Ⅹ THEORY OF RADIATION
放射線の理論

59. An assembly of bosons

We consider a dynamical system composed of w’ similar particles. We set up a representation for one of the particles with discrete basic kets |a®>, |x, jo >,.... Then, as explained in § 54, we get a sym- metrical representation of the assembly of w’ particles by taking as basic kets the products |a#> [a&> lag... lo> == lad of af...0%-> (1) in which there is one factor for each particle, the suffixes 1, 2, 3,..., w’ of the a’s being the labels of the particles and the indices a, b, ¢,...,9 denoting indices , ®, ®,... in the basic kets for one particle. Tf the particles are bosons, so that only symmetrical states occur in nature, then we need to work with only the symmetrical kets that can be constructed from the kets (1). The states corresponding to these symmetrical kets will form a complete set of states for the assembly of bosons. We can build up a theory of them as follows. We introduce the linear operator S defined by S=w!lt> P, (2) the sum being taken over all the wu’! permutations of the w’ particles. Then S applied to any ket for the assembly gives a symmetrical ket. We may therefore call S the symmetrizing operator. From (8) of § 55 it is real. Applied to the ket (1) it gives wl Plot of af...08,> == Sla%a%a®...a%>, (3) the labels of the particles being omitted on the right-hand side as they are no longer relevant. The ket (3) corresponds to a state for the assembly of u’ bosons with a definite distribution of the bosons among the various boson states, without any particular boson being assigned to any particular state. The distribution of bosons is specl- fied if we specify how many bosons are in each boson state. Let Ni, Ny, 24,... be the numbers of bosons in the states a, of), of)... respectively with this distribution. The n’’s are defined algebraically by the equation attabtott..f-ed = ny of? 4 Ny) } Ny oe (4) The sum of the ns is of course u’. The number of n”s is equal to the number of basic kets [a ), which in most applications of the 226 THEORY OF RADIATION § 59 theory is very much greater than u’, so most of the n’’s will be zero. If a, «®, of,..., a? are all different, i.e. if the n'’s are all 0 or 1, the ket (3) is normalized, since in this case the terms on the left-hand side of (3) are all orthogonal to one another and each contributes u'!-1 to the squared length of the ket. However, if a4, 0, 0%)..., a7 are not all different, those terms on the left-hand side of (3) will be equal which arise from permutations P which merely interchange bosons in the same state. The number of equal terms will be m,! ng! ng!..., so the squared length of the ket (3) will be Cote? ol,..o9| 8? lataae...o% == ny! ng! ng!..- (5) For dealing with a general state of the assembly we can introduce the numbers ”,, M», %3,... of bosons in the states o, a, o®),.., respectively and treat the n’s as dynamical variables or as observ- ables. They have the eigenvalues 0, 1, 2,..., u’. The ket (3) is a simultaneous eigenket of all the n’s, belonging to the eigenvalues M4, Ny, Ng,.... The various kets (3) form a complete set for the dynamical system consisting of u’ bosons, so the n’s all commute (see the converse to the theorem of § 13). Further, there is only one independent ket (3) belonging to any set of eigenvalues 7}, M9, %3,---- Hence the n’s form a complete set of commuting observables. If we normalize the kets (3) and then label the -resulting kets by the eigenvalues of the n’s to which they belong, ie. if we put (m4! 25! 13!...) “FS fataba®...o8) = [ny ng ng...>, (6) we get a set of kets |[n, ,73...>, with the n”s taking on all non-negative integral values adding up to wu’, which kets will form the basic kets of a representation with the n’s diagonal. The n’s can be expressed as functions of the observables Oy, Og, Q%g,.., &y Which define the basic kets of the individual bosons by means of the equations rg = >. 3a, ay? (7) or the equations d a flo%) = ¥ f(a) (8) holding for any function /. Let us now suppose that the number of bosons in the assembly is not given, but is variable. This number is then a dynamical variable or observable u, with eigenvalues 0, 1, 2,..., and the ket (3) is an eigenket of wu belonging to the eigenvalue u’. To get a complete set of kets for our dynamical system we must now take all the § 59 AN ASSEMBLY OF BOSONS 227 symmetrical kets (3) for all values of u’. We may arrange them in order thus ty Jaty, Slata®y, Satara”), ..., (9) where first is written the ket, with no label, corresponding to the state with no bosons present, then come the kets corresponding to states with one boson present, then those corresponding to states with two bosons, and so on. A general state corresponds to a ket which is a sum of the various kets (9). The kets (9) are all orthogonal to one another, two kets referring to the same number of bosons being orthogonal as before, and two referring to different numbers of bosons being orthogonal since they are eigenkets of wu belonging to different eigenvalues. By normalizing all the kets (9), we get a set of kets like (6) with no restriction on the 7’s (i.e. each n’ taking on all non- negative integral values) and these kets form the basic kets of a representation with the n’s diagonal for the dynamical system con- sisting of a variable number of bosons. If there is no interaction between the bosons and if the basic kets lo), |x®,... correspond to stationary states of a boson, the kets (9) will correspond to stationary states for the assembly of bosons. The number w of bosons is now constant in time, but it need not be a specified number, i.e. the general state is a superposition of states with various values for uw. If the energy of one boson is H(a), the energy of the assembly will be > (os) = 2 gH" (10) from (8), H* being short for the number H(a*). This gives the Hamiltonian for the assembly as a function of the dynamical variables n. 60. The connexion between bosons and oscillators In § 34 we studied the harmonic oscillator, a dynamical system of one degree of freedom describable in terms of a canonical q and p, such that the Hamiltonian is a sum of squares of g and p, with numerical coefficients. We define a general oscillator mathematically as a system of one degree of freedom describable in terms of a canonical g and p, such that the Hamiltonian is a power series in ¥ and p, and remains so if the system is perturbed in any way. We shall now study a dynamical system composed of several of these oscillators. We can describe each oscillator in terms of, instead of q and p, a complex dynamical variable 