(a) Determination of the position of a free particle. As a first example of the destruction of the knowledge of a particle’s momentum
FIG. 5
by an apparatus determining its position, we consider the use of a microscope.cb Let the particle be moving at such a distance from the microscope that the cone of rays scattered from it through the objective has an angular opening ε. If λ is the wave-length of the light illuminating it, then the uncertainty in the measurement of the x-co-ordinate (see Fig. 5) according to the laws of optics governing the resolving power of any instrument is:
(16)
But, for any measurement to be possible at least one photon must be scattered from the electron and pass through the microscope to the eye of the observer. From this photon the electron receives a Compton recoil of order of magnitude h/λ. The recoil cannot be exactly known, since the direction of the scattered photon is undetermined within the bundle of rays entering the microscope. Thus there is an uncertainty of the recoil in the x-direction of amount
(17)
and it follows that for the motion after the experiment
(18)
Objections may be raised to this consideration; the indeterminateness of the recoil is due to the uncertain path of the light quantum within the bundle of rays, and we might seek to determine the path by making the microscope movable and measuring the recoil it receives from the light quantum. But this does not circumvent the uncertainty relation, for it immediately raises the question of the position of the microscope, and its position and momentum will also be found to be subject to equation (18). The position of the microscope need not be considered if the electron and a fixed scale be simultaneously observed through the moving microscope, and this seems to afford an escape from the uncertainty principle. But an observation then requires the simultaneous passage of at least two light quanta through the microscope to the observer—one from the electron and one from the scale—and a measurement of the recoil of the microscope is no longer sufficient to determine the direction of the light scattered by the electron. And so on ad infinitum.
One might also try to improve the accuracy by measuring the maximum of the diffraction pattern produced by the microscope. This is only possible when many photons co-operate, and a calculation shows that the error in measurement of x is reduced to Δx = λ/√m sin ∈ when m photons produce the pattern. On the other hand, each photon contributes to the unknown change in the electron’s momentum, the result being Δpx = √mh sin ∈/λ (addition of independent errors). The relation (18) is thus not avoided.
It is characteristic of the foregoing discussion that simultaneous use is made of deductions from the corpuscular and wave theories of light, for, on the one hand, we speak of resolving power, and, on the other hand, of photons and the recoils resulting from their collision with the particle under consideration. This is avoided, in so far as the theory of light is concerned, in the following considerations.
If electrons are made to pass through a slit of width d (Fig. 6), then their co-ordinates in the direction of this width are known at the moment after having passed it with the accuracy Δx = d . If we assume the momentum in this direction to have been zero before
FIG. 6
passing through the slit (normal incidence), it would appear that the uncertainty relation is not fulfilled. But the electron may also be considered to be a plane de Broglie wave, and it is at once apparent that diffraction phenomena are necessarily produced by the slit. The emergent beam has a finite angle of divergence α, which is, by the simplest laws of optics,
(19)
where λ is the wave-length of the de Broglie waves. Thus the momentum of the electron parallel to the screen is uncertain, after passing through the slit, by an amount
(20)
since h/λ is the momentum of the electron in the direction of the beam. Then, since Δx = d, Δx Δp ∼ h.
Δx Δp ~ h.
In this discussion we have avoided the dual character of light, but have made extensive use of the two theories of the electron.
As a last method of determining position we discuss the well-known method of observing scintillations produced by α-rays when they are received on a fluorescent screen or of observing their tracks in a Wilson chamber. The essential point of these methods is that the position of the particle is indicated by the ionization of an atom; it is obvious that the lower limit to the accuracy of such a measurement is given by the linear dimension Δqs of the atom, and also that the momentum of the impinging particle is changed during the act of ionization. Since the momentum of the electron ejected from the atom is measurable, the uncertainty in the change of momentum of the impinging particle is equal to the range Δps within which the momentum of this electron varies while moving in its unionized orbit. This variation in momentum is again related to the size of the atom by the inequalityΔps Δqs ≥ h.
Later discussion will show, in fact, that quite generallycc Δps Δqs ∼ nh,
where n is the quantum number of the stationary state concerned (cf. § 2c below). Thus the uncertainty relation also governs this type of position measurement; here the dualism of treatment is relegated to the background, and the uncertainty relation appears rather to be the result of the Bohr quantum conditions determining the stationary state, but naturally the quantum conditions are themselves manifestations of the duality.
