The Dreams That Stuff is Made of
Page 20
Second among the requirements traditionally imposed on a physical theory is that it must explain all phenomena as relations between objects existing in space and time. This requirement has suffered gradual relaxation in the course of the development of physics. Thus Faraday and Maxwell explained electromagnetic phenomena as the stresses and strains of an ether, but with the advent of the relativity theory, this ether was dematerialized; the electromagnetic field could still be represented as a set of vectors in space-time, however. Thermodynamics is an even better example of a theory whose variables cannot be given a simple geometric interpretation. Now, as a geometric or kinematic description of a process implies observation, it follows that such a description of atomic processes necessarily precludes the exact validity of the law of causality—and conversely. Bohrcl has pointed out that it is therefore impossible to demand that both requirements be fulfilled by the quantum theory. They represent complementary and mutually exclusive aspects of atomic phenomena. This situation is clearly reflected in the theory which has been developed. There exists a body of exact mathematical laws, but these cannot be interpreted as expressing simple relationships between objects existing in space and time. The observable predictions of this theory can be approximately described in such terms, but not uniquely—the wave and the corpuscular pictures both possess the same approximate validity. This indeterminateness of the picture of the process is a direct result of the interdeterminateness of the concept “observation”—it is not possible to decide, other than arbitrarily, what objects are to be considered as part of the observed system and what as part of the observer’s apparatus. In the formulas of the theory this arbitrariness often makes it possible to use quite different analytical methods for the treatment of a single physical experiment. Some examples of this will be given later. Even when this arbitrariness is taken into account the concept “observation” belongs, strictly speaking, to the class of ideas borrowed from the experiences of everyday life.cm It can only be carried over to atomic phenomena when due regard is paid to the limitations placed on all space-time descriptions by the uncertainty principle.
The general relationships discussed here may be summarized in the followingcn diagrammatic form:
It is only after attempting to fit this fundamental complementarity of space-time description and causality into one’s conceptual scheme that one is in a position to judge the degree of consistency of the methods of quantum theory (particularly of the transformation theory). To mold our thoughts and language to agree with the observed facts of atomic physics is a very difficult task, as it was in the case of the relativity theory. In the case of the latter, it proved advantageous to return to the older philosophical discussions of the problems of space and time. In the same way it is now profitable to review the fundamental discussions, so important for epistemology, of the difficulty of separating the subjective and objective aspects of the world. Many of the abstractions that are characteristic of modern theoretical physics are to be found discussed in the philosophy of past centuries. At that time these abstractions could be disregarded as mere mental exercises by those scientists whose only concern was with reality, but today we are compelled by the refinements of experimental art to consider them seriously.
CLASSICAL THEORY
CAUSAL RELATIONSHIPS OF PHENONMENA DESCRIBEDIN TERMS OF SPACE AND TIME
QUANTUM THEORY
DISCUSSION OF IMPORTANT EXPERIMENTS
In the preceding chapters the principles of the quantum theory have all been discussed, but a real understanding of them is obtainable only through their relation to the body of experimental facts which the theory must explain. This is particularly true of the general principle of complementarity. A discussion of further experiments of a less idealized type than those previously used to illustrate the separate principles is therefore necessary at this point.
§ 1. THE C. T. R. WILSON EXPERIMENTS
The essential features of the C. T. R. Wilson photographs may be most easily explained with the help of the classical corpuscular picture. This explanation is also completely justified from the standpoint of the quantum theory. The uncertainty relations are not essential to the explanation of the primary fact of the rectilinearity of the tracks of α-particles. It is always correct to apply the classical theory to such semi-macroscopic phenomena, and the quantum theory is necessary only for the explanation of the finer features.
Nevertheless it will be profitable to discuss the quantum theory of the Wilson photograph. We encounter at once the arbitrariness in the concept of observation already mentioned, and it appears purely as a matter of expediency whether the molecules to be ionized are regarded as belonging to the observed system or to the observing apparatus. Consider first the latter alternative. The system to be observed then consists of one α-particle only, and the position measurement resulting from the ionization will be represented in the mathematical scheme of the theory by a probability packet |ψ(qʹ)|2 in the coordinate space q = x, y, z, of the α-particle. The calculation will be carried out only for one of the three degrees of freedom.