7, like the y of § 34, and its 228 THEORY OF RADIATION § 60 conjugate complex 7, satisfying the commutation relation (7) of § 34. We attach labels 1, 2, 3,... to the different oscillators, so that the whole set of oscillators is describable in terms of the dynamical variables 41,° 2, Nase Fi Te a--- Satisfying the commutation relations Na oN Na = 9; Nate Ta = 9, (11) Fa Meo Ha = Sab- Put Nata = Me (12) so that Fate = Mtl. (13) The n’s are observables which commute with one another and the work of § 34 shows that each of them has as eigenvalues all non- negative integers. For the ath oscillator there is a standard ket for the Fock representation, |0,> say, which is a normalized eigenket of n, belonging to the eigenvalue zero. By multiplying all these standard kets together we get a standard ket for the Fock representation for the set of oscillators, 10, }O,) 10) (14) which is a simultaneous eigenket of all the n’s belonging to the eigenvalues zero. We shall denote it simply by |0>. From (13) of §34 7jql0> = 0 (15) for any a. The work of §34 also shows that, if m,, mg, ”5,... are any non-negative integers, ott nt nB...|0> (16) is a simultaneous eigenket of all the n’s belonging to the eigenvalues Ny, Ng, My,... respectively. The various kets (16) obtained by taking different n’’s form a complete set of kets all orthogonal to one another and the square of the length of one of them is, from (16) of § 34, n,!ng!ng!.... From this we see, bearing in mind the result (5), that the kets (16) have just the same properties as the kets (9), so that we can equate each ket (16) to the ket (9) referring to the same n’ values without getting any inconsistency. This involves putting S focterPod’....0.7 = NaN Ter Iq|0>- (17) The standard ket |0> becomes equal to the first of the kets (9), corre- sponding to no bosons present. The effect of equation (17) is to identify the states of an assembly of bosons with the states of a set’ of oscillators. This means that the §60 CONNEXION BETWEEN BOSONS AND OSCILLATORS = 229 dynamical system consisting of an assembly of similar bosons 18 equiva- lent to the dynamical system consisting of a set of oscillators—the two systems are just the same system looked at from two different points of view, There is one oscillator associated with each independent boson state. We have here one of the most fundamental results of quantum mechanics, which enables a unification of the wave and corpuscular theories of light to be effected. Our work in the preceding section was built up on a discrete set of basic kets |x) for a boson. We could pass to a different discrete set of basic kets, [84> say, and build up a similar theory on them. The basic kets for the assembly would then be, instead of (9), I>, IB4>, SIB4B™>, SIB4B3B°, .... (18) The first of the kets (18), referring to no bosons present, is the same as the first of the kets (9). Those kets (18) referring to one boson present are linear functions of those kets (9) referring to one boson present, namely iB = Ss joa? for a boson there will be a new set of oscillator variables 7,4, and corresponding to (17) we shall have S|BAB¥BY...> = 1478 7-0). (20) Thus a ket 147 ,...[0> with wu’ factors 44, 7,,... must be a linear func- tion of kets 7, 7p...0> with w’ factors 7,, 7),.... It follows that each linear operator 74 must be a linear function of the 7,’s. Equation (19) gives 7.410> = ¥ tall> and hence na = > NXa|B>. (21) Thus the 7’s transform according to the same law as the basic kets for aboson. The transformed y’s satisfy, with their conjugate complexes, the same commutation relations (11) as the original ones. The trans- formed 7’s are on just the same footing as the original ones and hence, when we look upon our dynamical system as a set of oscillators, the different degrees of freedom have no invariant significance. The 7s transform according to the same law as the basic bras for a boson, and thus the same law as the numbers forming the representative of a state x. This similarity people often describe by 359557 . Q 230 THEORY OF RADIATION § 60 saying that the 7,’s are given by a process of second quantization applied to , meaning thereby that, after one has set up a quantum theory for a single particle and so introduced the numbers representing a state of the particle, one can make these num- bers into linear operators satisfying with their conjugate complexes the correct commutation relations, like (11), and one then has the . appropriate mathematical basis for dealing with an assembly of the particles, provided they are bosons. There is a corresponding proce- dure for fermions, which will be given in § 65. Since an assembly of bosons is the same as a set of oscillators, it must be possible to express any symmetrical function of the boson variables in terms of the oscillator variables 7 and 7. An example of this is provided by equation (10) with »,7, substituted for x,. Let us see how it goes in general. Take first the case of a function of the boson variables of the form Up = & U, (22) where each U. is a function only of the dynamical variables of the rth boson, so that it has a representative referring to the basic kets |a®> of the rth boson. In order that Up may be symmetrical, this representative must be the same for all r, so that it can depend only on the two eigenvalues labelled by a and b. We may therefore waite 1 = . (25) Since U, is symmetrical we can replace SU, by U, S and can then substitute for the symmetrical kets in (25) their values given by (17). We get in this way Up Nay Neos |0> = 2 2 Na "in Yar Mest |O>, (28) < na meaning that the factor 7, must be cancelled out. Now from (15) and the commutation relations (11) No Neer Qasr |0> = > "a Ne, Veer 10> 802, (27) Tr §60 CONNEXION BETWEEN BOSONS AND OSCILLATORS — 231 (note that 7, is like the operator of partial differentiation 0/d7,), so (26) becomes Op Hey Ne|0> = 2 Na Tp Ney Ney |0> form a complete set, and hence we can infer from (28) the operator equation O, = 2 nao] T [b> ip. (29) 'This gives us Up in terms of the 7 and 7 variables and the matrix elements . Now Jet us take a symmetrical function of the boson variables consisting of a sum of terms each referring to two bosons, Vp = > Veg: (30) rSer We do not need to assume ¥, = V,. Corresponding to (23), V, has matrix elements at ab|Trglagad> = (31) for brevity. Proceeding as before we get, corresponding to (25), SVpoPoF...> = YY Slofta¥e..cf..08..