in a state for which the «’s certainly have the values «’. The ket corresponding to this state is the basic ket |«’>. If there were no perturbation, i.e. if the Hamiltonian were E, this state would be stationary.. The perturba- tion causes the state to change. At time the ket corresponding to the state in Schrédinger’s picture will be T'lo’>, according to equation (1) of § 27. The probability of the «’s then having the values «” is P(o'n”) = [|*. (11) For «” ~«’, P(a’«’) is the probability of a transition taking place from state «' to state «” during the time interval f) > t, while P(a’a’) + Z.f. Physik, 35 (1925), 568. § 44 PERTURBATION CAUSING TRANSITIONS 173 is the probability of no transition taking place at all. The sum of P(a'x”) for all x” is, of course, unity. Let us now suppose that initially the system, instead of being certainly in the state «’, is in one or other of various states o’ with the probability P,, for each. The Gibbs density corresponding to this distribution is, according to (68) of § 33 p= > |a’>P will have changed to T'|a’> and each bra — > ot” | Ta! Py Cal |Tha"> = 5 Py Plo'a’) (14) with the help of (11). This result expresses that the probability of the system being in the state «” at time ¢ is the sum of the probabilities of the system being initially in any state «’ ~ «", and making a transi- tion from state «’ to state «” and the probability of its being initially in the state w” and making no transition. Thus the various transition probabilities act independently of one another, according to the ordinary laws of probability. The whole problem of calculating transitions thus reduces to the determination of the probability amplitudes <«"|7’|w’>. These can be worked out from the differential equation for 7’, equation (6) of § 27, or dT dt = HT = (E+V)1T. (15) The calculation can be simplified by working with P% ao gi El-tMn TZ, (16) We have dT */dt = et tolk(— ET +1 dT /dt) = CAM DT — VT, (17) where VE = eth th Ve -tEt yh, (18) ie. V* is the result of applying a certain unitary transformation to V. Equation (17) is of a more convenient form than (15), because (17) makes the change in 7* depend entirely on the perturbation V, and 174 PERTURBATION THEORY § 44 for V = 0 it would make 7'* equal its initial value, namely unity. We have from (16) Kal | PA |a) = et U-t0IN "| la’, so that P(a'e”) = |[2, (19) showing that 7'* and 7 are equally good for determining transition probabilities. Our work up to the present has been exact. We now assume V is a small quantity of the first order and express 7'* in the form T* = ATF TEH..., (20) where T* is of the first order, Tf is of the second, and so on. Substi- tuting (20) into (17) and equating terms of equal order, we get ihdT* [dt = V*, id TE [dt = V*T*, (21) From the first of these equations we obtain t T# = ih} ii V(t!) dt’, (22) to from the second we obtain t tv TS a= —hi-? ij V*(t’) dt’ { V*(t") dt”, (23) ty to and so on. For many practical problems it is sufficiently accurate to retain only the term 7*, which gives for the transition probability P(o'a") with a” a’ t Plo’ ot") == K-28] Cos" { V*(e') dt! |o!> ‘0 (24) 2 2 i = 8) [dal |V*() Ia") ae’ fo We obtain in this way the transition probability to the second order of accuracy. The result depends only on the matrix element of V*(t') referring to the two states concerned, with ¢’ going from é, to ¢. Since V* is real, like V, Col VCE) [a == Coe" |V*(t) |e” and hence P(o'n") == P(x"a’) . (25) to the second order of accuracy. § 44 PERTURBATION CAUSING TRANSITIONS 175 Sometimes one is interested in a transition «’ > «” such that the matrix element <«"|V*|a’> vanishes, or is small compared with other matrix elements of V*. It is then necessary to work to a higher accuracy. If we retain only the terms 7} and T#, we get, for x” 4 a’, P(a'a") = A? t [ dt” * (28) rr ta ‘The terms a” == o’ and a” = «” are omitted from the sum since they are small compared with other terms of the sum, on account of the smallness of . To interpret the result (26), we may suppose that the term t | Ko" IPE )Ia’> at! (27) ty gives rise to a transition directly from state «’ to state «”, while the term é v tit | Kat” | V*(E) |x”) dt’ | Kal" |VEE Ya’) dt” (28) to i, gives rise to a transition from state «’ to state «”, followed by a transition from state «” to state «”. The state «” is called an inter- mediate state in this interpretation. We must add the term (27) to the various terms (28) corresponding to different intermediate states and then take the square of the modulus of the sum, which means that there is interference between the different transition processes— the direct one and those involving intermediate states—and one can- not give a meaning to the probability for one of these processes by itself. For each of these processes, however, there is a probability amplitude. If one carries out the perturbation method to a higher degree of accuracy, one obtains a result which can be interpreted similarly, with the help of more complicated transition processes involving a succession of intermediate states. 