Ⅷ COLLISION PROBLEMS
衝突問題

48. General remarks

In this chapter we shall investigate problems connected with a par- ticle which, coming from infinity, encounters or ‘collides with’ some atomic system and, after being scattered through a certain angle, goes off to infinity again. The atomic system which does the scattering we shall call, for brevity, the scatterer. We thus have a dynamical system composed of an incident particle and a scatterer interacting with each other, which we must deal with according to the laws of quantum mechanics, and for which we must, in particular, calculate the probability of scattering through any given angle. The scatterer is usually assumed to be of infinite mass and to be at rest throughout the scattering process. The problem was first solved by Born by a method substantially equivalent to that of the next section. We must take into account the possibility that the scatterer, considered as a system by itself, may have a number of different stationary states and that if it is initially in one of these states when the particle arrives from infinity, it may be left in a different one when the particle goes off to infinity again. The colliding particle may thus induce transi- tions in the scatterer. The Hamiltonian for the whole system of scatterer plus particle will not involve the time explicitly, so that this whole system will have stationary states represented by periodic solutions of Schré- dinger’s wave equation. The meaning of these stationary states requires a little care to be properly understood. It is evident that for any state of motion of the system the particle will spend nearly all its time at infinity, so that the time average of the probability of the particle being in any finite volume will be zero. Now for a stationary state the probability of the particle being in a given finite volume, like any other result of observation, must be independent of the time, and hence this probability will equal its time average, which we have seen is zero. Thus only the relative probabilities of the particle being in different finite volumes will be physically significant, their absolute values being all zero. The total energy of the system has a continuous range of eigenvalues, since the initial energy of the particle can be anything. Thus a ket, |s> say, corresponding to a stationary state, 186 COLLISION PROBLEMS § 48 being an eigenket of the total energy, must be of infinite length. We can see a physical reason for this, since if |s> were normalized and if @ denotes that observable—a certain function of the position of the particle—that is equal to unity if the particle is in a given finite volume and zero otherwise, then would be zero, meaning that | the average value of Q, i.e. the probability of the particle being in the given volume, is zero. Such a ket |s> would not be a convenient one to work with. However, with |s> of infinite length, can be finite and would then give the relative probability of the particle being in the given volume. In picturing a state of a system corresponding to a ket |z> which is not normalized, but for which = n say, it may be convenient to suppose that we have n similar systems all occupying the same space but with no interaction between them, so that each one follows out its own motion independently of the others, as we had in the theory of the Gibbs ensemble in § 33. We can then interpret , Q being the observable defined above, which has the value unity when the particle is in the given volume and zero otherwise. If the ket is represented by a Schrédinger wave function involving the Cartesian coordinates of the particle, then the square of the modulus of the wave function could be interpreted directly as the density of particles in the picture. One must remember, however, that each of these particles has its own individual scatterer. Different particles may belong to scatterers in different states. There will thus be one particle density for each state of the scatterer, namely the density of those particles belonging to scatterers in that state. This is taken account of by the wave function involving variables describing the state of the scatterer in addition to those describing the position of the particle. For determining scattering coefficients we have to investigate stationary states of the whole system of scatterer plus particle. For instance, if we want to determine the probability of scattering in various directions when the scatterer is initially in a given stationary § 48 GENERAL REMARKS 187 state and the incident particle has initially a given velocity in a given direction, we must investigate that stationary state of the whole system whose picture, according to the above method, contains at great distances from the point of location of the scatterers only particles moving with the given initial velocity and direction and belonging each to a scatterer in the given