> (32) r,Sezr ap and corresponding to (26) Vor Mee Max |0> = Na 2 1 Ne NaN |9P8on, San 640 |V led>. (33) aber T,Ser We can deduce as an extension of (27) Fe Fa Des Nee !09 =D Mees Mer Deer [Sen Saas (34) so that (33) becomes Vor Me, Maye (OY = Na Ae Fa Ney Neo|026a2| Ved, abe giving us the operator equation Vo = >> Na 1o A. Fa- (35) ‘The method can readily be extended to give any symmetrical func- tion of the boson variables in terms of the y’s and 7’s. The foregoing theory can easily be generalized to apply to an assembly of bosons in interaction with some other dynamical system, which we shall call for definiteness the atom. We must introduce a set of basic kets, |£ say, for the atom alone. We can then get a set of basic kets for the whole system of atom and bosons together by multiplying each of the kets |£’> into each of the kets (9). We may write these kets iC, [C'a%, S[Lata®>, S[lata’a®>, o (36) 232 THEORY OF RADIATION § 60 We may look upon the system as composed of the atom in interaction with a set of oscillators, so that it can be described in terms of the atom variables and the oscillator variables ,, 7,. Using again the standard ket |0> for the set of oscillators, we have Sl ata° a... = Na My Ne lOD[L>, (37) corresponding to (17), as the equation expressing the basic kets (36) in terms of the oscillator variables. Any function of the atom variables and boson variables which is symmetrical between all the bosons is expressible as a function of the atom variables and the 7’s and 7’s. Consider first a function Up of the form (22) with U. a function only of the atom variables and the variables of the rth boson, so that it has a representative <£’o2|U,|C"02>. This representative must be independent of r in order that Up may be symmetrical between all the bosons, so we may write it CC'at|O|6"aS. Now let us define to be that function of the atom variables whose representative is (C'a?|T7|C’«°>, so that we have = 10">, (38) corresponding to (23). The equations (24)—(28) can now be taken over and applied to the present work if both sides of all these equations are multiplied by [¢’> on the right, with the result that formula (29) still holds. We can deal similarly with a symmetrical function VY of the form (30) with V, a function only of the atom variables and the variables of the rth and sth bosons. Defining to be that function of the atom variables whose representative is (Colt ab 08 08), we find that formula (35) still holds. 61. Emission and absorption of bosons Let us suppose that the oscillators of the preceding section are harmonic oscillators and there is no interaction between them. The energy of the ath oscillator is then, from (5) of § 34, i, = hwy Na Fat tiw,. We shall neglect the constant term }iw,, which is the energy of the oscillator in its lowest state—the so-called ‘zero-point energy’. This neglect does not have any dynamical consequences, as explained at the beginning of § 30, and merely involves a redefinition of H,. The total energy of ail the oscillators is now Hy = > A, = > hwy a Na = > heog Na (39) @ a a § 61 EMISSION AND ABSORPTION OF BOSONS 233 with the help of (12). This is of the same form as (10), with hw, for He, Thus a set of harmonic oscillators 1s equivalent to an assembly of bosons in stationary states with no interaction between them. If an oscillator of the set ts in tis n’th quantum state, there are n’ bosons in the associated boson state. In general the Hamiltonian for the set of oscillators will be a power series in the variables 7,, 7,, Say Hy = Hp+ 2 (OU, Na+ Datta) + > (Ue 1a +Vab Na Mot Van Fah) +--+ (40) where Hp, U,, Ugy, Vy ave numbers, Hp being real and U,, = O,,. If the set of oscillators are in interaction with an atom, as we had at the end of the preceding section, the total Hamiltonian will still be of the form (40), with Hp, U,, U,,, V,, functions of the atom variables, Hy, in particular being the Hamiltonian for the atom by itself. A general treatment of this dynamical system would be rather compli- cated and for practical applications one assumes that the terms Hp+ > Daa Nata (41) are large compared with the others and form by themselves an unperturbed system, the remaining terms being taken into account as a perturbation producing transitions in the unperturbed system, according to the theory of § 44. If, further, U,, is independent of the atom variables, the unperturbed system with Hamiltonian (41) con- sists merely of an atom with Hamiltonian Hp and an assembly of bosons in stationary states with Hamiltonian of the form (39), with no interaction. Let us consider what kinds of transitions are produced by the various perturbation terms in (40). Take a stationary state of the unperturbed system for which the atom is in a stationary state, ¢’ say, and bosons are present in the stationary boson states, a, 6, ¢,.... This stationary state for the unperturbed system corresponds to the ket Ta Mo Ner+!9>18°>, (42) like (37). If the term U,», of (40) is multiplied into this ket, the result is a linear combination of kets like : Nae Qa Nd Nor |OS”, : , (43) t” denoting any stationary state of the atom. The ket (43) refers to one more boson than the ket (42), the extra boson being in the state x. 234 THEORY OF RADIATION § 61 Thus the perturbation term U, 7, gives rise to transitions in which one boson is emitted into state 2 and the atom makes an arbitrary jump. If the term U, 4, of (40) is multiplied into (42), the result is zero unless (42) contains a factor 7, and is then a linear combination of Kets like Teta Mo te lO>1E">, referring to one boson less in state xz. Thus the perturbation term U,,9~ gives rise to transitions in which one boson is absorbed from state x, the atom again making an arbitrary jump. Similarly, we find that a perturbation term U,,, 7,7, ( 4 y) gives rise to processes in which a boson is absorbed from state y and one is emitted into state x, or, what is the same thing physically, one boson makes a transition from state y to state x. This kind of process would be produced by a term like the U, of (22) and (29) in the perturbation energy, pro- vided the diagonal elements vanish. Again, the perturbation terms Vey "2 Ny» Vey Te Ty give rise to processes in which two bosons are emitted or absorbed, and so on for more complicated terms. With any of these emission and absorption processes the atom can make. an arbitrary jump. Let us determine how the probability of occurrence of each of these . transition processes depends on the numbers of bosons originally present in the various boson states. From §§ 44, 46 the transition probability is always proportional to the square of the modulus of the matrix element of the perturbation energy referring to the two states concerned. Thus the probability of a boson being emitted into state x with the atom making a jump from state ¢’ to state £” is proportional to 1<"| <1 9--(Mip+ 1). [Uy Tel Mo--Mee->10">|?, (44) the n”s being the numbers of bosons initially present in the various boson states. Now from (6) and (17), with reference to (4), , [105 My Mg... > == (My! 5! 