45. Application to radiation In the preceding section a general theory of the perturbation of an atomic system was developed, in which the perturbing energy could vary with the time in an arbitrary way. A perturbation of this kind can be realized in practice by allowing incident electromagnetic 176 PERTURBATION THEORY § 45 radiation to fall on the system. Let us see what our result (24) reduces to in this case. If we neglect the effects of the magnetic field of the incident radia- tion, and if we further assume that the wave-lengths of the harmonic components of this radiation are all large compared with the dimen- sions of the atomic system, then the perturbing energy is simply the scalar product V = (D, 8), (29) where D is the total electric displacement of the system and € is the electric force of the incident radiation. We suppose € to be a given function of the time. If we take for simplicity the case when the incident radiation is plane polarized with its electric vector in a certain direction and let D denote the Cartesian component of D in this direction, the expression (29) for V reduces to the ordinary product V=Dé, where & is the magnitude of the vector &. The matrix elements of V are is independent of t From (18) Kai" |V*(t) or") == Cor” |D]o’pet BM HIRE (t), and hence the expression (24) for the transition probability becomes i J EE" EM’ 10h © (4') ar (30) to If the incident radiation during the time interval f, to ¢ is resolved into its Fourier components, the energy crossing unit area per unit frequency range about the frequency v will be, according to classical electrodynamics, t P(al'a") = | a" |DJa">|? 2 E,= So | erin E(t’) dt’| (31) 2a to Comparing this with (30), we obtain P(a'a") = 2acMh-*| in (32) plays the part of the ampli- tude of one of the Fourier components of D in the classical theory of a multiply-periodic system interacting with radiation. In fact it was the idea of replacing classical Fourier components by matrix elements which led Heisenberg to the discovery of quantum mechanics in 1925. Heisenberg assumed that the formulas describing the interaction with radiation of a system in the quantum theory can be obtained from the classical formulas by substituting for the Fourier components of the total electric displacement of the system the corresponding matrix elements. According to this assumption applied to spontaneous emis- sion, a system having an electric moment D will, when in the state + Einstein, Phys. Zeits. 18 (1917), 121. 178 PERTURBATION THEORY § 15 a’, spontaneously emit radiation of frequency v = (#’—E")/h, where EB" is an energy-level, less than E’, of some state «”, at the rate 4 (2av)* 3 8 The distribution of this radiation over the different directions of emission and its state of polarization for each direction will be the same as that for a classical electric dipole of moment equal to the real part of . To interpret this rate of emission of radiant energy as a transition probability, we must divide it by the quantum of energy of this frequency, namely hy, and call it the probability per unit time of this quantum being spontaneously emitted, with the atomic system simultaneously dropping to the state a” of lower energy. These assumptions of Heisenberg are justified by the present radiation theory, supplemented by the spontaneous transition theory of Chapter X. 46. Transitions caused by a perturbation independent of the time The perturbation method of § 44 is still valid when the perturbing energy V does not involve the time f explicitly. Since the total Hamiltonian H in this case does not involve ¢ explicitly, we could now, if desired, deal with the system by the perturbation method of § 43 and find its stationary states. Whether this method would be convenient or not would depend on what we want to find out about the system. If what we have to calculate makes an explicit reference to the time, e.g. if we have to calculate the probability of the system being in a certain state at one time when we are given that it is in a certain state at another time, the method of § 44 would be the more convenient one. Let us see what the result (24) for the transition probab’' lity becomes when V does not involve t explicitly and let us take ty = 0 to simplify the writing. The matrix element <«”|V|«’> is now independent of t, |<" |D Ja">|?. (34) and from (18) La" |V*(t') fox’ = Cou" |V |! ete” BWI, (35) * - t , , “ , ice" ~ EM 1 80 [ |?