The Dreams That Stuff is Made of

by Stephen Hawking

  The starting-point of the critique of the relativity theory was the postulate that there is no signal velocity greater than that of light. In a similar manner, this lower limit to the accuracy with which certain variables can be known simultaneously may be postulated as a law of nature (in the form of the so-called uncertainty relations) and made the starting-point of the critique which forms the subject matter of the following pages. These uncertainty relations give us that measure of freedom from the limitations of classical concepts which is necessary for a consistent description of atomic processes. The program of the following considerations will therefore be: first, to obtain a general survey of all concepts whose introduction is suggested by the atomic experiments; second, to limit the range of application of these concepts; and third, to show that the concepts thus limited, together with the mathematical formulation of quantum theory, form a self-consistent scheme.

  § 2. THE FUNDAMENTAL CONCEPTS OF QUANTUM THEORY

  The most important concepts of atomic physics can be induced from the following experiments:

  (a) Wilsonbp photographs. The α- and β-rays emitted by radioactive elements cause the condensation of minute droplets when allowed to pass through supersaturated water vapor. These drops are not distributed at random, but are arranged along definite tracks which, in the case of α-rays (Fig. 1), are nearly straight lines, in the case of β-rays, are irregularly curved. The existence of the tracks and their continuity show that the rays may appropriately be regarded as streams of minute particles moving at high speeds. As is well known, the mass and charge of these particles may be determined from the deflection of the rays by electric and magnetic fields.

  FIG. 1 Tracks of α-particles in Wilson Chamber

  (b) Diffraction of matter waves (Davisson and Germer;bq Thomson,br Ruppbs). After the conception of β-rays as streams of particles had remained unchallenged for more than fifteen years, another series of experiments was performed which indicated that they could be diffracted and were capable of interference as if they were waves. Typical of these experiments is that of G. P. Thomson, in which a narrow beam of artificial β-rays of moderate energy is passed through a thin foil of matter. The foil is composed of minute crystals oriented at random, but the atoms in each crystal are regularly arranged. A photographic plate receiving the emergent rays exhibits rings of blackening (Fig. 2), as though the rays were waves and were diffracted by the minute crystals. From the diameters of the rings and the structure of the crystals, the length of these waves may be determined and is found to be λ = h/mv, where m is the mass and v the velocity of the particles as determined by the above-mentioned experiments. Similar experiments were performed by Davisson and Germer, Kikuchi,bt and Rupp.

  FIG. 2 Diffraction of electrons on passing through a thin foil of matter.

  (c) The diffraction of X-rays. The same dual interpretation is necessary in the case of light and electromagnetic radiation in general. After Newton’s objections to the wave theory of light had been refuted and the phenomena of interference explained by Fresnel, this theory dominated all others for many years, until Einstein† pointed out that the experiments of Lenard on the photoelectric effect could only be explained by a corpuscular theory. He postulated that the momentum of the hypothetical particles was related to the wave-length of the radiation by the formula p = h/λ (cf. § 2b). The necessity for both interpretations is particularly clear in the case of X-rays: If a homogeneous beam of X-rays is passed through a crystalline mass, and the emergent rays received on a photographic plate (Fig. 3), the result is much like the result of G. P. Thomson’s experiment, and it may be concluded that X-rays are a form of wave motion, with a determinable wave-length.

  (d) The Compton-Simonbt experiment. When a beam of X-rays passes through supersaturated water vapor, it is scattered by the molecules. Secondary products of the scattering are the “recoil” electrons, which are apparently particles of considerable energy, since they form tracks of condensed droplets as do the β-rays. These tracks are not very long, however, and occur with random direction. They apparently originate within the region traversed by the primary X-ray beam. Other secondary products of the scattering are the photo-electrons, which again make themselves evident by longer tracks of condensed water droplets. Under suitable conditions these tracks originate at points outside the primary X-ray beam, but the two secondary products are not unrelated. If it be assumed that the X-ray beam consists of a stream of light-particles (photons) and that the scattering process is the collision of a photon with one of the electrons of the molecule, as a result of which the electron recoils in the observed direction, Einstein’s postulate regarding the energy and momentum of the photons enables the direction of the photon after the collision to be calculated. This photon then collides with a second molecule, and gives up its remaining energy to an electron (the photoelectron). This assumption has been quantitatively verified (Fig. 4).