5...) — Pt a7! 4... 109, (45) so that Ne |My Mg. Myee > = (1-1)? [24 N9..(n,+1)..>. (46) Hence (44) is equal to (m+ 1)/ CC U,12[?, (47) showing that the probability of a transition in which a boson is emitted into state x 1s proportional to the number of bosons originally in state x plus one. § 61 EMISSION AND ABSORPTION OF BOSONS © 235 The probability of a boson being absorbed from state x with the atom making a jump from state ¢’ to state ¢” is proportional to [<0" |< 23.-(M— 1). [Oy ply MaMa 10> [?, (48) the ns again being the numbers of bosons initially present in the various boson states. Now from (45) fol, Mya...) = n\n my.(m—1)..>, (49) so (48) is equal to nJco" 0169/2. (50) Thus the probability of a transition in which a boson is absorbed from state x is proportional to the number of bosons originally in state x. Similar methods may be applied to more complicated processes, and show that the probability of a process in which a boson makes a transition from state y to state x (x + y)is proportional to n,(n,+1). More generally, the probability of a process in which bosons are absorbed from states x, y,... and emitted into states a, b,... is propor- tional to nn. (Miy A) (myth)... (51) the n’’s being in each case the numbers of bosons originally present. These results hold both for direct transition processes and transition processes that take place through one or more intermediate states, in accordance with the interpretation given at the end of § 44. 62. Application to photons Since photons are bosons, the foregoing theory can be applied to them. A photon is in a stationary state when it is in an eigenstate of momentum. It then has two independent states of polarization, which may be taken to be two perpendicular states of linear polariza- tion. The dynamical variables needed to describe the stationary states are then the momentum p, a vector, and a polarization variable 1, consisting of a unit vector perpendicular to p. The variables p and 1 take the place of our previous «’s. The eigenvalues of p consist of all numbers from —co to co for each of the three Cartesian com- ponents of p, while for each eigenvalue p’ of p, | has just two eigenvalues, namely two arbitrarily chosen vectors perpendicular” to p’ and to one another. Owing to the eigenvalues of p forming a continuous range, there are a continuous range of stationary states, giving us the continuous basic kets |p'l’>. However, the fore- going theory was built up in terms of discrete basic kets |x’> for a boson. There are two formalisms which one may use for getting over this discrepancy. 236 THEORY OF RADIATION § 62 The first consists in replacing the continuous three-dimensional distribution of eigenvalues for p by a large number of discrete points lying very close together, forming a dust spread over the whole three- dimensional p-space. Let sp, be the density of the dust (the number of points per unit volume) in the neighbourhood of any point p’. Then sp must be large and positive, but is otherwise an arbitrary function of p’. An integral over the p-space may be replaced by a sum over the dust of points, in accordance with the formula Jf [PW dpi dp, dp, = ¥ fp')sp*, (52) which formula provides the basis of the passage from continuous p’ values to discrete ones and vice versa. Any problem can be worked out in terms of the discrete p’ values, for which the theory of §§ 59-61 can be used, and the results can be transformed back to refer to con- tinuous p’ values. The arbitrary density sp. should then disappear from the results. The second formalism consists in modifying the equations of the theory of §§ 59-61 so as to make them apply to the case of a con- ‘ tinuous range of basic kets |a’>, by replacing sums by integrals and replacing the 6 symbol in the commutation relations (11) by & func- tions, so far as concerns the variables with continuous eigenvalues. Each of these formalisms has some advantages and some disadvan- tages. The first is usually more convenient for physical discussion, the second for mathematical development. Both will be developed here and one or other will be used according to which is more suitable at the moment. The Hamiltonian describing an assembly of photons interacting with an atom will be of the general form (40), with the coefficients Hp, Uz, Uy, V,n involving the atom variables. This Hamiltonian may b : e€ written Hp = Hp-+Ho+Hp, (53) where Hp is the energy of the atom alone, H - is the energy of the assembly of photons alone, Hp = 2,?v op (54) vp being the frequency of a photon of momentum p’, and Hg is the interaction energy, which can be evaluated from analogy with the classical theory, as will be shown in the next section. The whole system can be treated by a perturbation method as discussed in the § 62 APPLICATION TO PHOTONS 237 preceding section, Hp, and H, providing the energy (41) of the unperturbed system and Hg being the perturbation energy, which gives rise to transition processes in which photons are emitted and absorbed and the atom jumps from one stationary state to another. We saw in the precéding section that the probability of an absorp- tion process is proportional to the number of bosons originally in the state from which a boson is absorbed. From this we can infer that the probability of a photon being absorbed from a beam of radiation incident on an atom is proportional to the intensity of the beam. We also saw that the probability of an emission process is propor- tional to the number of bosons originally in the state concerned plus one. To interpret this result we must make a careful study of the relations involved in replacing the continuous range of photon states by a discrete set. Let us neglect for the present the polarization variable 1. Let [p’p> be the normalized ket corresponding to the discrete photon state p’. Then from (22) of § 16 > |p’D> = [p'D>sp. (56) The connexion between |p’> and |p’D) is like the connexion between the basic kets when one changes the weight function of the representa- tion, as shown by (38) of § 16. With nj, photons in each discrete photon state p’, the Gibbs density p for the assembly of photons is, according to (68) of § 33, p => [p’D>n,n,, according to (73) of § 33. From (57) this equals (x'|plx'> = | given by (54) of § 23. Equation (58) expresses the number