For precision of thought we therefore assume that our measurements always give average values over a very small space region of volume ∂ v = (∂l )3, which depends on the method of measurement. Since it is a question of the measurement of the field strengths, light of wave-length λ much less than ∂l will not be detected by the experiment. The measurement gives, say, the values E and H for the field strengths (averaged over δv). If these values E and H were exactly known there would be a contradiction to the particle theory, since the energy and momentum of the small volume δv are
(37)
and the right-hand members could be made as small as desired by taking δv sufficiently small. This is inconsistent with the particle theory, according to which the energy and momentum content of the small volume is made up of discrete and finite amounts hν and hν/c, respectively. For the highest frequency detectable h ν ≤ (hc/δl) so that it is clear that the right-hand members of equation (37) must be uncertain by just the magnitudes of these quanta (hν and hν/c ) in order that there be no contradiction to the particle theory. Accordingly there must be uncertainty relations between the components of E and H which give rise to an uncertainty in the value of E of the order of magnitude hc/δl and in G of the order of magnitude h/δl when E and G are calculated by equations (37). Let ΔE and ΔH be the uncertainties in E and H; then the uncertainties in E and G are
with cyclic permutation for the y- and z-directions.
Since the most probable values of E and H may possibly be zero the terms on the right which contain only ΔE and ΔH must alone be sufficient to give the necessary uncertainty to E and G. This is attained if
(38)
with cyclic permutation for the other components. These uncertainty relations refer to a simultaneous knowledge of Ex and Hy in the same volume element; in different volume elements Ex and Hy can be known to any degree of accuracy.
The relations (38), as in the case of the particle theory, can also be derived directly from the exchange relations for E and H (cf. A, §§ 9, 12). If a division of space into finite cells of magnitude δv is used, the integration with respect to dv in the Lagrangian of A (97) becomes a sum over all the cells δv . The momentum conjugate to ψα(r ) in the rth cell is then [cf. A(104)]
(39)
and in place of A( 111 ),
(40)
where δrs , is now the usual δ-function,
In the limit δv → 0 (40) becomes A(111).
From (40) and A(134) applied to the case of electric and magnetic fields it follows that
(41)
When it is remembered that an uncertainty ΔΦk gives an uncertainty of order of magnitude ΔΦk /δl for the field strengths resulting from Φk, it will be seen that (41) leads immediately to the uncertainty relations (38).
Matter waves may be treated in an entirely similar way. It must be noted, however, that no experiment can ever measure the amplitude directly, as is evident from the fact that the de Broglie waves are complex. If exchange relations for the wave amplitudes are derived formally from those for ψ and ψ *, the result is, to be sure, a physically reasonable one in the case of the Bose-Einstein statistics. However, use of the experimentally correct Fermi-Dirac statistics gives the meaningless result that ψ and ψ * cannot be exactly measured simultaneously at different points of space. It is thus highly satisfactory that there is no experiment which will measure ψ at a given point at a given time. The mathematical reason for this is that even for the interaction of radiation and matter the part of the Lagrangian referring to matter contains only terms of the form ψ ψ *. From the considerations just given it can also be seen that the Bose-Einstein statistics is a physical necessity for light-quanta if one makes the apparently very natural assumption that measurements of the electric and magnetic fields at different points of space must be independent of each other.
§ 2. DISCUSSION OF AN ACTUAL MEASUREMENT OF THE ELECTROMAGNETIC FIELD
As in the case of the corpuscular picture, it must be possible to trace the origin of the uncertainty in a measurement of the electromagnetic field to its experimental source. We therefore discuss an experiment which is capable of simultaneously measuring Ex and Hz in the same element of volume δv. This can be accomplished by the observation of the deflection in the direction of x of two beams of cathode rays which traverse the volume in opposite directions along the y-axis (cf. Fig. 13). It may be assumed that the width of both beams in the z-direction is δl, i.e., the whole width of the volume element, but their widths in the perpendicular direction must be less than this, say d , so that they may traverse δv without mutual disturbance. If the distance between the two rays is of order of magnitude δl, the small inhomogeneities of the field in this direction are also averaged out; it would also be possible to vary the distance between them for this purpose. This experimental arrangement will enable the measurement of Ex and Hz in δl provided only that the fields are not too inhomogeneous; should this condition not be fulfilled, the method is incapable of giving a definite result, for the field must not vary appreciably across the width of the rays, or else these will become diffuse and no simple method of determining the deflections is then available.