[1—cos{(L"— E')t/ti}\/(H"— EB’). (36) § 46 TRANSITION PROBABILITIES 179 If #” differs appreciably from #’ this transition probability is small and remains so for all values of t. This result is required by the law of the conservation of energy. The total energy H is constant and hence the proper-energy # (i.e. the energy with neglect of the part V due to the perturbation), being approximately equal to H, must be approximately constant. This means that if # initially has the numerical value #’, at any later time there must be only a small probability of its having a numerical value differing considerably from £’. On the other hand, when the initial state a’ is such that there exists another state a” having the same or very nearly the same proper- energy J, the probability of a transition to the final state «” may be quite large. The case of physical interest now is that in which there is a continuous range of final states «” having a continuous range of proper-energy levels H” passing through the value ’ of the proper- energy of the initial state. The initial state must not be one of the continuous range of final states, but may be either a separate discrete state or one of another continuous range of states. We shall now have, remembering the rules of § 18 for the interpretation of probability amplitudes with continuous ranges of states, that, with P(w‘x”) having the value (36), the probability of a transition to a final state within the small range «” to «”-+da” will be P(a’ax”) dx” if the initial state a’ is discrete and will be proportional to this quantity if a’ is one of a continuous range. We may suppose that the o’s describing the final state consist of # together with a number of other dynamical variables 8, so that we have a representation like that of § 43 for the degenerate case. (The f’s, however, need have no meaning for the initial state «’.) We shall suppose for definiteness that the §’s have only discrete eigenvalues. The total probability of a transition to a final state «” for which the B’s have the values B” and # has any value (there will be a strong probability of its having a value near the initial value ZH’) will now be (or be proportional to) i P(a'ce") dE" = 2 f KBB" \a’>[*[1—cos{(E"— Ee (E"—B') dB" (37) = ah [ |< Hf’ +fia/t, B’|V ho’> |2[ 1 ~cos «]/a? dx 180 PERTURBATION THEORY § 46 if one makes the substitution (Z”— E’)i/ii = x. For large values of t this reduces to Ot CE'B'|V |x’) |? | [1—cosa]/x? da == Qt | CE" B"|V lo’ |? (38) Thus the total probability up to time ¢ of a transition to a final state for which the f’s have the values fp” is proportional to ¢. There is therefore a definite probability coefficient, or probability per unit time, for the transition process under consideration, having the value Qrh-* | B'B"|V Joe") |?. (39) It is proportional to the square of the modulus of the matrix element, associated with this transition, of the perturbing energy. If the matrix element is small compared with other matrix elements of Y, we must work with the more accurate formula (26). We have from (35) t t’ 6 0 Fa t — Lol" |V fe” > i eh” Et ih dt’ f eser—eyrin at’ 3 7 t _ ALAC XCMLALS | {eB -EWth eB" EB" dt’. 0 For Z£” close to EZ’, only the first term inthe integrand here gives rise to a transition probability of physical importance and the second term may be discarded. Using this result in (26) we get P(cx'c”) ; 9 on” 0’ 00” h c's [V lo > 2. — EB" — E’ ~ “fos This formula shows how intermediate states, differing from the initial state and final state, play a role in the determination of a probability coefficient. (40) § 46 TRANSITION PROBABILITIES 181 In order that the approximations used in deriving (39) and (40) may be valid, the time ¢ must be not too small and not too large. It must be large compared with the periods of the atomic system in order that the approximate evaluation of the integral (37) leading to the result (38) may be valid, while it must not be excessively large or else the general formula (24) or (26) will break down. In fact one could make the probability (38) greater than unity by taking ¢ large enough. The upper limit to tis fixed by the condition that the probability (24) or (26), or é times (39) or (40), must be small compared with unity. There is no difficulty in ¢ satisfying both these conditions simultaneously provided the perturbing energy V is sufficiently small. 