of photons per unit volume as an integral over the momentum space, so the inte- grand in (58) can be interpreted as the number of photons per unit of phase space. We obtain in this way the result that the number of photons per unit of phase space is equal to h- times the number of photons per discrete state, in other words, @ cell of volume h? in phase space is equivalent to a discrete state. This result is a general one, holding for any kind of particle. If the polarization variable of the photons is not neglected, the result holds for each of the two indepen- dent states of polarization. The momentum of a photon of frequency v is of magnitude hyv/c, so the element of momentum space dp, dp, dp, = Wc-v dvdu, dw being an element of solid angle for the direction of the vector p. Thus a distribution of photons with nj, per discrete state, which is equivalent to a distribution of h-nj,d?nd*x photons in an element of volume dz and an element of momentum space d°p, equals a distribution of nj, c~*v? dvdwd'x photons in an element of volume da and a frequency range @v and direction of motion dw. This corre- sponds to an energy density n, kc-*)> per unit solid angle per unit frequency range, or an intensity per unit frequency range (i.e. an energy crossing unit area per unit time per unit frequency range) of amount L, =n, hvIc2. (59) The result that the probability of a photon being emitted is pro- portional to n,,+1, 2,, being the number of photons initially present in the discrete state concerned, can now be interpreted as the proba- bility being proportional to £,,+-Av*/c?, where J,, is the intensity of the incident radiation per unit frequency range in the neighbourhood of the frequency of the emitted photon and having the same polariza- tion 1 as the emitted photon. Thus with no incident radiation there is still a certain amount of emission, but the emission is increased or § 62 APPLICATION TO PHOTONS 239 stimulated by incident radiation in the same direction and having the same frequency and polarization as the emitted radiation. The present theory of radiation thus completes the imperfect one of § 45 by giving both stimulated and spontaneous emission. The ratio it gives for the two kinds of emission, namely J,, : hv?/c?, is in agreement with that provided by Einstein’s theory of statistical equilibrium mentioned in § 45. The probability of a photon being scattered from the state p’l’ to the state p"l’ is proportional to ny1(Myy+1), the n’s being the numbers of photons initially in the discrete states concerned. We can interpret this result as the probability being proportional to Dry (Tye bho"9/c*). (60) Similarly for a more general radiative process in which several photons are emitted and absorbed, the probability is proportional to a factor J,, for each absorbed photon and a factor [,+-Av*/c? for each emitted photon. Thus the process is stimulated by incident radiation in the same direction and with the same frequency and polarization as any of the emitted photons. 63. The interaction energy between photons and an atom We shall now determine the interaction energy between an atom and an assembly of photons, ie. the Hg of equation (53), from analogy with the classical expression for the interaction energy between an atom and a field of radiation. For simplicity we shall suppose the atom to consist of a single electron moving in an electro- static field of force. The field of radiation may be described by a scalar and a vector potential. These potentials are to a certain extent arbitrary and may be chosen so that the scalar potential vanishes. The field is then completely described by the vector potential A,, A,, A,, or A. The change that the field. causes in the Hamiltonian describing the atom is now, as explained at the beginning of § 41, 1 -&a\7 nol @ Hg = 5,-|(p+54) P*| ag (Pe A) + e2 Qmc* AY (81) This is the classical interaction energy. The A that occurs here should be the value of the vector potential at the point where the electron is momentarily situated. It is, however, a good enough approximation if we take this A to be the vector potential at some fixed point in the atom, such as the nucleus, provided we are dealing with radiation whose wavelength is large compared with the dimensions of the atom. 240 THEORY OF RADIATION § 63 Let us first consider the field of radiation classically and ignore its interaction with the atom. The vector potential A satisfies, according to Maxwell’s theory, the equations A = 0, divA = 0, (62) being short for 6°/c? dt? 0?/éx*— @*/ay?— 6?/éz2, The first of these equations shows that A can be resolved into Fourier components in the form A= f fA, etka i2minel 4 A gitkx) nine 3p, (63) each Fourier component representing a train of waves moving with the velocity of light, described by a vector k whose direction gives the direction of motion of the waves and whose magnitude |k]| is connected with their frequency v, by Qavy, = c[k]. (64) The vector k is just the momentum of a photon which the quantum theory would associate with these waves, divided by #. For each value of k we have an amplitude A,, which is in general a complex vector, and the integral in (63) extends over the whole of the three- dimensional k-space. The second of equations (62) gives - (k, Ay) = 0, (65) showing that, for each value of k, A; is perpendicular to k. This expresses that the waves are transverse waves. A, 1s determined by its two components in two directions perpendicular to each other and to k, these two components corresponding to two independent states of linear polarization. The total energy of the radiation is given by the volume integral] Hy == (87)-4 [ (S24 #) Bx (66) taken over the whole of space, where the electric field € and the magnetic field # of the radiation are given by = Af = curlA. (67) Using standard formulas of vector analysis, we have div[A x Af] = (#, curl A)—(A, curl #) = Jf—(A, curl curl A) == A+ (A, VA) with the help of the second of equations (62). Thus (66) becomes, §63 INTERACTION ENERGY BETWEEN PHOTONS AND AN ATOM 241 with neglect of a term which can be transformed to a surface integral at infinity, (l/aA dA H. = 8 -i je ——;3 = = (Bn) } Fal at” Gt By substituting for A here its value given by (63), we can get the energy of the radiation in terms of the Fourier amplitudes A,. The energy of the radiation is constant (since we are now ignoring the interaction of the radiation and the atom), so in this calculation we may take ¢ = 0. This means taking A= | (Ay+A_Jei ah, (69) J vea)| de, (68) VA= — [ R( A, +A ,jete9 ask, @A/at = ic | Iki(A,, —A_, )e-@™ Bh, (70) Inserting these expressions in (68), we get Hy = (8) | { | (k/2(A, +A, Ay +A_.)