THE PRINCIPLE OF OBSERVABILITY IN PRINCIPLE IN MODERN PHYSICS

G. A. Gamov, Leningrad.

Bohr’s theory of the atom. During the almost twelve years since the creation of N. Bohr’s model of the atom, quantum mechanics has developed into a very extensive system.

Its fundamental propositions were the principle of adiabatic invariance and the correspondence principle. However, despite the highly developed theory of the quantization of conditionally periodic systems and the theory of perturbations, the fundamental requirement of integrality and the character of quantum jumps remained completely incomprehensible. Moreover, the theory encountered insurmountable difficulties in passing to infinitesimal motions. The helium atom and the complex Zeeman effect did not yield to treatment. At the end of 1925 and the beginning of 1926 two theories appeared almost simultaneously,¹ at first glance completely different from one another: Heisenberg’s matrix mechanics (W. Heisenberg)² and Schrödinger’s wave mechanics (E. Schrödinger).³

Heisenberg’s matrix mechanics. Heisenberg⁴ refuses to construct a model of atomic processes and builds a formal theory, substituting into the canonical equations of motion of the system, instead of coordinates and momenta, the corresponding matrices. The latter are infinite tables of quantities, long known in pure mathematics, with peculiar rules of operations that do not obey the law of commutativity of multiplication. The discreteness of the observed spectral frequencies is contained in the character of the matrices themselves.

Schrödinger’s wave mechanics. Schrödinger’s theory⁵ likewise refuses to describe the motion of material particles (electrons), but replaces them by waves of a certain scalar matter (Materieskalar), passing, according to definite recipes, from the Hamilton–Jacobi equation of ordinary mechanics to a certain second-order equation (Schrödinger’s equation), analogous to the equation of vibration of membranes, strings, plates, etc.

The existence of discrete states of the atom is explained by the presence of a series of discrete proper vibrations (solutions of our differential equation of the second order). The shortcoming of both theories must be recognized as their formal character and the absence of a visual interpretation within the framework of our ordinary spatio-temporal repre-

¹ See Uspekhi Fizicheskikh Nauk, vol. VI, issue 6; vol. VII, issue 1, p. 25; vol. VII, issues 3–4, p. 176.

² W. Heisenberg, ZS. f. Phys. 33, p. 879 (1925); Born–Jordan, ZS. f. Phys. 34, p. 858 (1925); Born–Jordan, Heisenberg, ZS. f. Phys. 35, p. 557 (1926).

³ E. Schrödinger, Ann. d. Phys. 79, pp. 361, 489; 80, p. 437 (1926).

...representations. From the purely formal side, the new theories have yielded brilliant results. In connection with the theory of the rotating electron that has appeared recently, it has been possible to solve the problem of helium, the complex structure of spectral lines, and the complex Zeeman effect. The question of anharmonic motions has also found its solution.

The connection between matrix and wave mechanics. Soon after the appearance of both new theories it was shown1) that, despite their seemingly fundamental difference, these theories are mathematically identical. The quantities entering the matrices are the coefficients of the expansion in series of the corresponding functions (coordinates, momenta, etc.) in the fundamental functions of the problem.

The solution of any problem can be translated from the language of matrices into the language of wave theory and back. Recently, in most works on quantum mechanics both methods of notation have been used simultaneously.

There arises, however, the question of which of these theories should be assigned physical meaning. How are we to imagine that the coordinate of a point is represented by an infinite table of quantities? What are waves in “coordinate space,” whose number of dimensions is equal to the number of degrees of freedom of our system? What connection does this have with our usual ideas about the motion of material points (electrons) in three-dimensional space? After all, the individual electron exists—we know this at least from the cloud-chamber photographs of Wilson.

The wave theory of matter of de Broglie–Schrödinger. The wave theory of matter tried to construct the electron by building the so-called wave packets, i.e. to find such solutions of the Schrödinger equation for the one-body problem that would vanish everywhere outside some small region of space. Such a wave packet could represent an electron (the electric density \(|\psi|^2\) is different from zero only inside the electron).

It turned out, however, that such packets spread out in motion, which was a decisive blow to the wave theory of matter.

The statistical interpretation of wave mechanics according to M. Born. In the works of M. Born2) the idea of the statistical character of wave mechanics was first expressed. The de Broglie–Schrödinger waves are probability waves of the corresponding atomic processes. One may say that “atomic processes are governed only by the laws of probability, but this probability itself changes according to certain exact laws.”

Wave mechanics makes it possible to solve only questions of the dynamics of an ensemble of a very large number of atoms, and its results have only a statistical meaning. The propagation of a plane de Broglie wave corresponds to a stream of particles (electrons) moving in parallel. If, after this wave passes through places occupied by an ensemble of atoms (described by some Schrödinger function), a diffraction process occurs and the wave breaks up into a number of plane waves going in different directions, then this is interpreted as the scattering of electrons as a result of collisions with atoms. The intensity of the wave going in a certain direction gives the mean number of electrons rebounding in that direction; the \(n\)-th coefficient of the solution of the Schrödinger equation for an atom determines the mean number of atoms from the given ensemble that are in the \(n\)-th quantum state, and so on.