initial stationary state, together with particles moving outward from the point of location of the scatterers and belonging possibly to scatterers in various stationary states. This picture corresponds closely to the actual state of affairs in an experimental determination of scattering coefficients, with the difference that the picture really describes only one actual system of scatterer plus particle. The distribution of outward moving particles at infinity in the picture gives us immediately all the infor- mation about scattering coefficients that could be obtained by experi- ment. For practical calculations about the stationary state described by this picture one may use a perturbation method somewhat like that of § 43, taking as unperturbed system, for example, that for which there is no interaction between the scatterer and particle. In dealing with collision problems, a further possibility to be taken into consideration is that the scatterer may perhaps be capable of absorbing and re-emitting the particle. This possibility arises when there exists one or more states of absorption of the whole system, a state of absorption being an approximately stationary state which is closed in the sense mentioned at the end of § 38 (i.e. for which the probability of the particle being at a greater distance than r from the scatterer tends to zero as r > c0). Since a state of absorption is only approximately stationary, its property of being closed will be only a transient one, and after a sufficient lapse of time there will be a finite probability of the particle being on its way to infinity. Physically this means there is a finite probability of spontaneous emission of the particle. The fact that we had to use the word ‘approximately’ in stating the conditions required for the phenomena of emission and absorption to be able to occur shows that these condi- tions are not expressible in exact mathematical language. One can give a meaning to these-phenomena only with reference to a perturbation method. They occur when the unperturbed system (of scatterer plus particle) has stationary states that are closed. The introduction of the perturbation spoils the stationary property of these states and gives rise to spontaneous emission and its converse absorption. 188 COLLISION PROBLEMS § 48 For calculating absorption and emission probabilities it is necessary to deal with non-stationary states of the system, in contradistinction to the case for scattering coefficients, so that the perturbation method of § 44 must be used. Thus for calculating an emission coefficient we must consider the non-stationary states of absorption described above. Again, since an absorption is always followed by a re-emission, it cannot be distinguished from a scattering in any experiment in- volving a steady state of affairs, corresponding to a stationary state of the system. The distinction can be made only by reference to a non-steady state of affairs, e.g. by use of a stream of incident particles that has a sharp beginning, so that the scattered particles will appear immediately after the incident particles meet the scatterers, while those that have been absorbed and re-emitted will begin to appear only some time later. This stream of particles would be the picture of a certain ket of infinite length, which could be used for calculating the absorption coefficient. 49, The scattering coefficient We shall now consider the calculation of scattering coefficients, taking first the case when there is no absorption and emission, which means that our unperturbed system has no closed stationary states. We may conveniently take this unperturbed system to be that for which there is no interaction between the scatterer and particle. Its Hamiltonian will thus be of the form E=H+W, (1) where H, is that for the scatterer alone and W that for the particle alone, namely, with neglect of relativistic mechanics, W = 1/2m.(pit+-py+P?)- (2) The perturbing energy V, assumed small, will now be a function of the Cartesian coordinates of the particle x, y, z, and also, perhaps, of its momenta p,, p,, pP,, together with dynamical variables describ- ing the scatterer. Since we are now interested only in stationary states of the whole system, we use a perturbation method like that of § 43. Our unper- turbed system now necessarily has a continuous range of energy- levels, since it contains a free particle, and this gives rise to certain modifications in the perturbation method. The question of the change in the energy-levels caused by the perturbation, which was the main § 49 THE SCATTERING COEFFICIENT 189 question of § 43, no longer has a meaning, and the convention in § 43 of using the same number of primes to denote nearly equal eigen- values of # and H now drops out. Again, the splitting of energy- levels which we had in§ 43 when the unperturbed system is degenerate cannot now arise, since if the