— —{k||k'|(A,—A_,, Ay —A_, le et) hed dBa: =? [[ {k%(A, +A y, Ay tA) — |k||k’|(A, —A_,, Ay A, )}JS(K-+k') Bhd’, with the help of formula (49) of §23, 6(k+k’) being the product of three factors, one for each component of k. Hence Hp = 7 f K(A,+A_,, Ady Ay) —(Ay—A_4, A_.—A,)} d*ke d == dn? | k(A,, A,,) dk. . (71) We can replace the continuous distribution of k-values by a dust of discrete k-values,, like we did with the p-values in the preceding section. The integral (71) then goes over, according to formula (52), into the sum Hp = Aq? >) k?(A,, A,)sg', k s, being the density of the discrete k-values. We may also write this as Hy = 4? 3 WA, Ay sz}, (72) kl A,, being a component of A, in a direction 1 perpendicular to k and 242 THEORY OF RADIATION § 63 ‘the summation with respect to | referring to two directions | perpen- dicular +o each other, Thus there is one term in (72) for each inde- pendent stationary state for a photon. The field quantities € and # at any point x can be looked upon as dynamical variables. The quantities — Qrivyt A xe 4... gD RL Aggy = Aggy 87S, Ayy = Ayer are then dynamical variables at time #, since they are connected with & and # at various points x at time ¢ by equations which do not involve #, as follows from (63) and (67). A,, is constant, so A, varies with ¢ according to the simple harmonic law. Thus A,, is like the 7, of a harmonic oscillator, defined by (3) of § 34, the w of the oscillator being 27v,. We may take each A,,, to be proportional to the », of some harmonic oscillator and then the field of radiation becomes a set of harmonic oscillators. Let us now pass over to the quantum theory and take the A,yy, Ax to be dynamical variables in the Heisenberg picture. The expression (72) for the energy may be retained unchanged, the order in which the factors A,,, Ay, there occur being the correct one to give no zero- point energy. The A,, then still vary with time according to the e law and may still be taken to be proportional to the 7,’s of harmonic oscillators. The factor of proportionality may be obtained by equat- ing (72) to the expression (39) for the energy, with the label a replaced by the two labels k and I and with Av, for iw,. This gives de® > KA yy Ayy Se? = 2 hy Meu Tu: the suffix ¢ being inserted to show that we are dealing with Heisenberg dynamical variables (as we should when transferring equations of the classical theory to the quantum theory). Hence, using (64), 47? Ay = ch yy Sh, (73) with neglect of an unimportant arbitrary phase factor. In this way the Heisenberg dynamical variables 1,y, which describe the field of radiation as a set of oscillators, are introduced. The commutation relations between the 7, and 7, are known, being given by (11), so equation (73) fixes the commutation relations between the A,, and Ay, It thus fixes the commutation relations between the potentials A and the field quantities € and # at various points x at the time J. (Incidentally, the commutation relations of the Ay, 4 are fixed, §63 INTERACTION ENERGY BETWEEN PHOTONS AND AN ATOM 243 so the commutation relation of two potential or field quantities at two different times is also fixed.) We can still use (73) when the interaction between the field of radiation and the atom is taken into account. This involves assuming that the interaction does not affect the commutation relations . between the potentials and field quantities at a given time. The interaction causes the 7,,,s to cease to vary according to the simple harmonic law and the oscillators to cease to be harmonic. Thus it may affect the commutation relation between two potential or field quantities at two different times. We can now take over the interaction energy (61) into the quantum. theory, putting p, for p to show it is a Heisenberg dynamical variable. Taking the atomic nucleus to be at the origin we get, by substituting (63) with x = 0 into (61), é if on me al (Bp Aut Ay) ad +503 =f Aut Ags Ay t Ay 1) BhdPk’ e ~ 4, 8 ~ oo => > (Dy Aggy + Ang) 8k t+ ames > (Awt+ Ans Age + Anis set E ke if we pass from continuous to discrete k-values. Thus é 7 VK Ao, = me > PYWAxut Anus + >, (Anu Ayy)(A vvit All sg tse, py being the component of p, in the direction 1. With the help of (73) we may express Hp, in terms of the y, and 7,4, and we can then drop the suffix ¢ (which means going over.to Schrédinger dynamical variables), so that we obtain finally cht yn pot A,,)eci- 42m > PMc (Mert Hua) se 4 i eh 4 + 324m Mm Vic" KEI With the model of the atom we are using, the interaction energy appears as a linear plus a quadratic function in the y’s and 7's. The linear terms give rise to emission and absorption processes, the Hy = ‘mat tha) mert+ ter) Ws tse*. (74) 244 THEORY OF RADIATION § 63 quadratic ones to scattering processes and processes in which two photons are absorbed or emitted simultaneously. The order of the factors 7 and 7 in the quadratic terms is not determined by the procedure of working from the classical theory, but this order is unimportant, since a change in it merely changes Hy by a constant. The matrix element of Hg referring to the emission of a photon into the discrete state KI, or into the discrete state p’l, as it may also be labelled, with the atom jumping from state «° to state a’, is eh é€ ’ Ne ee Cy! -% 4772my't oe [plo >sy — mh(2av' jt Co [p,]o>s5 since s, == s,/%. The p, occurring here, referring to the momentum of the electron, is, of course, quite distinct from the other letters p, referring to the momentum of the emitted photon. To avoid con-— fusion we shall replace the electron momentum p by mx, these two dynamical variables being the same for the unperturbed atom. Pass- ing over to continuous photon states by means of the conjugate imaginary of equation (56), we get = é h(2zv')! Similarly, the matrix element of Hy referring to the absorption of a photon from the continuous state p% with the atom jumping from state «° to state a’ is = ¢P'lo’|Hg|0®> = Coe" |, |X%9>- (75) é A(2av%)t and the matrix element referring to the scattering of a photon from the continuous state p°l® to the continuous state p’l’ with the atom jumping from state «” to state a’ is , (76) nee e , = Orhimly't (LT) by-aas (77) there being two terms in (74) which contribute to it. These matrix elements will be used in the next section. The matrix elements referring to the simultaneous absorption or emission of two photons may be written down in the same way, but they lead to physical effects too small to be of practical importance. 