A formal justification of the statistical interpretation of matrix mechanics was given in the work of P. Jordan3).

1) E. Schrödinger, Ann. d. Phys. 79, p. 734 (1926).
2) M. Born, ZS. f. Phys. 38, p. 803 (1926); 40, p. 167 (1926).
3) P. Jordan, ZS. f. Phys. 40, p. 809 (1927).

Since the new quantum mechanics¹) is only a statistics of atomic processes, it is, of course, natural to pose the question: what, then, are the laws of the elementary phenomena whose consequences are our statistical conclusions? From the given results of the statistics one must find the most elementary phenomena and the exact laws governing them. This has not yet been done. Attempts to construct an atomic model for the new quantum mechanics have not succeeded. But, perhaps, such a model cannot be constructed at all? Must a theory bearing a statistical character necessarily be based on exact laws and models? W. Heisenberg attempts to solve the question precisely in this direction in his latest article.²)

For the consistent pursuit of this point of view, a deep critique is needed of our notions of the material point, its coordinates, and its velocity. The principal role in this critique is played by the question of the principial observability of a given physical quantity, which has already played a large role in the construction of the theory of relativity (the principial unobservability of absolute motion and simultaneity).

On elementary processes.

In constructing a physical theory one usually has to operate with a number of elementary processes, for example, the motion of electrons along ellipses inside atoms.

These elementary processes may also be unobservable experimentally, but their existence is made highly probable in view of the agreement of theory with experience.

Recently experimentalists have become especially interested in elementary processes. The paths of individual electrons and protons knocked out of an atom in a collision, etc., can be observed in Wilson cloud-chamber photographs. The works of Geiger and Bothe make it possible to register the elementary act of light scattering (“the collision of an electron with a light quantum”) in the Compton effect. The experiments of Stern and Gerlach with an atomic beam give information about the magnetic moment of the atom.

However, the question arises: is it possible to carry this investigation through to the end and to observe the whole picture of intra-atomic processes—motions along orbits, electronic jumps, etc.—as it presents itself to the mental eye of the modern physicist?

On the principial observability of physical quantities.

Here the concept of the principial observability of a physical phenomenon or quantity comes onto the stage.

A definite physical quantity is called principially observable if one can indicate a method—perhaps not executable with the present state of technology, but physically possible—by means of which our quantity can be measured.

The principial unobservability of simultaneity and the special theory of relativity.

The principle of principial observability says: in constructing a physical theory one may use only quantities that are principially observable. If a principially unobservable quantity is found to be present in a theory, then the theory must be rebuilt on new foundations so that in its new form it does not contain this quantity.

The principial unobservability of simultaneity also gave, as is known, Einstein the foundation for a critique of our notions of space and time, leading to the special theory of relativity, which does not contain the concept of absolute time.

¹) Here the expression “new quantum mechanics” should be understood as the totality of matrix and wave mechanics, as opposed to Bohr’s “old quantum mechanics.”

²) W. Heisenberg, ZS. f. Phys. 43, p. 172 (1927).

On the principled observability of intra-atomic processes.

Let us turn to the representations of Bohr’s theory from the point of view of our new principle. Can we indicate such a conceivable, physically possible method by means of which one might determine the coordinate and the velocity of a moving electron?

We can determine the position of an electron by illuminating it and observing it under a microscope; its velocity, on the other hand, is easily calculated from the Doppler effect of the light scattered by the electron. The limit of accuracy in determining the position is set by the wavelength of the incident light. We can, however, in principle use an arbitrarily short wave (penetrating radiation!), construct a “microscope for γ-rays,” and attain any desired accuracy in determining the coordinate. Here, however, one must take into account a very important phenomenon—the Compton effect. At the moment when the light is scattered by the electron (i.e. at the moment for which the coordinate is determined), the latter undergoes a sharp jump in velocity. The shorter the wavelength, the greater this jump, which introduces an uncertainty into the magnitude of the electron’s velocity.

Let us find the relation between the uncertainty \(\Delta q\) in the coordinate and the uncertainty \(\Delta p\) in the mechanical momentum of the electron. Consider the collision of a light quantum with an electron (here we shall use, for clarity, the hypothesis of light quanta, although the derivation can in essence be carried out quite formally from the basic formula for the Compton effect). The two extreme possible cases are depicted in Fig. 1. Neglecting the change in the frequency of the reflected quantum, we have, from the law of conservation of quantity of motion, for the first case:

Fig. 1.

\[ \Delta p=\frac{2h\nu}{c}=\frac{2h}{\lambda}, \]

for the second

\[ \Delta p=0. \]

For other cases we shall have intermediate values. On the average, taking into account that \(\Delta q \sim \lambda\), we obtain:

\[ \Delta p \Delta q \sim h. \]

If we wish to depict the state of motion of the electron in phase space \((pq)\), Fig. 2, then we shall be able to indicate only a certain area of magnitude \(h\), “inside which the desired point lies.”