unperturbed system is degenerate the perturbed one, which must also have a continuous range of energy- levels, will also be degenerate to exactly the same extent. We again use the general scheme of equations developed at the beginning of § 43, equations (1) to (4) there, but we now take our unperturbed stationary state forming the zero-order approximation to belong to an energy-level EH’ just equal to the energy-level H’ of our perturbed stationary state. Thus the a’s introduced in the second of equations (3) § 43 are now all zero and the second of equations (4) there now reads (B’—B)|1) = V0). (3) Similarly, the third of equations (4) § 43 now reads We shall proceed to solve equation (3) and to obtain the scattering coefficient to the first order. We shall need equation (4) in § 51. Let « denote a complete set of commuting observables describing the scatterer, which are constants of the motion when the scatterer is alone and may thus be used for labelling the stationary states of the scatterer. This requires that H, shall commute with the «’s and be a function of them. We can now take a representation of the whole system in which the a’s and x, y, z, the coordinates of the particle, are diagonal. This will make H, diagonal. Let |0> be represented by éxa’|0> and |1> by , the single variable x being written to denote x, y, z and the prime being omitted from x for brevity. Also the single differential d?x will be written to denote the product dadydz. Equation (3), written in terms of representatives, becomes, with the help of (1) and (2), {E! —H,(o') +-h?/2m V7} xa! |L> = > | xa’ |V | x"a"S da” x"x" 09. (5) Suppose that the incident particle has the momentum p® and that the initial stationary state of the scatterer is «°. The stationary state of our unperturbed system is now the one for which p = p® and a == «, and hence its representative is = | xo" |V | xx 3x0 eilp?. x0 or (Kk? +-V?)(xa'|l> = F, (7) where 2 = 2mh-*{ HE’ — H,(a')} (8) and F = 2mh- | (X00 |V |X 3x9 etx, (9) a definite function of x, y, z, and «’. We must also have HE’ = H,(x)+ p%/2m. (10) Our problem now is to obtain a solution of (7) which, for values of x, y, z denoting points far from the scatterer, represents only outward moving particles. The square of its modulus, |(x«'|1)]°, will then give the density of scattered particles belonging to scatterers in the state «’ when the density of the incident particles is [(xw|0)|2, which is unity. If we transform to polar coordinates r, 8, 6, equation (7) becomes a ! 20 1 Qo. o 1 a j}-— — ++, — sin? — -++- ar" r Or r* sin @ 26 26 ° r*sin?@ ad? ke (rOba' jl) = F. (11) Now F must tend to zero as 7 + 00, on account of the physical re- quirement that the interaction energy between the scatterer and particle must tend to zero as the distance between them tends to infinity. If we neglect F in (11) altogether, an approximate solution for large r is CO pal! |) == U(Ogou" r-het*r, (12) where w is an arbitrary function of 6, ¢, and «’, since this expression substituted in the left-hand side of (11) gives a result of order r-3. When we do not neglect F, the solution of (11) will still be of the form (12) for large r, provided F tends to zero sufficiently rapidly as roo, but the function uw will now be definite and determined by the solution for smaller values of r. For values «’ of the «’s such that k?, defined by (8), is positive, the k in (12) must be chosen to be the positive square root of &, in order that (12) may represent only outward moving particles, i.e. particles for which the radial component of momentum, which from § 38 equals p,—thr-1 or —th(d/er+r-1), has a positive value. We now have that the density of scattered particles belonging to scatterers in state a’, equal to the square of the modulus of (12), falls off with increasing y according to the inverse square law, as is physically § 49 THE SCATTERING COEFFICIENT 191 necessary, and their angular distribution is given by |u(6¢a’)|?. Further, the magnitude, P’ say, of the momentum of these scattered particles must equal kf, the momentum ‘being radial for large 7, so that their energy is equal to 12 242 FO EE 2 BY Bol) = Ho!) Hiya’) + with the help of (8) and (10). This is just the energy of an incident particle, namely p%/2m, reduced by the increase in energy of the scatterer, namely H,(a’)—H,(a°)s in agreement with the law of con- servation of energy. For values «’ of the «’s such that 4? is negative there are no scattered particles, the total initial energy being insuffi- cient for the scatterer to be left in the state a’. We must now evaluate u(#da’) for a set of values «’ for the a’s such that k? is positive, and obtain the angular distribution of the scattered particles belonging to scatterers in state a’. It is sufficient to evaluate uw for the direction @ = 0 of the pole of the polar coordinates, since this direction is arbitrary. We make use of Green’s theorem, which states that for any two functions of position A and B the volume integral i] (AV?