64. Emission, absorption, and scattering of radiation We can now determine directly the coefficients of emission, absorp- tion, and scattering of radiation by substituting in the formulas of § 64 EMISSION, ABSORPTION, AND SCATTERING OF RADIATION 245 Chapter VIII the values for the matrix elements given by (75), (76), and (77). For determining the emission probability we can use formula (56) of §53. This shows that for an atom in a state a? the proba- bility per unit time per unit solid angle of its spontaneously emitting a photon and dropping to a state a’ of lower energy is 2 2 Now the energy and momentum of a photon of frequency v are W = hy, P = hy/e. Again, from the Heisenberg law (20) of § 29, Cot! |e == — 2rrtv(a?a') |”. (79) [ |. 246 THEORY OF RADIATION § 64 of § 45, in which the radiation field is treated as an external perturba- ton, gives the correct value for the absorption coefficient. This agreement between the elementary theory and the present theory could be inferred from general arguments. The two theories differ only in that the field quantities all commute with one another in the elementary theory and satisfy definite commutation relations in the present theory, and this difference becomes unimportant for strong fields. Thus the two theories must give the same absorption and emission when strong fields are concerned. Since both theories give the rate of absorption proportional to the intensity of the inci- dent beam, the agreement must hold also for weak fields in the case of absorption. In the same way the stimulated part of the emission in the present theory must agree with the emission in the elementary theory. Let us now consider scattering. The direct scattering coefficient is given by formula (38) of § 50. Such scattering of photons will not be accompanied by any change of state of the atom on account of the factor 6,0 in the expression for the matrix element (77). Thus the final energy W’ of the photon will equal its initial energy W°. The scattering coefficient now reduces to e4/méct. (1/19). This is the same as that given by classical mechanics for the scattering of radiation by a free electron. We thus see that the direct scatter- ing of radiation by an electron in an atom is independent of the atom and is correctly given by the classical theory. This result, it should be remembered, holds only provided the wavelength of the radiation is large compared with the dimensions of the atom. The direct scattering is a mathematical concept and cannot be separated out experimentally from the total scattering, given by formula (44) of § 51. Let us see what this total scattering is in the case of photons. We must be careful in our application of formula (44) of §51. The summation > in this formula may be considered as representing the contribution to the scattering of double transitions consisting of transitions firstly from the initial state to state k and secondly from state & to the final state. The first transition may be an absorption of the incident photon and the second an emission of the required scattered photon, but it is also possible for the first transition to be the emission and the second the absorption. It is clear from the general nature of the method used for deriving formula § 64 EMISSION, ABSORPTION, AND SCATTERING OF RADIATION 247 (44) of § 51 that both these kinds of double transitions must be in- cluded in the summation ¥ when this formula is applied to photons, i although only the first of them appears in the actual derivation given in § 51, as the possibility of the particle being created or annihilated was not taken into account there. We use zero, single prime, and double prime to refer to the initial, tinal, and intermediate states of the atom respectively, and zero and single prime to refer to the absorbed and emitted photons respec- - tively. Then, for the double transition of absorption followed by emission, we must take for the matrix elements , — of the formula (44) of § 51 (k|V |p = = = = and E' —E, = b+ Hp(«°)—Hp(2")—h hv’ = —hfp' + (00°) ], there being now two photons, of frequencies v® and v’, in existence for the intermediate state. Substituting in (44) of § 51 the values of the matrix elements given by (75), (76), and (77), we get for the sates coefficient i aI OD) Barae + +> fe faa ble ola SIP can If we write (81) in terms of x instead of 2, we get rey a (l 1) 8rg0 — > voxel” )o( 0” (= ria ae aad eel) oy We can simplify (82) with the help of the quantum conditions. We have Ly Lyo—LyoXy = 0, 248 THEORY OF RADIATION § 64 which gives > {ox" [ty] <” ary fe} = 0, (83) and also Cy Lyp— Lp y = 1/m. (ty Pp— Pye Ly) = whim. (1h), which gives > {dcx lay |ae”> . vac") —v oat” Ken latyo ja” . Lo" [ary [oe >} a 90) 8 igo = = (VP) Byrne. (84) 27m, Qari m Multiplying (83) by v’ and adding to (84), we obtain ¥ {6a ep.” [’ +-v(a"a®)] — Ca ee” a®>[v" + v(a!a")} == hf/2am. (V1) by-40- If we substitute this expression for #/27m.(11°)5,..0 in (82), we obtain, after a straightforward reduction making use of identical relations between the v’s, (27re)* yee ee ee hict v9 —p(a"n®) v’-+v(a"e®) This gives the scattering coefficient in the form of the effective area that a photon has to hit per unit solid angle of scattering. It is known as the Kramers-Heisenberg dispersion formula, having been first obtained by these authors from analogies with the classical theory of dispersion. The fact that the various terms in (82) can be combined to give the result (85) justifies the assumption made in deriving formula (44) of § 51, that the matrix elements of the interaction energy are of the second order of smallness compared with the ones, at any rate when the scattered particles are photons. 2 poy’3 . (85) a” 65. An assembly of fermions An assembly of fermions can be treated by a method similar to that used in §§ 59 and 60 for bosons. With the kets (1) we may use the antisymmetrizing operator A defined by A=w!4D4P, (2') summed over all permutations P, the + or — sign being taken according to whether P is eyen or odd. Applied to the ket (1) it gives WIE S + Plot ob o§...09> = AlatoPa®...0%>, (3') a ket corresponding to a state for an assembly of u’ fermions. The § 65 AN ASSEMBLY OF FERMIONS 249 ket (3’) is normalized provided the individual fermion kets |a”, |a”,... are all different, otherwise it is zero. In this respect the ket (3’) is simpler than the ket (3). However, (3’) is more complicated than (3) in that (3') depends on the order in which a”, 2, o”,... occur in it, being subject to a change of sign if an odd permutation is applied to this order. We can, as before, introduce the numbers 7), No, 3,... of fermions in the states ao, o, «,... and treat them as dynamical variables or observables. They each have as eigenvalues only 0 and 1. They form a complete set of commuting observables for the assembly of fermions. The basic kets of a representation with the n’s diagonal may be taken to be connected with the kets (3’) by the equation Alotaboe...a%> = |, ngng...