  FIG. 3 Diffraction of X-rays by MgO powder

  FIG. 4 Photograph showing recoil electron and associated photo electron liberated by X-rays. The upper photograph is retouched.

  (e) The collision experiments of Franck and Hertz.bu When a beam of slow electrons with homogeneous velocity passes through a gas, the electronic current as function of the velocity changes discontinuously at certain values of the velocity (energy). The analysis of these experiments leads to the conclusion that the atoms in the gas can only assume discrete energy values (Bohr’s postulate). When the energy of the atom is known, one speaks of a “stationary state of the atom.” When the kinetic energy of the electron is too small to change the atom from its stationary state to a higher one, the electron makes only elastic collisions with the atoms, but when the kinetic energy suffices for excitation some electrons will transfer their energy to the atom, so the electronic current as a function of the velocity changes rapidly in the critical region. The concept of stationary states, which is suggested by these experiments, is the most direct expression of the discontinuity in all atomic processes.

  From these experiments it is seen that both matter and radiation possess a remarkable duality of character, as they sometimes exhibit the properties of waves, at other times those of particles. Now it is obvious that a thing cannot be a form of wave motion and composed of particles at the same time—the two concepts are too different. It is true that it might be postulated that two separate entities, one having all the properties of a particle, and the other all the properties of wave motion, were combined in some way to form “light.” But such theories are unable to bring about the intimate relation between the two entities which seems required by the experimental evidence. As a matter of fact, it is experimentally certain only that light sometimes behaves as if it possessed some of the attributes of a particle, but there is no experiment which proves that it possesses all the properties of a particle; similar statements hold for matter and wave motion. The solution of the difficulty is that the two mental pictures which experiments lead us to form—the one of particles, the other of waves—are both incomplete and have only the validity of analogies which are accurate only in limiting cases. It is a trite saying that “analogies cannot be pushed too far,” yet they may be justifiably used to describe things for which our language has no words. Light and matter are both single entities, and the apparent duality arises in the limitations of our language.

  It is not surprising that our language should be incapable of describing the processes occurring within the atoms, for, as has been remarked, it was invented to describe the experiences of daily life, and these consist only of processes involving exceedingly large numbers of atoms. Furthermore, it is very difficult to modify our language so that it will be able to describe these atomic processes, for words can only describe things of which we can form mental pictures, and this ability, too, is a result of daily experience. Fortunately, mathematics is not subject to this limitation, and it has been possible to invent a mathematical scheme—the quantum theory—which seems entirely adequate for the treatm
ent of atomic processes; for visualization, however, we must content ourselves with two incomplete analogies—the wave picture and the corpuscular picture. The simultaneous applicability of both pictures is thus a natural criterion to determine how far each analogy may be “pushed” and forms an obvious starting-point for the critique of the concepts which have entered atomic theories in the course of their development, for, obviously, uncritical deduction of consequences from both will lead to contradictions. In this way one obtains the limitations of the concept of a particle by considering the concept of a wave. As N. Bohrbv has shown, this is the basis of a very simple derivation of the uncertainty relations between co-ordinate and momentum of a particle. In the same manner one may derive the limitations of the concept of a wave by comparison with the concept of a particle.

  It must be emphasized that this critique cannot be carried through entirely without using the mathematical apparatus of the quantum theory, for the development of the latter preceded the clarification of the physical principles in the historic sequence. In order to avoid obscuring the essential relationships by too much mathematics, however, it has seemed advisable to relegate this formalism to the Appendix. The exposition of mathematical principles given there does not pretend to be complete, but only to furnish the reader with those formulas which are essential for the argument of the text. References to this Appendix are given as A (16), etc.

