The Dreams That Stuff is Made of
Page 56
(17)
which is identical with (13) [except for the fact that in (17) the primes have been omitted from the N’s, which is permissible when we do not require to refer the N’s as q-numbers]. We have thus established that the Hamiltonian (11) describes the effect of a perturbation on an assembly satisfying the Einstein-Bose statistics.
§ 4. THE REACTION OF THE ASSEMBLY ON THE PERTURBING SYSTEM.
Up to the present we have considered only perturbations that can be represented by a perturbing energy V added to the Hamiltonian of the perturbed system, V being a function only of the dynamical variables of that system and perhaps of the time. The theory may readily be extended to the case when the perturbation consists of interaction with a perturbing dynamical system, the reaction of the perturbed system on the perturbing system being taken into account. (The distinction between the perturbing system and the perturbed system is, of course, not real, but it will be kept up for convenience.)
We now consider a perturbing system, described, say, by the canonical variables Jk, ωk, the J’s being its first integrals when it is alone, interacting with an assembly of perturbed systems with no mutual interaction, that satisfy the Einstein-Bose statistics. The total Hamiltonian will be of the formHT = HP(J) + Σn H(n),
where Hp is the Hamiltonian of the perturbing system (a function of the J’s only) and H(n) is equal to the proper energy H0(n) plus the perturbation energy V(n) of the nth system of the assembly. H(n) is a function only of the variables of the nth system of the assembly and of the J’s and w’s, and does not involve the time explicitly.
The Schrödinger equation corresponding to equation (14) is now ih ḃ (Jʹ, r1r2...) = ΣJ”Σs1,s2...Hr (Jʹ, r1r2 ...; J”, s1s2 ...) b (J”, s1s2 ...), in which the eigenfunction b involves the additional variables Jʹk. The matrix element HT(Jʹ, r1r2 ...;J”, s1s2 ...) is now always a constant. As before, it vanishes when more than one sn differs from the corresponding rn. When sm differs from rm and every other sn equals rn, it reduces to H (Jʹrm ; J” sm), which is the (Jʹrm; J”sm) matrix element (with the time factor removed) of H = H0 + V, the proper energy plus the perturbation energy of a single system of the assembly; while when every sn equals rn, it has the value HP (Jʹ)δJʹJ” + Σn H(Jʹrn ; J”rn). If, as before, we restrict the eigenfunctions to be symmetrical in the variables r1, r2 ..., we can again transform to the variables N1, N2 ..., which will lead, as before, to the result
(18)
This is the Schrödinger equation corresponding to the Hamiltonian function
(19)
in which Hrs is now a function of the J’s and w’s, being such that when represented by a matrix in the (J) scheme its (JʹJ”) element is H(Jʹr;J”s). (It should be noticed that Hrs still commutes with the N’s and θ’s.)
Thus the interaction of a perturbing system and an assembly satisfying the Einstein-Bose statistics can be described by a Hamiltonian of the form (19). We can put it in the form corresponding to (11ʹ) by observing that the matrix element H(Jʹr;J”s) is composed of the sum of two parts, a part that comes from the proper energy H0, which equals Wr when J”k = Jʹk and s = r and vanishes otherwise,and a part that comes from the interaction energy V which may be denoted by v( Jʹr ; J”s ). Thus we shall haveHrs = Wr δrs + νrs,
where νrs is that function of the J’s and w’s which is represented by the matri whose ( JʹJ”) element is v ( Jʹr ; J”s), and so (19) becomes
(20)
The Hamiltonian is thus the sum of the proper energy of the perturbing system Hp (J), the proper energy of the perturbed systems Σr Wr Nr and the perturbation energy (Ns + 1 − δrs)
§ 5. THEORY OF TRANSITIONS IN A SYSTEM FROM ONE STATE TO OTHERS OF THE SAME ENERG
Before applying the results of the preceding sections to light-quanta, shall consider the solution of the problem presented by a Hamiltonian of t type (19). The essential feature of the problem is that it refers to a dynamic system which can, under the influence of a perturbation energy which do not involve the time explicitly, make transitions from one state to others the same energy. The problem of collisions between an atomic system and electron, which has been treated by Born,kf is a special case of this type. Born method is to find a periodic solution of the wave equation which consists so far as it involves the co-ordinates of the colliding electron, of plane was representing the incident electron, approaching the atomic system, which are scattered or diffracted in all directions. The square of the amplitude of the waves scattered in any direction with any frequency is then assumed by Born to be the probability of the electron being scattered in that direction with the corresponding energy.
This method does not appear to be capable of extension in any simple manner to the general problem of systems that make transitions from one state to others of the same energy. Also there is at present no very direct and certain way of interpreting a periodic solution of a wave equation to apply to a non-periodic physical phenomenon such as a collision. (The more definite method that will now be given shows that Born’s assumption is not quite right, it being necessary to multiply the square of the amplitude by a certain factor.)
