We will now apply these considerations to the q-space waves. We select, at a definite time t, a definite point P of q-space, through which the parcel of waves passes in a given direction R, at that time. In addition let the mean frequency ν or the mean E-value for the packet be also given. These conditions correspond exactly to postulating that at a given time the mechanical system is starting from a given configuration with given velocity components. (Energy plus direction is equivalent to velocity components.)
In order to carry over the optical construction, we require firstly one set of wave surfaces with the desired frequency, i.e. one solution of the Hamilton-Jacobi equation (1ʹ) for the given E-value. This solution, W, say, is to have the following property: the surface of the set which passes through P at time t, which we may denote by
(14)
must have its normal at P in the prescribed direction R. But this is still not enough. We must be able to vary to an infinitely small extent this set of waves W in an n-fold manner (n = number of degrees of freedom), so that the wave normal will sweep out an infinitely small (n − 1) dimensional space angle at the point P, and so that the frequency will vary in an infinitely small one-dimensional region, whereby care is taken that all members of the infinitely small n-dimensional continuum of sets of waves meet together at time t in the point P in exactly agreeing phase. Then it is a question of finding at any other time where that point lies at which this agreement of phases occurs.
To do this, it will be sufficient if we have at our disposal a solution W of the Hamilton-Jacobi equation, which is dependent not only on the constant E, here denoted by α1, but also on (n − 1) additional constants α2, α3 . . . αn, in such a way that it cannot be written as a function of less than n combinations of these n constants. For then we can, firstly, bestow on α1 the value prescribed for E, and, secondly, define α2, α3 . . . αn, so that the surface of the set passing through the point P has at P the prescribed normal direction. Henceforth we understand by α1, α2 . . . αn, these values, and take (14) as the surface of this set, which passes through the point P at time t. Then we consider the continuum of sets which belongs to the αk-values of an adjacent infinitesimal αk-region. A member of this continuum, i.e. therefore a set, will be given by
(15)
for a fixed set of values of d α1, dα2 ...dαn, and varying constant. That member of this set, i.e. therefore that single surface, which goes through P at time t will be defined by the following choice of the const.,
(15ʹ)
where , etc., are the constants obtained by substituting in the differential coefficients the co-ordinates of the point P and the value t of the time (which latter really only occurs in ).
The surfaces (15ʹ) for all possible sets of values of dα1, dα2 . . .dαn, form on their part a set. They all go through the point P at time t, their wave normals continuously sweep out a little (n − 1) dimensional solid angle and, moreover, their E-parameter also varies within a small region. The set of surfaces (15ʹ) is so formed that each of the sets (15) supplies one representative to (15ʹ), namely, that member which passes through P at time t.
We will now assume that the phase angles of the wave functions which belong to the sets (15) happen to agree precisely for those representatives which enter the set (15ʹ). They agree therefore at time t at the point P.
We now ask: Is there, at any arbitrary time, a point where all surfaces of the set (15ʹ) cut one another, and in which, therefore, all the wave functions which belong to the sets (15) agree in phase? The answer is: There exists a point of agreeing phase, but it is not the common intersection of the surfaces of set (15’), for such does not exist at any subsequent arbitrary time. Moreover, the point of phase agreement arises in such a way that the sets (15) continuously exchange their representatives given to (15ʹ).
That is shown thus. There must hold
(16)
simultaneously for the common meeting point of all members of (15ʹ) at any time, because the dα1’s are arbitrary within a small region. In these n + 1 equations, the right-hand sides are constants, and the left are functions of the n + 1 quantities q1, q2, . . . qn, t. The equations are satisfied by the initial system of values, i.e. by the co-ordinates of P and the initial time t. For another arbitrary value of t, they will have no solutions in q1 . . .qn, but will more than define the system of these n quantities.
We may proceed, however, as follows. Let us leave the first equation, W = W0, aside at first, and define the qk‘s as functions of the time and the constants according to the remaining n equations. Let this point be called Q. By it, naturally, the first equation will not be satisfied, but the left-hand side will differ from the right by a certain value. If we go back to the derivation of system (16) from (15’), what we have just said means that though Q is not a common point for the set of surfaces (15ʹ), it is so, however, for a set which results from (15’), if we alter the right-hand side of equation (15ʹ) by an amount which is constant for all the surfaces. Let this new set be (15”). For it, therefore, Q is a common point. The new set results from (15’), as stated above, by an exchange of the representatives in (15ʹ). This exchange is occasioned by the alteration of the constant in (15), by the same amount, for all representatives. Hence the phase angle is altered by the same amount for all representatives. The new representatives, i.e. the members of the set we have called (15”), which meet in the point Q, agree in phase angle just as the old ones did. This amounts therefore to saying:
The point Q which is defined as a function of the time by the n equations
(17)
continues to be a point of agreeing phase for the whole aggregate of wave sets (15).