The angular deflection, α, of the rays in the distance δl is to be observed, and the field can be calculated from the formulas
Because of the natural spreading of the matter rays, the accuracy of the measurements is given by
(42)
FIG. 13
One essential factor remains to be considered, however. Each of the two electrons which pass through δv simultaneously modifies the field, and hence the path of the other electron. The amount of this modification is uncertain to some extent, since it is not known at which point in the cathode ray the electron is to be found. The uncertainty as to the actual fields which arises from this fact is thus
(43)
whence
which was to be shown. It is to be noted that the simultaneous consideration of both the corpuscular and wave picture of the process taking place is again fundamental. If the corpuscular picture of the cathode rays had not been invoked, and a continuous distribution of charge assumed as the picture of the rays, then the uncertainty (43) would have disappeared.
THE STATISTICAL INTERPRETATION OF QUANTUM THEORY
§ 1. MATHEMATICAL CONSIDERATIONS
It is instructive to compare the mathematical apparatus of quantum theory with that of the theory of relativity. In both cases there is an application of the theory of linear algebras. One can therefore compare the matrices of quantum theory with the symmetric tensors of the special theory of relativity. The greatest difference is the fact that the tensors of quantum theory are in a space of infinitely many dimensions, and that this space is not real but imaginary. The orthogonal transformations are replaced by the so-called “unitary” transformations. In order to obtain a picture of this space, we abstract from such differences, fundamental though they be. Then every quantum theoretical “quantity” is characterized by a tensor whose principal directions may be drawn in this space (cf. Fig. 14). In order to obtain a clear picture, one may recall the tensor of the moments of inertia of a rigid body. The principal directions are, in general, different for each quantity; only matrices which commute with one another have coincident principal directions. The exact knowledge of the numerical value of any dynamical variable corresponds to the determination of a definite direction in this space, in the same manner as the exact knowledge of the moment of inertia of a solid body determines the principal direction to which this moment belongs (it is assumed that there is no degeneracy). This direction is thus parallel to the k th principal axis of the tensor T , along which the component Tkk has the value measured. The exact knowledge of the direction (except for a factor of absolute magnitude unity) in unitary space is the maximum information regarding the quantum dynamical variable which can be obtained. Weylcj has called this degree of knowledge a determinate case (reiner Fall ). An atom in a (non-degenerate) stationary state presents such a determin
ate case: The direction characterizing it is that of the k th principal axis of the tensor E, which belongs to the energy value Ekk. There is obviously no significance to be attached to the terms “value of the coordinate q,” etc., in this direction, just as the specification of the moment of inertia about an axis not coinciding with one of the principal directions is insufficient to determine any type of motion of the rigid body, no matter how simple. Only tensors whose principal axes coincide with those of E have a value in this direction. The total angular momentum of the atom, for example, can be determined simultaneously with its energy. If a measurement of the value of q is to be made, then the exact knowledge of the direction must be replaced by inexact information, which can be considered as a “mixture” of the original directions Ekk, each with a certain probability coefficient. For example, the indeterminate recoil of the electron when its position is measured by a microscope converts the determinate case E kk into such a mixture (cf. chap. ii, § 2a ). This mixture must be of such a kind that it may also be considered as a mixture of the principal directions of q , though with other probability coefficients. The measurement singles a particular value qʹ out of this as being the actual result. It follows from this discussion that the value of q’ cannot be uniquely predicted from the result of the experiment determining E, for a disturbance of the system, which is necessarily indeterminate to a certain degree, must occur between the two experiments involved.
FIG. 14
This disturbance is qualitatively determined, however, as soon as one knows that the result is to be an exact value of q. In this case, the probability of finding a value qʹ after E has been measured is given by the square of the cosine of the angle between the original direction Ek and the direction q’. More exactly one should say by the analogue to the cosine in the unitary space, which is |S(Ek, qʹ)|. This assumption is one of the formal postulates of quantum theory and cannot be derived from any other considerations. It follows from this axiom that the values of two dynamical quantities are causally related if, and only if, the tensors corresponding to them have parallel principal axes. In all other cases there is no causal relationship. The statistical relation by means of probability coefficients is determined by the disturbance of the system produced by the measuring apparatus. Unless this disturbance is produced, there is no significance to be given the terms “value” or “probable value” of a variable in a given direction of unitary space which is not parallel to a principal axis of the corresponding tensor. Thus one becomes entangled in contradictions if one speaks of the probable position of the electron without considering the experiment used to determine it (cf. the paradox of negative kinetic energy, chap. ii, § 2d). It must also be emphasized that the statistical character of the relation depends on the fact that the influence of the measuring device is treated in a different manner than the interaction of the various parts of the system on one another. This last interaction also causes changes in the direction of the vector representing the system in the Hilbert space, but these are completely determined. If one were to treat the measuring device as a part of the system—which would necessitate an extension of the Hilbert space—then the changes considered above as indeterminate would appear determinate. But no use could be made of this determinateness unless our observation of the measuring device were free of indeterminateness. For these observations, however, the same considerations are valid as those given above, and we should be forced, for example, to include our own eyes as part of the system, and so on. The chain of cause and effect could be quantitatively verified only if the whole universe were considered as a single system—but then physics has vanished, and only a mathematical scheme remains. The partition of the world into observing and observed system prevents a sharp formulation of the law of cause and effect. (The observing system need not always be a human being; it may also be an inanimate apparatus, such as a photographic plate.)