> (6’) corresponding to (6), the n’s being connected with the variables at, a, of... by equation (4). The -- sign is needed in (6’) since, for given n’”’s, the occupied states «%, «”, a°,... are fixed but not their order, so that the sign of the left-hand side of (6’) is not fixed. To set up a rule which determines the sign in (6’), we must arrange all the states « for a fermion arbitrarily in some standard order. The a's oceurring in the left-hand side of (6’) form a certain selection from all the a’s and the standard order for all the a’s will give a standard order for this selection. We now make the rule that the + sign should occur in (6’) if the «’s on the left-hand side can be brought into their standard order by an even permutation and the — sign if an odd permutation is required. Owing to the complexity of this rule, the representation with the basic kets |n,ngn3...> is not a very useful one. If the number of fermions in the assembly is variable, we can set up the complete set of kets Id, Ja, Ajatx®>, Alatarae>, (9’) corresponding to (9). A general ket is now expressible as a sum of the various kets (9’). To continue with the development we introduce a set of linear operators n, 7, one pair 7,, 7, corresponding to each fermion state «*, satisfying the commutation relations Na Not No Na = 9 Na ot No fa = 0, (11’) Dao Qa = Sap: 250 THEORY OF RADIATION § 65 These relations are like (11) with a -+ sign instead of a — on the left- hand side. They show that, for a + 6, 7, and 7, anticommute with n» and 7, while, putting b = a, they give m= 9, = 0, FaMat Nata = 1. (11’) To verify that the relations (11’) are consistent, we note that linear operators y, 7 satisfying the conditions (11’) can be constructed in the following way. For each state a we take a set of linear operators Fxq> Sy Fcq like the o,, o,, o, introduced in § 37 to describe the spin of an electron and such that o,,, Gyq; 6,4 commute with 9,,, Op, %» for 6 a. We also take an independent set of linear operators €,, one for each state «*, which all anticommute with one another and have their squares unity, and commute with all the o variables. Then, putting Na = 2$a(Fra—tya)> Na = HalOrattya), we have all the conditions (11’) satisfied. From (11") ("4 ja)” = Naa Na Ie = Nall—e Na) a = Na Na This is an algebraic equation for 7,7,, showing that »,7, is an observable with the eigenvalues 0 and 1. Also y, 7, commutes with 1, i, for 6 4a. These results allow us to put Nata = Na (12') the same as (12). From (11”) we get now Na Ve = I—n,, (13) the equation corresponding to (13), Let us write the normalized ket which is an eigenket of all the n’s belonging to the eigenvalues zero as |0>. Then Ng\O> = 0, so from (12’) CO| M4 Fg|O> = 0. Hence FqlO> = 9, . (15') like (15). Again showing that 7,,|0> is normalized, and showing that »,|0> is an eigenket of n, belonging to the eigenvalue unity. It is an eigenket of the other n’s belonging to the eigenvalues zero, since the other n’s commute with »,. By generalizing the § 65 AN ASSEMBLY OF FERMIONS 251 argument we see that 4_ 7p Ne---4y|07 18 normalized and is a simul- taneous eigenket of all the n’s, belonging to the eigenvalues unity for Mg Nyy Ney, R, and zero for the other n’s. This enables us to put Alatoal...07> = 14 Ne Mg|O> (17) both sides being antisymmetrical in the labels a, 6, c,...,g. We have here the analogue of (17). If we pass over to a different set of basic kets [8+> for a fermion, we can introduce a new set of linear operators 74 correspor.ding to them. We then find, by the same argument as in the case of bosons, that the new 7’s are connected with the original ones by (21). This shows that there is a procedure of second quantization for fermions, similar to that for bosons, with the only difference that the commu- tation relations (11’) must be employed for fermions to replace the commutation relations (11) for bosons. A symmetrical linear operator Up of the form (22) can be expressed in terms of the y, 7 variables by a similar method to that used for bosons. Equation (24) still holds, and so does (25) with S replaced by A. Instead of (26) we now have Oy Ney Marg (0 = pa 2 (—)9 1 Nes Ney |0> = 2 Na 2 (—)’-*n;,* Nay Nag? |O>85x,<4| U\b>, (26’) nz,i meaning that the factor y,, must be cancelled out, without its position among the other 7,’s being changed before the cancellation. Instead of (27) we have Fe Nay Nag LOD =D (ng? Ney Nae |9 Spe, (27') so (28) holds unchanged and thus (29) holds unchanged. We have the same final form (29) for Up in the fermion case as in the boson case. Similarly, a symmetrical linear operator Vp of the form (30) can be expressed as v= Se ny (ab |V led) ua Fes (35"} auc the same as one of the ways of writing (35). The foregoing work shows that there is a deep-seated analogy between the theory of fermions and that of bosons, only slight changes having to be made in the general equations of the formalism when one passes from one to the other. There is, however, a development of the theory of fermions that has no analogue for bosons. For fermions there are only the two 252 THEORY OF RADIATION § 65 alternatives of a state being occupied or unoccupied and there is symmetry between these two alternatives. One can demonstrate the symmetry mathematically by making a transformation which inter- changes the concepts of ‘occupied’ and ‘unoccupied’, namely Na = Nas Na = Nar The creation operators of the unstarred variables are the annihilation operators of the starred variables, and vice versa. Thestarred variables are now seen to satisfy the same quantum conditions and to have all the same properties as the unstarred ones.. If there are only a few unoccupied states, a convenient standard ket to work with would be the one for which every state is occupied, namely |0*> satisfying Mq|0*> = [O*). It thus satisfies nilo*> = 0, or ial0*) = 0. Other states for the assembly will now be represented by Ta 1 Ne-|0*), in which variables appear referring to the unoccupied fermion states a,b,c.... We may look upon these unoccupied fermion states as holes among the occupied ones and the 7* variables as the operators of creation of such holes. The holes are just as much physical things as the original particles and are also fermions.