Of all the n-surfaces, of which Q is shown by (17) to be the common point, only the first is variable; the others remain fixed (only the first of equations (17) contains the time). The n − 1 fixed surfaces determine the path of the point Q as their line of intersection. It is easily shown that this line is the orthogonal trajectory of the set W = const. For, by hypothesis, W satisfies the Hamilton-Jacobi equation (1ʹ) identically in α1, α2 . . . αn. If we now differentiate the Hamilton-Jacobi equation with respect to αk (k = 2, 3, . . . n), we get the statement that the normal to a surface, = const., is perpendicular, at every point on it, to the normal of the surface, W = const., which passes through that point, i.e. that each of the two surfaces contains the normal to the other. If the line of intersection of the n − 1 fixed surfaces (17) has no branches, as is generally the case, then must each line element of the intersection, as the sole common line element of the n − 1 surfaces, coincide with the normal of the W-surface, passing through the same point, i.e. the line of intersection is the orthogonal trajectory of the W-surfaces. Q.E.D.
We may sum up the somewhat detailed discussion, which has led us to equations (17), in a much shorter or (so to speak) shorthand fashion, as follows: W denotes, apart from a universal constant , the phase angle of the wave function. If we now deal not merely with one, but with a continuous manifold of wave systems, and if these are continuously arranged by means of any continuous parameters αi, then the equations = const. express the fact that all infinitely adjacent individuals (wave systems) of this manifold agree in phase. These equations therefore define the geometrical locus of the points of agreeing phase. If the equations are sufficient, this locus shrinks to one point; the equations then define the point of phase agreement as a function of the time.
Since the system of equations (17) agrees with the known second system of equations of Jacobi, we have thus shown:
The point of phase agreement for certain infinitesimal manifolds of wave systems, containing n parameters, moves according to the same laws as the image point of the mechanical system.
I consider it a very difficult task to give an exact proof that the superposition of these wave systems really produces a noticeable disturbance in only a relatively small region surrounding the point of phase agreement, and that everywhere else they practically destroy one another through
interference, or that the above statement turns out to be true at least for a suitable choice of the amplitudes, and possibly for a special choice of the form of the wave surfaces. I will advance the physical hypothesis, which I wish to attach to what is to be proved, without attempting the proof. The latter will only be worth while if the hypothesis stands the test of trial and if its application should require the exact proof.
On the other hand, we may be sure that the region to which the disturbance may be confined still contains in all directions a great number of wave lengths. This is directly evident, firstly, because so long as we are only a few wave lengths distant from the point of phase agreement, then the agreement of phase is hardly disturbed, as the interference is still almost as favourable as it is at the point itself. Secondly, a glance at the three-dimensional Euclidean case of ordinary optics is sufficient to assure us of this general behaviour.
What I now categorically conjecture is the following:
The true mechanical process is realised or represented in a fitting way by the wave processes in q-space, and not by the motion of image points in this space. The study of the motion of image points, which is the object of classical mechanics, is only an approximate treatment, and has, as such, just as much justification as geometrical or “ray” optics has, compared with the true optical process. A macroscopic mechanical process will be portrayed as a wave signal of the kind described above, which can approximately enough be regarded as confined to a point compared with the geometrical structure of the path. We have seen that the same laws of motion hold exactly for such a signal or group of waves as are advanced by classical mechanics for the motion of the image point. This manner of treatment, however, loses all meaning where the structure of the path is no longer very large compared with the wave length or indeed is comparable with it. Then we must treat the matter strictly on the wave theory, i.e. we must proceed from the wave equation and not from the fundamental equations of mechanics, in order to form a picture of the manifold of the possible processes. These latter equations are just as useless for the elucidation of the micro-structure of mechanical processes as geometrical optics is for explaining the phenomena of diffraction.
Now that a certain interpretation of this micro-structure has been successfully obtained as an addition to classical mechanics, although admittedly under new and very artificial assumptions, an interpretation bringing with it practical successes of the highest importance, it seems to me very significant that these theories—I refer to the forms of quantum theory favoured by Sommerfeld, Schwarzschild, Epstein, and others—bear a very close relation to the Hamilton-Jacobi equation and the theory of its solution, i.e. to that form of classical mechanics which already points out most clearly the true undulatory character of mechanical processes. The Hamilton-Jacobi equation corresponds to Huygens’ Principle (in its old simple form, not in the form due to Kirchhoff). And just as this, supplemented by some rules which, are not intelligible in geometrical optics (Fresnel’s construction of zones), can explain to a great extent the phenomena of diffraction, so light can be thrown on the processes in the atom by the theory of the action-function. But we inevitably became involved in irremovable contradictions if we tried, as was very natural, to maintain also the idea of paths of systems in these processes; just as we find the tracing of the course of a light ray to be meaningless, in the neighbourhood of a diffraction phenomenon.