As examples of cases in which causal relations do exist the following may be mentioned: The conservation theorems for energy and momentum are contained in the quantum theory, for the energies and momenta of different parts of the same system are commutative quantities. Furthermore, the principal axes of q at time t are only infinitesimally different from the principal axes of q at time t+ dt. Hence, if two position measurements are carried out in rapid succession, it is practically certain that the electron will be in almost the same place both times.
§2. INTERFERENCE OF PROBABILITIES
Many paradoxical conclusions may be deduced from the foregoing principles if the perturbation introduced by measuring instruments is not adequately considered. The following idealized experiment furnishes a typical example of such a paradox.
A beam of atoms, all of which are initially in the state n, is directed through a field F1 (Fig. 15). This field will cause transitions to other states if it is inhomogeneous in the direction of the beam, but will not separate atoms of one state from those in another. Let be the transformation function for the transitions in the field F1so that
FIG. 15
is the probability of finding an atom in the state m after it has emerged from the field F1. Farther on the atoms encounter a second field F2, similar in properties to F1 for which the corresponding transformation function is . This field is again incapable of separating the atoms in different states, but beyond F2 a determination of the stationary state is made by means of a third field of force. Now, for those atoms that are in the state m after passing through F1 the probability of a transition to state l on passing F2 is given by [Sml]2. Hence the probable fraction of the atoms in the state l beyond F2 should be given by
(44)
On the other hand, according to equation A(69), the transformation function for the combined fields F1 and F2 is , which results in the value
(45)
for the same probability as represented by equation (44).
The contradiction disappears when it is remarked that the formulas (44) and (45) really refer to two different experiments. The reasoning leading to (44) is correct only when an experiment permitting the determination of the stationary state of the atom is performed between F1 and F2. The performance of such an experiment will necessarily alter the phase of the de Broglie wave of the atom in state m by an unknown amount of order of magnitude one, as has been shown in chapter ii, § 2d. In applying (45) to this experiment each member in the summation must thus be multiplied by the arbitrary factor exp(iϕm ) and then averaged over all values of ϕm . This phase average agrees with (44), which thus applies to this experiment. The rules of the calculus of probabilities can be applied to |Snm |2 only when the causal chain has actually been broken by an observation in the manner explained in the foregoing section. If no break of this sort has occurred it is not reasonable to speak of the atom as having been in a stationary state between F1 and F2, and the rules of quantum mechanics apply.
Three general cases may be illustrated by this experiment, and they must be carefully distinguished in any application of the general principles. They are:
CASE I: The atoms remain undisturbed between F1 and F2. The probability of observing the state l beyond F2 is then
CASE II: The atoms are disturbed between F1 and F2 by the performance of an experiment which would have made possible the determination of the stationary state. The result of the experiment is not observed, however. The probability of the state l is then
CASE III: The additional experiment of Case II is performed and its result is observed. The atom is known to have been in state m while passing from F1 to F2. The probability of the state l is then given by
The difference between Cases II and III is recognized in all treatments of the theory of probability, but the difference between I and II does not exist in classical theories which assume the possibility of observation without perturbation. When stated in a sufficiently generalized form, this distinction is the center of the whole quantum theory.
§ 3. BOHR’S CONCEPT OF COMPLEMENTARITYck
With the advent of Ein
stein’s relativity theory it was necessary for the first time to recognize that the physical world differed from the ideal world conceived in terms of everyday experience. It became apparent that ordinary concepts could only be applied to processes in which the velocity of light could be regarded as practically infinite. The experimental material resulting from modern refinements in experimental technique necessitated the revision of old ideas and the acquirement of new ones, but as the mind is always slow to adjust itself to an extended range of experience and concepts, the relativity theory seemed at first repellantly abstract. None the less, the simplicity of its solution for a vexatious problem has gained it universal acceptance. As is clear from what has been said, the resolution of the paradoxes of atomic physics can be accomplished only by further renunciation of old and cherished ideas. Most important of these is the idea that natural phenomena obey exact laws—the principle of causality. In fact, our ordinary description of nature, and the idea of exact laws, rests on the assumption that it is possible to observe the phenomena without appreciably influencing them. To co-ordinate a definite cause to a definite effect has sense only when both can be observed without introducing a foreign element disturbing their interrelation. The law of causality, because of its very nature, can only be defined for isolated systems, and in atomic physics even approximately isolated systems cannot be observed. This might have been foreseen, for in atomic physics we are dealing with entities that are (so far as we know) ultimate and indivisible. There exist no infinitesimals by the aid of which an observation might be made without appreciable perturbation.
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