(b) Measurement of the velocity or momentum of a free particle. The simplest and most fundamental method of measuring velocity depends on the determination of position at two different times. If the time interval elapsing between the position measurements is sufficiently large, it is possible to determine the velocity before the second was made with any desired accuracy, but it is the velocity after this measurement which alone is of importance to the physicist, and this cannot be determined with exactness. The change in momentum which is necessarily produced by the last observation is subject to such an indeterminateness that the uncertainty relation is again fulfilled, as has been shown in the last section.
Another common method of determining the velocity of charged particles makes use of the Doppler effect. Figure 7 shows the experimental arrangement in its essentials. The component, px, of the
(22)
FIG. 7
electron’s momentum may be supposed to be known with ideal exactness, its x-co-ordinate therefore completely unknown. On the other hand, the y-co-ordinate of the electron will be assumed to have been accurately determined, and p y correspondingly unknown. The problem is therefore to determine the velocity in the y-direction, and it is to be shown that the knowledge of the y-co-ordinate is destroyed by this measurement to the extent demanded by the uncertainty relation. The light may be supposed incident along the x-axis, and the scattered light observed in the y-direction. (It is to be noted that the Doppler effect vanishes, under these conditions, if the electron moves along the straight line x − y = 0.) The theory of the Doppler effect is in this case identical with that of the Compton effect, and it is only necessary to use the laws of conservation of energy and momentum of the electron and light quantum. Letting E denote the energy of the electron, ν the frequency of the incident light, and using primes to distinguish the same quantity before and after the collision, we have
(21)
whence
Since it is assumed that px and ν are known, the accuracy of the determination of py is conditioned only by the accuracy with which the frequency νʹ of the scattered light is measured:
(23)
To determine νʹ with this accuracy, it is necessary to observe a train of waves of finite length, which in turn demands a finite time:
As it is unknown whether the photon collided with the electron at the beginning or at the end of this time interval, it is also unknown whether the electron moved with the velocity (1/µ)py or (1/µ)pʹy during this time. The uncertainty in the position of the electron which is produced by this cause is thus
whenceΔpyΔy ∼ h.
A third method of ve
locity measurement depends on the deflection of charged particles by a magnetic field. For this purpose a beam must be defined by a slit, whose width will be designated by d. This ray then enters a homogeneous magnetic field, whose direction is to be taken perpendicular to the plane of Figure 8. The length of that part of the
FIG. 8
ray which lies in the region of the field may be a; after leaving this region, the ray traverses a field-free region of length l and then passes through a second slit also of width d, whose position determines the angle of deflection α. The velocity of the particles in the direction of the beam is to be determined from the equation
(24)
The corresponding errors in measurement are related by
It may be supposed that the position of the particle in the direction of the ray was initially known with great accuracy. This may be achieved, for example, by opening the first slit only during a very brief interval. It will again be shown that this knowledge is lost during the experiment in such a manner that the relation Δp Δq ∼ h is fulfilled after the experiment. To begin with, the accuracy with which the angle α can be determined is obviously d/(l+a), but even this accuracy can only be attained if the natural de Broglie scattering of the ray is less than this. Therefore
whence
The uncertainty in the position of the particle in the ray after the experiment is equal to the product of the time required to pass through the field and reach the second slit and the uncertainty in the velocity. Thus
whence
The terms in the parentheses are equal to v/α and λ = h/µv, whence
since equation (24) is valid only for small values of α. For large angles of deflection, this derivation requires radical modification. One must remember, among other things, that the experiment as described here would not distinguish between α = 0 and α = 2π.
(c) Bound electrons. If it be required to deduce the uncertainty relations for the position, q, and momentum, p, of bound electrons, two problems must be clearly distinguished. The first assumes that the energy of the system, i.e., its stationary state, is known, and then inquires what accuracy of knowledge of p and q is implied in, or is compatible with, this knowledge of the energy. The second, distinct problem disregards the possibility of determining the energy of the system and merely inquires what the greatest accuracy is with which p and q may simultaneously be known. In this second case, the experiments necessary for the measurement of p and q may produce transitions from one stationary state to another; in the first case, the methods of measurement must be so chosen that transitions are not induced.