We can argue as follows. I will, however, not yet give a conclusive picture of the actual process, which positively cannot be arrived at from this starting-point but only from an investigation of the wave equation; I will merely illustrate the matter qualitatively. Let us think of a wave group of the nature described above, which in some way gets into a small closed “path”, whose dimensions are of the order of the wave length, and therefore small compared with the dimensions of the wave group itself. It is clear that then the “system path” in the sense of classical mechanics, i.e. the path of the point of exact phase agreement, will completely lose its prerogative, because there exists a whole continuum of points before, behind, and near the particular point, in which there is almost as complete phase agreement, and which describe totally different “paths”. In other words, the wave group not only fills the whole path domain all at once but also stretches far beyond it in all directions.
In this sense do I interpret the “phase waves” which, according to de Broglie, accompany the path of the electron; in the sense, therefore, that no special meaning is to be attached to the electronic path itself (at any rate, in the interior of the atom), and still less to the position of the electron on its path. And in this sense I explain the conviction, increasingly evident to-day, firstly, that real meaning has to be denied to the phase of electronic motions in the atom; secondly, that we can never assert that the electron at a definite instant is to be found on any definite one of the quantum paths, specialised by the quantum conditions; and thirdly, that the true laws of quantum mechanics do not consist of definite rules for the single path, but that in these laws the elements of the whole manifold of paths of a system are bound together by equations, so that apparently a certain reciprocal action exists between the different paths.du
It is not incomprehensible that a careful analysis of the experimentally known quantities should lead to assertions of this kind, if the experimentally known facts are the outcome of such a structure of the real process as is here represented. All these assertions systematically contribute to the relinquishing of the ideas of “place of the electron” and “path of the electron”. If these are not given up, contradictions remain. This contradiction has been so strongly felt that it has even been doubted whether what goes on in the atom could ever be described within the scheme of space and time. From the philosophical standpoint, I would consider a conclusive decision in this sense as equivalent to a complete surrender. For we cannot really alter our manner of thinking in space and time, and what we cannot comprehend within it we cannot understand at all. There are such things—but I do not believe that atomic structure is one of them. From our standpoint, however, there is no reason for such doubt, although or rather because its appearance is extraordinarily comprehensible. So might a person versed in geometrical optics, after many attempts to explain diffraction phenomena by means of the idea of the ray (trustworthy for his macroscopic optics), which always came to nothing, at last think that the Laws of Geometry are not applicable to diffraction, since he continually finds that light rays, which he imagines as rectilinear and independent of each other, now suddenly show, even in homogeneous media, the most remarkable curvatures, and obviously mutually influence one another. I consider this analogy as very strict. Even for the unexplained curvatures, the analogy in the atom is not lacking—think of the “non-mechanical force”, devised for the explanation of anomalous Zeeman effects.
In what way now shall we have to proceed to the undulatory representation of mechanics for those cases where it is necessary? We must start, not from the fundamental equations of mechanics, but from a wave equation for q-space and consider the manifold of processes possible according to it. The wave equation has not been explicitly used or even put forward in this communication. The only datum for its construction is the wave velocity, which is given by (6) or (6ʹ) as a function of the mechanical energy parameter or frequency respectively, and by this datum the wave equation is evidently not uniquely defined. It is not even decided that it must be definitely of the second order. Only the striving for simplicity leads us to try this to begin with. We will then say that for the wave function ψ we have
(18)
valid for all processes which only depend on the time through a factor e2πi νt. Therefore, considering (6), (6ʹ), and (11), we get, respectively,
(18ʹ)
and
(18”)
The differential operations are to be understood with regard to the line element (3). But even under the postulation of second order, the above is not the only equation consistent with (6). For it is possi
ble to generalize by replacing div grad ψ by
(19)
where f may be an arbitrary function of the q’s, which must depend in some plausible way on E, V(qk), and the coefficients of the line element (3). (Think, e.g., of f = u.) Our postulation is again dictated by the striving for simplicity, yet I consider in this case that a wrong deduction is not out of the question.dv
The substitution of a partial differential equation for the equations of dynamics in atomic problems appears at first sight a very doubtful procedure, on account of the multitude of solutions that such an equation possesses. Already classical dynamics had led not just to one solution but to a much too extensive manifold of solutions, viz. to a continuous set, while all experience seems to show that only a discrete number of these solutions is realised. The problem of the quantum theory, according to prevailing conceptions, is to select by means of the “quantum conditions” that discrete set of actual paths out of the continuous set of paths possible according to classical mechanics. It seems to be a bad beginning for a new attempt in this direction if the number of possible solutions has been increased rather than diminished.
The Dreams That Stuff is Made of Page 27