47. The anomalous Zeeman effect One of the simplest examples of the perturbation method of § 43 is the calculation of the first-order change in the energy-levels of an atom caused by a uniform magnetic field. The problem of a hydrogen atom in a uniform magnetic field has already been dealt with in § 41 and was so simple that perturbation theory was unnecessary. The case of a general atom is not much more complicated when we make a few approximations such that we can set up a simple model for the atom. We first of all consider the atom in the absence of the magnetic field and look for constants of the motion or quantities that are approximately constants of the motion. The total angular momen- tum of the atom, the vector j say, is certainly a constant of the motion. This angular momentum may be regarded as the sum of two parts, the total orbital angular momentum of all the electrons, | say, and the total spin angular momentum, s say. Thus we have j =I-+s. Now the effect of the spin magnetic moments on the motion of the electrons is small compared with the effect of the Coulomb forces and may be neglected as a first approximation. With this approximation the spin angular momentum of each electron is a constant of the motion, there being no forces tending to change its orientation. Thus s, and hence also 1, will be constants of the motion. The magnitudes, 1, s, and j say, of I, s, and j will be given by I+Hi = (B+R+E+20), s+ = (BSB ot + dH), i+ = GI), $595.57 N 182 PERTURBATION THEORY § 47 corresponding to equation (39) of § 36. They commute with each other, and from (47) of § 36 we see that with given numerical values for L and s the possible numerical values for 7 are l+s, l+s—f, ..., |l—s|. Let us consider a stationary state for which J, s, and j have definite numerical values in agreement with the above scheme. The energy of this state will depend on /, but one might think that with neglect of the spin magnetic moments it would be independent of s, and also of the direction of the vector s relative to J, and thus of 7. It will be found in Chapter IX, however, that the energy depends very much on the magnitude s of the vector s, although independent of its direction when one neglects the spin magnetic moments, on account of certain phenomena arising from the fact that the electrons are indistinguishable one from another. There are thus different energy- levels of the system for each different value of / and s. This means that | and s are functions of the energy, according to the general definition of a function given in § 11, since the / and s of a stationary state are fixed when the energy of that state is fixed. We can now take into account the effect of the spin magnetic moments, treating it as a small perturbation according to the method of § 43. The energy of the unperturbed system will still be approxi- mately a constant of the motion and hence / and s, being functions of this energy, will still be approximately constants of the motion. The directions of the vectors 1 and s, however, not being functions of the unperturbed energy, need not now be approximately constants of the motion and may undergo large secular variations. Since the vector j is constant, the only possible variation of | and s is a pre- cession about the vector j. We thus have an approximate model of the atom consisting of the two vectors | and s of constant lengths precessing about their sum j, which is a fixed vector. The energy is determined mainly by the magnitudes of 1 and s and depends only slightly on their relative directions, specified by j. Thus states with the same / and s and different j will have only slightly different energy-levels, forming what is called a multiplet term. Let us now take this atomic model as our unperturbed system and suppose it to be subjected to a uniform magnetic field of magnitude # in the direction of the z-axis. The extra energy due to this magnetic field will consist of a term eF#/2me.