We consider the first problem in some detail, and assume an atom in a given stationary state. As Bohr has shown,cd the corpuscular theory then forces one to conclude that Δp Δq is in general greater than h . For it is obvious that we are concerned with the variation of p and q as the electron moves in its orbit, and it follows from
(25)
that
(26)
This may most readily be comprehended from a diagram of the orbit in phase space as given by classical mechanics (Fig. 9). The integral is nothing else than the area inclosed by the orbit, and Δp s Δqs is obviously of the same order of magnitude. The index s which accompanies these uncertainties is to indicate that they are not the absolute minima of these quantities, but are the special values which are assumed by them when the stationary state of the atom is known simultaneously and exactly. This uncertainty is of practical importance, for example, in the discussion of the scintillation method of counting α-particles (chap. ii, § 2a). In the classical theory, it would seem strange to consider this as an essential uncertainty, for further experiments could be made without disturbing the orbit. The quantum theory, however, shows that a knowledge of the energy is a “determinate case” (reiner Fall),ce i.e., a case which is represented in the mathematical scheme by a definite wave packet (in configuration space) which does not involve any undetermined constants. This wave packet is the Schrödinger function of the stationary state. If the calculation of pages 16–19 is carried through for this packet, the value of Δps Δqs is found to be greater in proportion to the number of nodes possessed by the characteristic function. If we consider a function s in equation (12) which possesses n nodes, the calculation would show thatΔps Δqs ∼ nh .
FIG. 9
To pass on to the second problem: The maximum accuracy is obviously given by ΔpΔq ∼ h if all knowledge of the stationary states be disregarded. Then the measurements can be carried out by such violent agents that the electron can be regarded as free (acted on only by negligible forces). The momentum of the electron can most readily be measured by suddenly rendering the interaction of the electron with the nucleus and neighboring electrons negligible. It will then execute a straight-line motion and its momentum can be measured in the manner already explained. The disturbance necessary for such a measurement is therefore obviously of the same order of magnitude as the binding energy of the electron.
The relation [eq. (6)] is of importance, as Bohr points out, for the equivalence of classical and quantum mechanics in the limit of large quantum numbers. This is seen when the validity of the concept of an “orbit” is examined. As the highest accuracy attainable is Δp Δq ∼ h , the orbit must be the path of a probability packet whose cross-section (|S(pʹ)|2|S(qʹ)|2) is approximately h. Such a packet can describe a well-defined, approximately closed path only if the area in-closed by this path is much greater than the cross-section of the wave packet. This, according to equation (26), is possible only in the limit of large quantum numbers; for small n, on the other hand, the concept of an orbit loses all significance, in phase space as well as in configuration space. It is thus seen to be essential for this limiting equivalence of the two theories that the factor n occurs on the right side of equation (26).
The inapplicability of the concept of an orbit in the region of small quantum numbers can be made clear from direct physical considerations in the following manner: The orbit is the temporal sequence of the points in space at which the electron is observed. As the dimensions of the atom in its lowest state are of the order 10−8 cm, it will be necessary to use light of wave-length not greater than 10−9 cm in order to carry out a position measurement of sufficient accuracy for the purpose. A single photon of such light is, however, sufficient to remove the electron from the atom, because of the Compton recoil. Only a single point of the hypothetical orbit is thus observable. One can, however, repeat this single observation on a large number of atoms, and thus obtain a probability distribution of the electron in the atom. According to Born, this is given mathematically by ψ ψ * (or, in the case of several electrons, by the average of this expression taken over the co-ordinates of the other electrons in the atom). This is the physical significance of the statement that ψ ψ * is the probability of observing the electron at a given point. This result is stranger than it seems at first glance. As is well known, ψ diminishes exponentially with increasing distance from the nucleus; there is thus always a small but finite probability of finding the electron at a great distance from the center of the atom. The potential energy of the electrons is negative at such a point, but very small. The kinetic energy is always positive; so that the total energy is therefore certainly greater than the energy of the stationary state under consideration. This paradox finds its resolution when the energy imparted to the electron by the photon used in making the position measurement is taken into account. This energy is considerably greater than the ionization energy of the electron, and thus suffices to prevent any violation of the law of conservation of energy, as is readily calculated explicitly from the theory of the Compton effect.
This paradox also serves as a warning against carrying out the “statistical interpretation” of quantum mechanics too schematically. Because of the exponential behavior of the Schrödinger function at infinity, the electron will sometimes be found as much as, say, 1 cm from the nucleus. One might suppose that it would be possible to verify the presence of the elect
ron at such a point by the use of red light. This red light would not produce any appreciable Compton recoil and the foregoing paradox would arise once more. As a matter of fact, the red light will not permit such a measurement to be made; the atom as a whole will react with the light according to the formulas of dispersion theory, and the result will not yield any information regarding the position of a given electron in the atom. This may be made plausible if one remembers that (according to the corpuscular theory) the electron will execute a number of rotations about the nucleus during one period of the red light. The statistical predictions of quantum theory are thus significant only when combined with experiments which are actually capable of observing the phenomena treated by the statistics. In many cases it seems better not to speak of the probable position of the electron, but to say that its size depends upon the experiment being performed.
The Dreams That Stuff is Made of Page 17