-B—BV?A) d'x taken over any volume equals the surface iutegral il (Adb/én— BOA/on) dS taken over the boundary of the volume, @/én denoting differentiation along the normal to the surface. We take A= e—ikr cos 7 B= and apply the theorem to a large sphere with the origin as centre. The volume integrand is thus e-ikr eos 8 V2— = e~tkroosé from (7) or (11), while the surface integrand is, with the help of (12), p® Im’ . o., e-tbreond = robot! | LDb—~ a’ of the scatterer. Tt depends on that matrix element (p’o’|V|p%x°> of the perturbing energy V whose column p%° and whose row p’«’ refer respectively to the initial and fina] states of the unperturbed system, between which the scattering transition process takes place. The result (15) is thus in some ways analogous to the result (24) of § 44, although the numerical coefficients are different in the two cases, corresponding to the different natures of the two transition processes. 50. Solution with the momentum representation The result (15) for the scattering coefficient makes a reference only to that representation in which the momentum p is diagonal. One would thus expect to be able to get a more direct proof of the result by working all the time in the p-representation, instead of working im the x-representation and transforming at the end to the p-repre- sentation, as was done in § 49. This would not at first sight appear to be a great improvement, as the lack of directness of the x-repre- sentation method is offset by more direct applicability, it being possible to picture the square of the modulus of the x-representative of a state as the density of a stream of particles in process of being scattered. The x-representation method has, however, other more serious disadvantages. One of the main applications of the theory of collisions is to the case of photons as incident particles. Now a photon is not a simple particle but has a polarization. It is evident from classical electromagnetic theory that a photon with a definite momentum, i.e. one moving in a definite direction with a definite frequency, may have a definite state of polarization (linear, circular, etc.), while a photon with a definite position, which is to be pictured as an electromagnetic disturbance confined to a very small volume, cannot have any definite polarization. These facts mean that the polarization observable of a photon commutes with its momentum but not with its position. This results in the p-representation method being immediately applicable to the case of photons, it being only necessary to introduce the polarizing variable into the representatives and treat it along with the o’s describing the scatterer, while the 194 COLLISION PROBLEMS § 50 x-representation method is not applicable. Further, in dealing with photons, it is necessary to take relativistic mechanics into account. This can easily be done in the p-representation method, but not so easily in the x-representation method. Equation (3) still holds with relativistic mechanics, but W is now given by W2/c2 == m2c2-+ P? = me2+Lp2-+p2-+p? (16) instead of by (2). Written in terms of p-representatives, equation (3) gives {B' —H,(o')—W} = = , (17) where W = H’—H,{«’) (18) and is the energy required by the law of conservation of energy for a scattered particle belonging to a scatterer in state «’. The ket |0> is represented by (6) in the x-representation and the basic ket |p°a®» is represented by (xa! [Pa = Byigo(X|P% = Syryo herr, from the transformation function (54) of § 23. Hence |0> = hi|p%®>, (19) and equation (17) may be written (W'—W) == hi pa’ |V |p). (20) We now meke a transformation from the Cartesian coordinates Das Pys Pz Of p to its polar coordinates P, w, x, given by Dy = Pcosw, Py = Psinw cos x, p, = Psinwsiny. If in the new representation we take the weight function P?sinw, then the weight attached to any volume of p-space will be the same as in the previous p-representation, so that the transformation will mean simply a relabelling of the rows and columns of the matrices without any alteration of the matrix elements. Thus (20) will become in the new representation (W'— W)< Puyo’ |L> = MC Paya’ |V| Paya, (21) W being now a function of the single variable P. § 50 SOLUTION WITH MOMENTUM REPRESENTATION 195 The coefficient of (Pwya’|1>, namely W’—W, is now simply a multiplying factor and not a differential operator as it was with the x-representation method. We can therefore divide out by this factor and obtain an explicit expression for . When, however, «’ is such that W’, defined by (18), is greater than mec?, this factor will have the value zero for a certain point in the domain of the variable P, namely the point P = P’, given in terms of W’ by (16). The function will then have a singularity at this point. This singularity shows that represents an infinite number of particles