(m,+ hio,), (41) § 47 THE ANOMALOUS ZEEMAN EFFECT 183 like the last term in equation (89) of § 41, contributed by each electron, and will thus be altogether eH /2me. > (m,--ha,) = eF#/2mc.(l,-+-28,) = e#/2me.(9,+-8,). (42) This is our perturbing energy VY. We shall now use the method of § 43 to determine the changes in the energy-levels caused by this V. The method will be legitimate only provided the field is so weak that V is small compared with the energy differences within a multiplet. Our unperturbed system is degenerate, on account of the direction of the vector j being undetermined. We must therefore take, from the representative of V in a Heisenberg representation for the un- perturbed system, those matrix elements that refer to one particular energy-level for their row and column, and obtain the eigenvalues of the matrix thus formed. We can do this best by first splitting up V into two parts, one of which is a constant of the unperturbed motion, so that its representative contains only matrix elements referring to the same unperturbed energy-level for their row and column, while the representative of the other contains only matrix elements refer- ring to two different unperturbed energy-levels for their row and column, so that this second part does not affect the first-order per- turbation. The term involving j, in (42) is a constant of the un- perturbed motion and thus belongs entirely to the first part. For the term involving s, we have SAPE+IEAD) = JASrJz SyJy S2Jz) 1 (825 D282) Ix (S2Jy Ja Sy)dy or = Ie ij tn) —Ul-+h) +9(5-+8)]—Lyyhe—Yelu 3 Sz Gan )—Ul+-h)+8(s+h)]—[yyJu Yeu oem (43) where Vn = S:dy—Je8y = Szly—L sy = ly s,—l, sy, | as = IS Sidq = L, 85 Syl, = 1, 8,—~l, 8. The first term ‘in this expression for s, is a constant of the unperturbed motion land thus belongs entirely to the first part, while the second term, as we shall now see, belongs entirely to the second part. Corresponding to (44) we can introduce Y_ == 1 8y—ly 8. It can now easily be verified that Ja Vatdy Vy tee =0 and from (30) of § 35 [jes x] = %Yy [yes Yu] = V2 [ie Ya = 0. 184 PERTURBATION THEORY § 47 These relations connecting j,, j,,J, and y,, Yy, yz, are of the same form as the relations connecting m,, m,, m, and x, y, z in the calculation in § 40 of the selection rule for the matrix elements of z in a repre- sentation with k diagonal. From the result there obtained that all matrix elements of z vanish except those referring to two & values differing by +/, we can infer that all matrix elements of y,, and similarly of y, and y,, in a representation with j diagonal, vanish except those referring to two j values differing by -#. The coeffi- cients of y, and y, in the second term on the right-hand side of (43) commute with j, so the representative of the whole of this term will contain only matrix elements referring to two j values differing by +4, and thus referring to two different energy-levels of the unper- turbed system. Hence the perturbing energy V becomes, when we neglect that part of it whose representative consists of matrix elements referring to two different unperturbed energy-levels, et (5 JGR —UL+A) +5(8-+h) Ime!*\ 2G-FR) The eigenvalues of this give the first-order changes in the energy- levels. We can make the representative of this expression diagonal by choosing our representation such that j, is diagonal, and it then gives us directly the first-order changes in the energy-levels caused by the magnetic field. This expression is known as Landé’s formula. The result (45) holds only provided the perturbing energy V is small compared with the energy differences within a multiplet. For larger values of V a more complicated theory is required. For very strong fields, however, for which V is large compared with the energy differ- ences within a multiplet, the theory is again very simple. We may now neglect altogether the energy of the spin magnetic moments for the atom with no external field, so that for our unperturbed system the vectors | and s themselves are constants of the motion, and not merely their magnitudes 7 and s. Our perturbing energy V, which is still e#/2me.(j,+s,), is now a constant of the motion for the unper- turbed system, so that its eigenvalues give directly the changes in the energy-levels. These eigenvalues are integral or half-odd integral multiples of eAh/2mc according to whether the number of electrons in the atom is even or odd. (45)