An alternative method of solving a collision problem is to find a non-periodic solution of the wave equation which consists initially simply of plane waves moving over the whole of space in the necessary direction with the necessary frequency to represent the incident electron. In course of time waves moving in other directions must appear in order that the wave equation may remain satisfied. The probability of the electron being scattered in any direction with any energy will then be determined by the rate of growth of the corresponding harmonic component of these waves. The way the mathematics is to be interpreted is by this method quite definite, being the same as that of the beginning of § 2.
We shall apply this method to the general problem of a system which makes transitions from one state to others of the same energy under the action of a perturbation. Let H0 be the Hamiltonian of the unperturbed system and V the perturbing energy, which must not involve the time explicitly. If we take the case of a continuous range of stationary states, specified by the first integrals, αk say, of the unperturbed motion, then, following the method of § 2, we obtain
(21)
corresponding to equation (4). The probability of the system being in a state for which each αk lies between α‘k and αʹk + dα’k at any time is when a(α’)is properly normalised and satisfies the proper initial conditions. If initially the system is in the state α0, we must take the initial value of a (α′) to be of the form a0. δ (α’–α0). We shall keep α0 arbitrary, as it would be inconvenient to normalise a (α’) in the present case. For a first approximation we may substitute for a (α") in the right-hand side of (21) its initial value.
This gives
where v (α′α0) is a constant and W(α’) is the energy of the state α′. Hence
(22)
For values of the α’ k such that W (α′)differs appreciably from W (α0), a (α’) is a periodic function of the time whose amplitude is small when the perturbing energy V is small, so that the eigenfunctions corresponding to these stationary states are not excited to any appreciable extent. On the other hand, for values of the such that W(α’) = W (α0), and for some k, a (α’)increases uniformly with respect to the time, so that the probability of the system being in the state α′ at any time increases proportionally with the square of the time. Physically, the probability of the system being in a state with exactly the same proper energy as the initial proper energy W (α0) is of no importance, being infinitesimal. We are interested only in the integral of the probability through a small range of proper energy values about the initial proper energy, which, as we shall find, increases linearly with the time, in agreement with the ordinary probability laws.
We transform from the variables α1, α2 ... αu to a set of variables that are arbitrary independent functions of the α’s such that one of them is the proper energy W, say, the variables W, γ1, γ2, . . . γu−1. The
probability at any time of the system lying in a stationary state for which each γ k lies between and + is now (apart from the normalising factor) equal to
(23)
For a time that is large compared with the periods of the system we shall find that practically the whole of the integral in (23) is contributed by values of Wʹ very close to W0 = W (α0). Put
a(α’) = a (W’, γ’) and ∂(α’1, α’2 . . . α’u) / ∂ (W , γ’1 . . . γ’u−1) = J (Wʹ , γ’) Then for the integral in (23) we find, with the help of (22) (provided for some k)
if one makes the substitution (Wʹ − W0)t/h = x . For large values of t this reduces to
The probability per unit time of a transition to a state for which each γk lies between and is thus (apart from the normalising factor)
(24)
which is proportional to the square of the matrix element associated with that transition of the perturbing energy.
To apply this result to a simple collision problem, we take the α’s to be the components of momentum px , p y , pz of the colliding electron and the γ’s to be θ and φ, the angles which determine its direction of motion. If, taking the relativity change of mass with velocity into account, we let P denote the resultant momentum, equal to , and E the energy, equal to , of the electron, m being its rest-mass, we find for the Jacobian
Thus the J (W0 , γ’) of the expression (24) has the value
(25)
where Eʹ and Pʹ refer to that value for the energy of the scattered electron which makes the total energy equal the initial energy W0 (i.e., to that value required by the conservation of energy).
We must now interpret the initial value of a(α’), namely, a0 δ (α’ − α0), which we did not normalise. According to § 2 the wave function in terms of the variables αk is b (α‘) = a (α’) e−i W′t/h , so that its initial value is
If we use the transformation functionkg
and the transformation rule
we obtain for the initial wave function in the co-ordinates x, y, z the value
This corresponds to an initial distribution of |a 0|2 (2π h )−3 electrons per unit volume. Since their velocity is P0c2/E0, the number per unit time striking a unit surface at right-angles to their direction of motion is |a0 |2P0c2 /(2πh)3E0 . Dividing this into the expression (24) we obtain, with the help of (25),
(26)
This is the effective area that must be hit by an electron in order that it shall be scattered in the solid angle sin θ’ dθ’ dφʹ with the energy E′. This result differs by the factor (2π h)2/2mEʹ. Pʹ/P0 from Born’s.kh The necessity for the factor P’/P0 in (26) could have been predicted from the principle of detailed balancing, as the factor |v(pʹ; p0)|2 is symmetrical between the direct and reverse processes.ki
§ 6. APPLICATION TO LIGHT-QUANTA.