moving about at great distances from the scatterers with energies indefinitely close to W’ and it is therefore this singularity that we have to study to get the angular distribution of the particles at infinity. The result of dividing out (21) by the factor W’— W is, according to (18) of § 15, (Paya’ |) = AK Paya’ |V | Powx%a>/(W’— W)+A(wxe’) 8(W'— W), (22) where A is an arbitrary function of w, y, and «’. To give a meaning to the first term on the right-hand side of (22), we make the conven- tion that its integral with respect to P over a range that includes the value P’ is the limit when ¢->0 of the integral when the small domain P’—« to P’-+« is excluded from the range of integration. This is sufficient to make the meaning of (22) precise, since we are interested effectively only in the integrals of the representatives of states when the representation has continuous ranges of rows and columns. We see that equation (21) is inadequate to determine the representative completely, on account of the arbitrary function A occurring in (22). We must choose this A such that represents only outward moving particles, since we want_ the only inward moving particles to be those corresponding to |0>. Let us take first the general case when the representative of a state of the particle satisfies an equation of the type (W’—W) = f(Pex), (23) where f(Pwx) is any function of P, w, and x, and W" is a number greater than mc?, so that is of the form (Pwx|> = f(Pex)|(W'—W)+A(wox) 8(W'"— W), (24) and let us determine now what A must be in order that may 196 . COLLISION PROBLEMS § 50 represent only outward moving particles. We can do this by trans- forming to the x-representation, or rather the (r6¢)-repre- sentation, and comparing it with (12) for large values of r. The transformation function is — f-icip.xi — hy-ietPricos w cos 0+ sin w sin § coa(y—S)yii For the direction 9 = 0 we find co ar T = h-# ii P?dP ii dy J sinw dw ePrewseli¢ Pury | 0 0 ~ an iPr cos colts w= — P2dP | dr ~ | pea Pwrxl>| + w=0 0 Qa ; [a etPrevsaf 9 (Pu | TOTP rR Ba OX! f- 0 The second term in the { } brackets is of order 7-2, as may be verified by further partial integrations with respect to w, and can therefore be neglected. We are left with foe) 2ar = thar) | P dP | dy {ert Pry |y—etP POy|)} 9 0 = th-iy-} { P dP {e*Prlh¢ Prry|y— et Prt Poy ly}, (25) 0 When we substitute for its value given by (24), the first term in the integrand in (25) gives ih} [e dP e*Paill (Py) |(W!—W)-+-Amy) 8(W'—W)}.. (26) The term involving 6(W’— IW) here may be integrated immediately and gives, when one uses the relation PdP = W aW/c?, which follows from (16), ihote-2p-1 f W dW ePr\(ry)8(W'—W) me = th-te~ 1 W'D(ary)etPh, (27) To integrate the other term in (26) we use the formula e-iPrih Nf extbrlh | mrs per = ae) [ 8 ae, (28) 0 . 0 § 50 SOLUTION WITH MOMENTUM REPRESENTATION 407 with neglect of terms involving r-1, for any continuous function g(P), which formula holds since | K(P)e-Prlt dP is of order r-! for any 0 continuous function K(P) and since the difference g(P)|(P'—P)—g(P')/(P’— P) is continuous. The right-hand side of (28), when evaluated with neglect of terms involving r—1, and also with neglect of the small domain P’—« to P’-+-e in the domain of integration, gives foe) foe) Pp e7tPrit dP Pp iPr eP'—Pyrlh ap = ig(P’etPrn | map "ap = ing(P’)ePl, (29) In our present example g(P) is g(P) = thr P f(Prx)(P'— P)/(W'—W), which has the limiting value when P = P’, g(P’) = th-3 PY (P’ay) WP’? = ther Wf (P'2x). Substituting this in (29) and adding on the expression (27), we obtain the following value for the integral (26) h-te27 Wf — af (P’ary) +iN(ax) pe, (30) Similarly the second term in the integrand in (25) gives h-tc2r Wf — a f(P’ 0x) —iA(Ox jer". (31) The sum of these two expressions is the value of <70¢/> when r is large. We require that shall represent only outward moving particles, and hence it must be of the form of a multiple of eP”, Thus (30) must vanish, so that \ ary) = —inf(P'rx). (32) We see in this way that the condition that shall represent only outward moving particles in the direction 0 = 0 fixes the value of A for the opposite direction @ = 7. Since the direction @ = 0 or w = 0 of the pole of our polar coordinates is not in any way singular, we can generalize (32) to Ney) = —inf(P’ox), (33) 3595.57 fe) 198 COLLISION PROBLEMS § 50 which gives the value of A for an arbitrary direction. This value substituted in (24) gives a result that may be written (Pwx|> = f(Pwx)/(W'— W)—in 8(W'—W)}, (34) since one can substitute P’ for P in the coefficient of a term involving 8(W’—W) as a factor without changing the value of the term. Zhe condition that shall represent only outward moving particles is thus that it shall contain the factor O/(W'—W)—in 8(W'—W)}. (35) It is interesting to note that this factor is of the form of the right- hand side of equation (15) of § 15. With A given by (33), expression (30) vanishes and the value of for large r is given by expression (31) alone, thus (r0$|> = —2th-to-tr AW 'F(P'Oy err. This may be generalized to == —Qah-te~2e 1 W'f(P’wy)elP 7h, giving the value of for any direction @, d in terms of f(P’wy) for the same direction labelled by w, y. This is of the form (12) with u(Oh) = —2rh-*c* W'f(P’wx) and thus represents a distribution of outward moving particles of momentum P’ whose number is CP’. 