If the time of this determination be taken as t = 0, and if a previous determination at a known time is also available, the momentum of the particle at time t = 0 may be determined: let p and q denote the most probable values of the momentum and co-ordinate at this time, and Δp , Δq the probable errors. The value of the uncertainty product will be considerably greater than h in any actual case, but we may assume that Δp Δq = h/2π (cf. the remarks concerning scintillation measurements, chap. ii, § 2a). This is a determinate case; it is then known [eq. (15)] that
(The index 0 indicates that qʹ0 is the value of the coordinate at t = 0.) The quantum theoretical equations of motion are then
Although p and q do not commute, the latter equation may nevertheless be integratedco to
To obtain the probability amplitude ψ (qʹ) at time t the transformation function must be calculated from A(41) and A(42):
The solution of this equation is
(46)
by A(69) the distribution at time t is then to be found from
which becomes, on evaluation of the integral,
(47)
where
It follows that
(48)
The most probable value for qʹ is thus (t/µ)p + q, which is the result to be expected on the classical theory. The mean square error (Δq)2 + (tΔp/µ)2 for q’ is made up of two terms corresponding to the uncertainties in and ; its value again agrees with that which would be calculated classically.
If these methods are applied to all three degrees of freedom, x, y, z, it is seen at once that the path of the center of the probability packet is a straight line. It is to be noted, however, that this result applies only while the α-particle is undisturbed in its motion. Each successive ionization of a water molecule transforms the packet (48) into an aggregate of such packets (Case II, p. 61). If the ionization is accompanied by an observation of the position, a smaller probability packet of the same form as (48) but with new parameters is separated out of the aggregate (Case III, p. 61). This forms the starting-point of a new orbit—and so on. The angular deviations between successive orbital segments are determined by the relative momenta of the particle and the atomic electron with which it interacts, which accounts for the differences between the paths of α- and β -particles.
As regards the formal aspect of the foregoing calculations, it may be noted that the transformation from qʹ0 to q’ can also be carried out by way of the energy. By equation A(70):
and therefore
The functions S(q‘E), S(E) are the normalized Schrödinger wave functions for the free electron; the function ψ(q’) can thus be built up by superposition of such Schrödinger functions. This method has been used by Darwin in an investigation of the motion of probability packets.
To complete this discussion we shall finally carry through a mathematical treatment of the Wilson photographs under the assumption that the molecules to be ionized are reg
arded as part of the system. This procedure is more complicated than the preceding method, but has the advantage that the discontinuous change of the probability function recedes one step and seems less in conflict with intuitive ideas. In order to avoid complication we consider only two molecules and one α-particle, and suppose the centers of mass of the former to be fixed at the points x1, y1, z1; x2, y2, z2. The α-particle is in motion with the momenta px , py , pz, and its co-ordinates are x, y, z. The co-ordinates of the electrons in the molecules may be denoted by the single symbols q1 and q2, respectively; the configuration space will thus involve only x , y , z, q1 , and q2 . We inquire for the probability that both molecules will be ionized and show that it is negligibly small unless the line joining them has nearly the same direction as the vector (px py pz). All interaction between the two molecules will be neglected, and their interaction with the α-particle will be treated as a perturbation;cp the energy of this interaction may be written
(49)
regarded as operators acting on the Schrödinger function. The wave equation is then
(50)
in which ∇2 = ∂2 /∂ x 2 + ∂2 / ∂y2 + ∂2 / ∂z2 , H0 (qi) is the energy operator of the molecule i, and ε is the perturbation parameter in powers of which the wave function is to be expanded: ψ = ψ(0) + εψ(1) + ε2ψ(2). . . . Substituting this series into the wave equation and equating each power of ε to zero, we obtain
(51)
The characteristic solutions of the first equation are
(52)
where
(53)
and
(54)
These solutions correspond to the case in which the momentum of the α-particle is known to be exactly p, its position therefore entirely unknown, while the molecules are known to be in the states n1, n2, respectively. All interaction is neglected, and the problem is to determine how the interaction modifies this state of affairs.
This may be solved by determining ψ(1), ψ(2) according to the method of Born. These quantities are first expanded in terms of the orthogonal functions ϕm1 (q1) ϕm2 (q2),
(55)
in which the 2 are of course functions of x, y, z, and t. The significance of these quantities is that
(56)
is the probability of observing the molecule 1 in the state m 1, molecule 2 in the state m2, and the electron at x, y, z.