We shall now apply the theory of § 4 to the case when the systems of the assembly are light-quanta, the theory being applicable to this case since lightquanta obey the Einstein-Bose statistics and have no mutual interaction. A light-quantum is in a stationary state when it is moving with constant momentum in a straight line. Thus a stationary state r is fixed by the three components of momentum of the light-quantum and a variable that specifies its state of polarisation. We shall work on the assumption that there are a finite number of these stationary states, lying very close to one another, as it would be inconvenient to use continuous ranges. The interaction of the light-quanta with an atomic system will be described by a Hamiltonian of the form (20), in which HP (J) is the Hamiltonian for the atomic system alone, and the coefficients vrs are for the present unknown. We shall show that this form for the Hamiltonian, with the vrs arbitrary, leads to Einstein’s laws for the emission and absorption of radiation.
The light-quantum has the peculiarity that it apparently ceases to exist when it is in one of its stationary states, namely, the zero state, in which its momentum, and therefore also its energy, are zero. When a light-quantum is absorbed it can be considered to jump into this zero state, and when one is emitted it can be considered to jump from the zero state to one in which it is physically in evidence, so that it appears to have been created. Since there is no limit to the number of light-quanta that may be created in this way, we must suppose that there are an infinite number of light-quanta in the zero state, so that the N0 of the Hamiltonian (20) is infinite. We must now have θ 0, the variable canonically conjugate to N0, a constant, since
and W0 is zero. In order that the Hamiltonian (20) may remain finite it is necessary for the coefficients v r0, v 0r to be infinitely small. We shall 1 suppose that they are infinitely small in sucha way as to make and finite, in order that the transition probability coefficients may be finite. Thus we put
where νr are and finite and conjugate imaginaries. We may consider the νr and to be functions only of the J’s and w’s of the atomic system, since their factors and are practically constants, the rate of change of N0 being very small compared with N0. The Hamiltonian (20) now becomes
(27)
The probability of a transition in which a light-quantum in the state r is absorbed is proportional to the square of the modulus of that matrix element of the Hamiltonian which refers to this transi tion. This matrix element must come from the term in the Hamiltonian, and must therefore be proportional to where is the number of light-quanta in state r before the process. The probability of the absorption process is thus proportional to . In the same way the probability of a light-quantum in state r being emitted is proportional to , and the probability of a light-quantum in state r being scattered into state s is proportional to Radiative processes of the more general type considered by Einstein and Ehrenfest,* in which more than one light-quantum take part simultaneously, are not allowed on the present theory.
To establish a connection between the number of light-quanta per stationary state and the intensity of the radiation, we consider an enclosure of finite volume, A say, containing the radiation. The number of stationary states for light-quanta of a given type of polarisation whose frequency lies in the range νr to νr + dνr and whose direction of motion lies in the solid angle dωr about the direction of motion for state r will now be The energy of the light-quanta in these stationary states is thus This must equal Ac −1Ir d νr d ωr , where Ir is the intensity per unit frequency range of the radiation about the state r. Hence
(28)
so that Nʹr is proportional to Ir and is proportional to Ir + We thus obtain that the probability of an absorption process is proportional to Ir , the incident intensity per unit frequency range, and that of an emission process is proportional to which are just Einstein’s laws.kj In the same way the probability of a process in which a light-quantum is scattered from a state r to a state s is proportional to , which is Pauli’s law for the scattering of radiation by an electron.kk
§ 7. THE PROBABILITY COEFFICIENTS FOR EMISSION AND ABSORPTION.
We shall now consider the interaction of an atom and radiation from the wave point of view. We resolve the radiation into its Fourier components, and suppose that their number is very large but finite. Let each component be labelled by a suffix r, and suppose there are σ r components associated with the radiation of a definite type of polarisation per unit solid angle per unit frequency range about the component r. Each component r can be described by a vector potential kr chosen so as to make the scalar potential zero. The perturbation term to be added to the Hamiltonian will now be, according to the classical theory with neglect of relativity mechanics, c −1Σr κr Ẋr, where Xr is the component of the total polarisation of the atom in the direction of κr , which is the direction of the electric vector of the component r.
We can, as explained in § 1, suppose the field to be described by the canonical variables Nr, θr , of which Nr is the number of quanta of energy of the component r, and θr is its canonically conjugate phase, equal to 2πhνr times the θr of § 1. We shall now have κr = ar cos θr/h, where ar is the amplitude of κr , which can be connected with Nr as fo
llows:—The flow of energy per unit area per unit time for the component r is Hence the intensity per unit frequency range of the radiation in the neighbourhood of the component r is Comparing this with equation (28), we obtain , and hence
The Hamiltonian for the whole system of atom plus radiation would now be, according to the classical theory,
(29)
where HP( J) is the Hamiltonian for the atom alone. On the quantum theory we must make the variables Nr and θr canonical q-numbers like the variables Jk, wk that describe the atom. We must now replace the cos θr /h in (29) by the real q-number
so that the Hamiltonian (29) becomes
(30)
This is of the form (27), with and
(31)
The wave point of view is thus consistent with the light-quantum point of view and gives values for the unknown interaction coefficient vrs in the light-quantum theory. These values are not such as would enable one to express the interaction energy as an algebraic function of canonical variables. Since the wave theory gives vrs = 0 for r, s ≠ 0, it would seem to show that there are no direct scattering processes, but this may be due to an incompleteness in the present wave theory.