4a*W'P’ 1 2 ° per unit solid angle per unit time. This distribution is the one represented by the (Pwy|> of (34). From this general result we can infer that, whenever we have a representative representing only outward moving particles and satisfying an equation of the type (23), the number per unit solid angle per unit time of these particles is given by (36). If this occurs in a problem in which the number of incident particles is one per unit volume, it will correspond to a scattering coefficient of amount 2Wow p: eT Pont (37) It is only the value of the function f(Pwy) for the point P = P’ that is of importance. - § 50 SOLUTION WITH MOMENTUM REPRESENTATION 199 If we now apply this general theory to our equations (21) and (22), we have f{(Poy) = hi Paya’ |V | P%wy%q%, Hence from (37) the scattering coefficient is Soh? WW! P’ [ct P®. |< P’ axa! |V | Pa (7. (38) If one neglects relativity and puts W°W’/c* = m2, this result reduces to the result (15) obtained in the preceding section by means of Green’s theorem. 51. Dispersive scattering We shall now determine the scattering when the incident particle is capable of being absorbed, that is, when our unperturbed system of scatterer plus particle has closed stationary states with the particle absorbed. The, existence of these closed states for the unperturbed system will be found to have a considerable effect on the scattering for the perturbed system, and indeed an effect that depends very much on the energy of the incident particle, giving rise to the pheno- menon of dispersion in optics when the incident particle is taken to be a photon. | We use a representation for which the basic kets correspond to the stationary states of the unperturbed system, as was the case with the p-representation of the preceding section. We take these station- ary states to be the states (p’a’) for which the particle has a definite momentum p’ and the scatterer is in a definite state «’, together with the closed states, & say, which form a separate discrete set, and assume that these states are all independent and orthogonal. This assumption is not accurate when the particle is an electron or atomic nucleus, since in this case for an absorbed state & the particle will still certainly be sormewhere, so that one would expect to be able to expand [k> in terms of the eigenkets |x’a’> of x, y, 2, and the a’s, and hence also in terms of the |p’a’>’s. On the other hand, when the particle is a photon it will no longer exist for the absorbed states, which are then certainly independent of and orthogonal to the states (p'«’) for which the particle does exist. Thus the assumption is valid in this case, which is an important practical one. Since we are concerned with scattering, we must still deal with - stationary states of the whole system. We shall now, however, have to work to the second order of accuracy, so that we cannot use merely 200 COLLISION PROBLEMS § 51 the first-order equation (3), but must use also (4). Equation (3) becomes, when written in terms of representatives in our present representation, (W'—W) = = = = . Expanding the right-hand sides by matrix multiplication, we get (W’—W) = XJ dp" p's" [D+ -+ ¥ is still given by (19), so (39) may be written (W'—W) = hic pa’ |V |p), (42) (B’—E,) of V vanish, since these matrix elements are not essential to the phenomena under investigation, and if they did not vanish it would mean simply that the absorbed states & had not been suitably chosen. We shall further assume that the matrix elements (p’a’|V |p’w”> are of the second order of smallness when the matrix elements , are taken to be of the first order of smallness. This assumption will be justified for the case of photons in § 64. We now have from (43) and. (42) that ¢k|1) is of the first order of smallness, provided E’ does not he near one of the discrete set of energy-levels E,, and (pa'|1> is of the second order. The value of (pa’|2> to the second order will thus be given, from the first of equations (41), by (W'—W) = A 2 Ba" |V |k"> plus (pa’|2>, therefore satisfies (W’—W){< pa’ |1>-+< pa’ |29} = Apo’ |V |pw>+ p3 {pa’|V |k>/(£’ —E,)}. This equation is of the type (23), provided «’ is such that W’ > me?, which means that «’ as a final state for the scatterer is not incon- sistent with the law of conservation of energy. We can therefore infer from the general result (37) that the scattering coefficient is 47h Wow' P|, , ‘a! |¥ |A>CR|V | px |? NE cpa’ pao) +S Pe ey (44) k The