The fact that the magnitude of this area in phase space is equal to \(h\) is closely connected with the division of phase space into cells of magnitude \(h\) in the old quantum mechanics. It can also be shown that this relation corresponds to the requirement:

\[ pq-qp=\frac{h}{2\pi i}\,1 \tag{A'} \]

under the statistical interpretation of matrix mechanics according to Jordan.

The same relation (A) is also obtained by another method of determining the coordinate of the electron, when the electron collides not with a light quantum, but with another electron. Thus we see that the simultaneous value of the coordinate and

G. A. GAMOV

the velocity of the electron in principle cannot be determined: an increase of accuracy in measuring the coordinate decreases the accuracy in measuring the velocity, and conversely. Consequently, in a rigorous exposition of the dynamics of atomic processes (micro-mechanics), the concept of a “pair of conjugate dynamical coordinates of a point” \((p, q)\) must not enter, just as exactly as the concept of absolute time does not enter into the theory of relativity. Wave mechanics provides precisely such a system. (Before Heisenberg’s critique of the concept of coordinates, wave mechanics was just as incomprehensible as the Lorentz transformations before Einstein’s critique of the concepts of space and time.)

In wave mechanics we cannot construct the concepts of coordinate and velocity with complete exactness. These concepts can be obtained only approximately, and the character of the uncertainty will fully correspond to our relation \(\Delta\). The concept of a trajectory as a geometrical line falls away. This is not contradicted by the existence of Wilson’s cloud photographs, since there we have, strictly speaking, not a trajectory, but a cylinder of very large cross-section, indicating only approximately the “path” of our electron.

Fig. 2.

Let us note that Schrödinger’s theory does not make it possible to predict exactly the position of the electron after a given interval of time. We obtain only a result of a statistical character: such-and-such a probability that it will be here, such-and-such that it will be there, etc.

Meanwhile, in performing an experiment, we always find the position of the electron exactly. In accepting Schrödinger’s theory as final, we must abandon determinism in the domain of micro-processes.

In this, the transition to wave mechanics differs from the transition to the theory of relativity, where the fall of the concept of absolute time did not affect the law of causality.

On the fundamental observability of electronic orbits in the atom.

Let us turn to the question of the observability of the electronic orbit in the atom.

“Electronic orbits” (N. Bohr) have dimensions of the order of \(10^{-8}\) cm. Consequently, in order to determine an orbit exactly, it is necessary to illuminate the atom with light whose wavelength is no greater than this quantity (hard \(\gamma\)-rays). A light quantum of such frequency, striking an electron in the atom, will of course knock it out beyond the limits of the atom. Thus we can determine only one point on a given orbit. The very determination of this one point already destroys the atom and makes the measurement of subsequent points impossible. It is a different matter if we have a set of atoms in an identical quantum state (the separation of atoms with a definite quantum state is in principle possible by the method of Stern and Gerlach) and carry out mass measurements. Here we obtain a certain function of the coordinates, which is nothing other than \(|\psi|^2\) of wave mechanics and determines, one might say to an adherent of the old theory, the probability for the electron to be in a given place.

On the spreading of wave packets.

The question of the spreading of wave packets also receives here a simple interpretation. A wave packet, from the point of view of the old mechanics, is nothing other than an indication that “in all probability” the electron lies in the region occupied by the packet, but we do not know its exact location. The velocity is likewise not entirely determined. Therefore, if after some interval of time we are asked about the probable position of the electron, we shall have to indicate some other region, of a size, as is easy to see,

THE PRINCIPLE OF OBSERVABILITY IN PRINCIPLE

the greater, the more time has elapsed since the initial measurement. After each new measurement of the electron’s position, the magnitude of the packet is reduced to its initial value. There is nothing surprising in this, since the wave packet is the result of our measurements.

Conclusion. Briefly summarizing the foregoing, we arrive at the following results:

The new quantum mechanics is not the result of a statistical description of atomic models as yet unknown to us.

The construction of a model of the atom on the basis of our ordinary kinematics is impossible, since in passing to the world of very small quantities many kinematic concepts (for example, trajectory) lose their meaning. Our usual ideas of motion are constructed by us on the basis of experience on a large scale, and from this there follows in no way the possibility (and necessity) of extrapolating them to the domain of atomic processes.

Modern physics shows that such extrapolation must not take place.

The kinematics and mechanics describing motion in the world of atoms differ essentially from our ordinary mechanics; the fundamental concepts and representations of motion here are quite different. Only in the case of the motion of very large masses does this micro-mechanics approach the old macro-mechanics, and we obtain the possibility of constructing the concepts of coordinates, velocities, trajectories, etc.

The analysis of the foundations of micro-mechanics cannot yet be considered complete.

The future will show whether the ideas set forth above are destined to become a firm foundation of physics.