Substituting equation (55) for i = 1 into the first of equations (51), we obtain
in which the abbreviations
(57)
have been used. The co-ordinates q1 and q2 have thus been eliminated from further consideration; the functions h(1), h(2) are functions of x, y, z, and of x1, y1, z1 or x2, y2, z2, respectively. These equations may be further simplified by writing
whence
(58)
where
(59)
In this expression the kinetic energy of the α-particle is so much greater than the other terms that, to a sufficient approximation, we may take
(60)
Equations (58) are then all of the form
(61)
which is the ordinary equation of wave-motion; ρm1m2 (xyz) is the density of the oscillators producing the wave, and, as it is complex, also determines their phase. The solution of equation (61) is given by Huyghen’s principle:
where R is the distance from xʹ, yʹ, zʹ to x, y, z.
Since, according to (58), ρm1m2 is zero unless m1 = n1 or m2 = n2, all the will be zero except and ; to the first approximation, only one of the two molecules will be excited. This is in agreement with the classical theory, which says that the probability of two collisions is of second order. The character of the functions and is readily determined qualitatively; by equation (57)
The (fictitious) oscillators producing the wave are thus all located in the region Γ1 about x1, y1, z1 (cf. Fig. 16) in which hn2m1 is appreciably different from zero. They vibrate coherently, their phase being determined essentially by the factor ; in the figure the lines of equal phase are drawn perpendicular to p. They are spaced at distances λ0. According to equation (61) the wave-length emitted by the oscillators is also λ0, and a simple application of Huyghen’s principle shows that the wave disturbance will have an appreciable amplitude only in the conical region which is shaded and whose axis is in the direction of p. The cross-section of this region near x1, y1, z1 is determined by the cross-section of the molecule: Γ1. Its angular opening also depends on Γ1, being greater when Γ1 is small—i.e., the uncertainty relation Δpx Δx ∼ h / 2π is fulfilled. Similar consid erations apply to ; it is different from zero only in a beam originating in Γ2 and also having the direction p.
FIG. 16
We now pass to the second approximation: may also be written exp(−2π i / h ) E0 t and equation (51) reduces to
(62)
The right-hand side of this equation will always be practically zero unless one of the two molecules lies in the beam originating at the other, for is different from zero only in the beam originating in Γ2 and hn1m1 (1) only in Γ1. Unless these two regions intersect, the first term will be zero; similarly the second term. Thus the probability of simultaneous ionization or excitation of the two atoms will vanish even in the second approximation unless the line joining their centers of gravity is practically parallel to the direction of motion of the α-particle. These considerations may be extended to the case of any number of molecules without essential modification. For each additional molecule the approximation must be carried one step farther, but the principles and results will be the same. It has thus been proved that the ionized molecules will lie practically on straight lines, and that the deviations from rectilinearity satisfy the uncertainty relations. In thus including the molecules in the observed system, it has not been necessary to introduce the discontinuously changing probability packet, but if we wish to consider the methods by which the excitation of the molecule can actually be observed, these discontinuous changes (now of a probability packet in the configuration space x, y,z,q1,q2) will again play a role.
§ 2. DIFFRACTION EXPERIMENTS
The diffraction of light or matter (Davisson-Germer, Thomson, Rupp, Kikuchi) by gratings may be explained most simply by the aid of the classical wave theories. The application of space-time wave theories to these experiments is justified from the point of view of the quantum theory, since the uncertainty relations do not in any way affect the purely geometrical aspects of the waves, but only their amplitude (cf. chap. iii, § 1). The quantum theory need only be invoked when discussing the dynamical relations involving the energy and momentum content of the waves.
The quantum theory of the waves being thus certainly in agreement with the classical theory in so far as the geometric diffraction pattern is concerned, it seems useless to prove it by detailed calculation. On the other hand, Duane has given an interesting treatment of diffraction phenomena from the quantum theory of the corpuscular picture. We imagine for simplicity that the corpuscle is reflected from a plane ruled grating, whose constant is d.
Let the grating itself be movable. Its translation in the x-direction may be looked upon as a periodic motion, in so far as only the interaction of the incident particles with the grating is considered; for the displacement of the whole grating by an amount d will not change this interaction. Thus we may conclude that the motion of the grating in this direction is quantized and that its momentum p x may assume only the values nh/d (as follows at once from the earlier form of the theory: ∫ pdq = nh). Since the total momentum of grating and particle must remain unchanged, the momentum of the particle can be changed only by an amount mh/d (m an integer):
Furthermore, because of its large mass, the grating cannot take up any appreciable amount of energy, so that
If θ is the angle of incidence, θʹ that of reflection, we have
whence
From equation A(83) for the wave-length of the wave associated with a particle it then follows that