scattering may now be considered as composed of two parts, a part that arises from the matrix element of the per- turbing energy and a part that arises from the matrix elements {p’a’|V|k> and . The first part, which is the same as our previously obtained result (38), may be called the direct scattering. The second part may be considered as arising from an absorption of the incident particle into some state k, followed immediately by a re-emission in « different direction, and is like the transitions through an intermediate state considered in § 44. The fact that we have to add the two terms before taking the square of the modulus denotes interference between the two kinds of scattering. There is no experi- mental way of separating the two kinds, the distinction between them being only mathematical. 52. Resonance scattering Suppose the energy of the incident particle to be varied con- tinuously while the initial state «° of the scatterer is kept fixed, so that the total energy H’ or H’ varies continuously. The formula (44) now shows that as HZ’ approaches one of the discrete set of energy- levels H,, the scattering becomes very large. In fact, according to formula (44) the scattering should be infinite when H’ is exactly equal to an #,. An infinite scattering coefficient is, of course, physically impossible, so that we can infer that the approximations used in deriving (44) are no longer legitimate when £’ is close to an H,. To investigate the scattering in this case we must therefore go back to the exact equation (E’ —B)|H’) = VIB’) equation (2) of § 43 with #’ written for H’, and use a different method 202 COLLISION PROBLEMS § 52 of approximating to its solution. This exact equation, written in terms of representatives like (41), becomes (’ —W) = >| dp" (p'a"|H) +S , (H!~By) Bp" pa") + IVR RY. Let us take one particular H,, and consider the case when Z’ is close toit. The large term in the scattering coefficient (44) now arises from those elements of the matrix representing V that lie in row & or in column &, i.e. those of the type <&£|V|pa’> or (pa’|V[k>. The scatter- ing arising from the other matrix elements of V is of a smaller order of magnitude. This suggests that in our exact equations (45) we should make the approximation of neglecting all the matrix elements of V except the important ones, which are those of the type or , where a’ is a state of the scatterer that has not too much energy to be disallowed as a final state by the law of conservation of energy. These equations then reduce to (W'—W) = = al Ck|V |pa’> dp Cpe’ |H">, the «’ summation being over those values of «’ for which W’ given by (18) is > mc. These equations are now sufficiently simple for us to be able to solve exactly without further approximation. From the first of equations (46) we obtain by division (pa’ HE") = |(W'—W)+28(W'—W). (47) We must choose A, which may be any function of the momentum pand «’, such that (47) represents the incident particles corresponding to |0> or A?|p%®) together with only outward moving particles. [The representative of h!fp°a) is actually of the form Ad(W’— W), since the conditions «’ = «® and p= p°® for it not to vanish lead to W’ = E'—H,(o') = EB’ —H,(o°) = W = W.] Thus (47) must be = Apo" | pa) + +{1/(W'— W)—in d(W'—W)}, (48) and from the general formula (37) the scattering coefficient will be 4a? WW" P' het P®. | p'a’ |V kD |? |Ck| A. (49) (45) (46) § 52 RESONANCE SCATTERING 203 It remains for us to determine the value of . We can do this by substituting for in the second of equations (46) its value given by (48). This gives (B!—B,)k| "> = WARY (pa) + + > ) [kV | po’> [2(1/(W’ — W)—in 8(W'— WY) ap = hi(a—16), where a= > | KkIV |po’>|? d®p/(W'—W) (50) Ot and yaad | {k|V pw >a’ —W) Mp = 9 > Hy [|?8(W'— W)P2dP sin w dud = 7 > P We ff [k|V|P’wya'>|?sin w dedy. (51) Thus ki") = h8Ck|V |p%°)/(E’ —E,—a-+ib). (52) Note that a and 6 are real and that 0 is positive. This value for [PF 1Ce|V | pon |? cp? (H' —H,,—a)?+-6? : One can obtain the total effective area that the incident particle must hit in order to be scattered anywhere by integrating (53) over all directions of scattering, i.e. by integrating over all directions of the vector p’ with its magnitude kept fixed at P’, and then summing over all «’ that are to be taken into consideration, ie. for which W’ > me. This gives, with the help of (51), the result Ach? W b], and determine the probability that at some later time the particle shall be on its way to infinity with a definite momentum. The method of § 46 can now be applied. From the result (39) of that section we see that the probability per unit time per unit range of w and y, of the particle being emitted in any direction w’, y’ with the scatterer being left in state «’ is Qa CW ew’ ya |V [> |, (55) provided, of course, that «’ is such that the energy W’, given by (18), of the particle is greater than mc”. For values of «’ that do not satisfy this condition there is no emission possible. The matrix element here must refer to a representation in which W, w, y, and « are diagonal with the weight function unity. The matrix elements of V appearing in the three preceding sections refer to a repre- sentation in which p,, p,, p, are diagonal with the weight function unity, or P, w, x are diagonal with the weight function P?sinw. They would thus refer to a representation in which W, w, y are diagonal with the weight function dP/dW.P*?sinw = WP/c?.sinw. Thus the matrix element in (55) is equal to (W'P'/c? sin wv’)? times our previous matrix element or (p’a'|V|k>, so that (55) is equal to 2a W'P" hk oe sin w’||?. (56) To obtain the total probability per unit time of the particle being emitted in any direction, with any final state for the scatterer, we § 53 EMISSION AND ABSORPTION 205 must integrate (56) over all angles w’, x’ and sum over all states a’ whose energy 4,(«’) is such that H,(a’)+mce? < #,. The result is just 2b/%, where b is defined by (51). There ts thus this simple rela- tion between the total emission coefficient and the half-width b of the absorption line. Let us now consider absorption. This requires that we shall take an initial state for which the particle is certainly not absorbed but is incident with a definite momentum. Thus the ket corresponding to the initial state must be of the form (19). We must now determine the probability of the particle being absorbed after time 4. Since our final state & is not one of a continuous range, we cannot use directly the result (39) of § 46. If, however, we take |0> = |p», (57) as the ket corresponding to the initial state, the analysis of §§ 44 and 46 is still applicable as far as equation (36) and shows us that the proba- bility of the particle being absorbed into state & after time ¢ is 2| |-[1 —cos{(H,—B")t/B} | E,— EB’). This corresponds to a distribution of incident particles of density h-8, owing to the omission of the factor #! from (57), as compared with (19). The probability of there being an absorption after time t when there is one incident particle crossing unit area per unit time is therefore 2h W/c2®P?. [ |?[1—cos{(H,— #’)i/i} (L,Y. (58) To obtain the absorption coefficient we must consider the incident particles not all to have exactly the same energy W° = H'—H,(a°), but to have a distribution of energy values about the correct value E,,—H,(o®) required for absorptidn. If we take a beam of incident particles consisting of one crossing unit area per unit time per unit energy range, the probability of there being an absorption after time t will be given by the integral of (58) with respect to H’. This integral may be evaluated in the same way as (37) of § 46 and is equal to 4h We? P. ||?. The probability per unit time of an absorption taking place with an incident beam of one particle per unit area per unit time per unit energy range is therefore 42h? W/e2P?. [Ck|V | p%) |2, (59) which is the absorption coefficient. 206 COLLISION PROBLEMS § 53 The connexion between the absorption and emission coefficients (59) and (56) and the resonance scattering coefficients calculated in the preceding section should be noted. When the incident beam does not consist of particles all with the same energy, but consists of a unit distribution of particles per unit energy range crossing unit area per unit time, the total number of incident particles with energies near an absorption line that get scattered will be given by the integral of (54) with respect to #’. If one neglects the dependence of the numerator of (54) on H’, this integral will, since r b | (HB, — ape have just the value (59). Thus the total number of scattered particles in the neighbourhood of an absorption line is equal to the total number absorbed. We can therefore regard all these scattered particles as absorbed particles that are subsequently re-emitted in a different direction. Further, the number of particles in the neighbourhood of the absorption line that get scattered per unit solid angle about a given direction specified by p’ and then belong to scatterers in state a’ will be given by the integral with respect to E’ of (53), which integral has in the same way the value 2h2T70 yy" Pp! SE Kp'a' IFIED ICEIV tp %> This is just equal to the absorption coefficient (59) multiplied by the emission coefficient (56) divided by 26/#, the total emission coefficient. This is in agreement with the point of view of regarding the resonance scattered particles as those that are absorbed and then re-emitted, with the absorption and emission processes governed independently each by its own probability law, since this point of view would make the fraction of the total number of absorbed particles that are re-emitted in a unit solid angle about a given direction just the emission coefficient for this direction divided by the total emission coefficient. df’ = a,