  CRITIQUE OF THE PHYSICAL CONCEPTS OF THE CORPUSCULAR THEORY OF MATTER

  § 1. THE UNCERTAINTY RELATIONS

  The concepts of velocity, energy, etc., have been developed from simple experiments with common objects, in which the mechanical behavior of macroscopic bodies can be described by the use of such words. These same concepts have then been carried over to the electron, since in certain fundamental experiments electrons show a mechanical behavior like that of the objects of common experience. Since it is known, however, that this similarity exists only in a certain limited region of phenomena, the applicability of the corpuscular theory must be limited in a corresponding way. According to Bohr,bw this restriction may be deduced from the principle that the processes of atomic physics can be visualized equally well in terms of waves or particles. Thus the statement that the positionbx of an electron is known to within a certain accuracy Δx at the time t can be visualized by the picture of a wave packet in the proper position with an approximate extension Δx. By “wave packet” is meant a wavelike disturbance whose amplitude is appreciably different from zero only in a bounded region. This region is, in general, in motion, and also changes its size and shape, i.e., the disturbance spreads. The velocity of the electron corresponds to that of the wave packet, but this latter cannot be exactly defined, because of the diffusion which takes place. This indeterminateness is to be considered as an essential characteristic of the electron, and not as evidence of the inapplicability of the wave picture. Defining momentum as px = µvx (where µ = mass of electron, vx = x-component of velocity), this uncertainty in the velocity causes an uncertainty in px of amount Δpx; from the simplest laws of optics, together with the empirically established law λ = h/p, it can readily be shown that

  (1)

  Suppose the wave packet made up by superposition of plane sinusoidal waves, all with wave-lengths near λ0. Then, roughly speaking, n = Δx/λ0 crests or troughs fall within the boundary of the packet. Outside the boundary the component plane waves must cancel by interference; this is possible if, and only if, the set of component waves contains some for which at least n + 1 waves fall in the critical range. This gives

  where Δλ is the approximate range of wave-lengths necessary to represent the packet. Consequently

  (2)

  On the other hand, the group velocity of the waves (i.e., the velocity of the packet) is by A (85)

  (3)

  so that the spreading of the packet is characterized by the range of velocities

  By definition Δpx = µΔvg and therefore by equation (2),Δx Δpx ≥ h.

  This uncertainty relation specifies the limits within which the particle picture can be applied. Any use of the words “position” and “velocity” with an accuracy exceeding that given by equation (1) is just as meaningless as the use of words whose sense is not defined.by

  The uncertainty relations can also be deduced without explicit use of the wave picture, for they are readily obtained from the mathematical scheme of quantum theory and its physical interpretation.bz Any knowledge of the co-ordinate q of the electron can be expressed by a probability amplitude S(qʹ), |S(qʹ)|2dqʹ being the probability of finding the numerical value of the co-ordinate of the electron between qʹ and qʹ + dqʹ. Let

  (4)

  be the average value of q . Then Δq defined by

  (5)

  can be called the uncertainty in the knowledge of the electron’s position. In an exactly analogous way |T(pʹ)|2dpʹ gives the probability of finding the momentum of the electron between pʹ and pʹ + dpʹ; again and Δp may be defined as

  (6)

  (7)

  By equation A(169), the probability amplitudes are related by the equations

  (8)

  where R(qʹ pʹ) is the matrix of the transformation from a Hilbert space in which q is a diagonal matrix to one in which p is diagonal. From equation A(41) we have∫ p(qʹq”)R(q” pʹ)dq” = ∫ R(qʹ p”)p(p” pʹ)d p”,

  and by equation A(42) this is equivalent to

  (9)

  whose solution is

  (10)

  Normalizing gives c the value 1/√h. The values of Δp , Δq are thus not independent. To simplify further calculations, we introduce the following abbreviations:

  (11)

  Then equations (5) and (7) become

  (5a)

  (7a)

  while equations (8) become

  (8a)

  Combining (5a), (7a), and (8a), the expression for (Δp )2 may be transformed, giving

  or

  (12)

  Now

  (13)

  as may be proved by rearranging the obvious relation

  (13a)

  Hence it follows from equation (12) that or

  (14)

  which was to be proved. The equality can be true in (14) only when the left side of (13a) vanishes, i.e., when or

  (15)

  where c is an arbitrary constant. Thus the Gaussian probability distribution causes the product Δp Δq to assume its minimum value.

  It must be emphasized again that this proof does not differ at all in mathematical content from that given at the beginning of this section on the basis of the duality between the wave and corpuscular pictures of atomic phenomena. The first proof, if carried through precisely, would also involve all the equations (4)–(14). Physically, the last proof appears to be more general than the former, which was proved on the assumption that x was a cartesian co-ordinate and applies specifically only to free electrons because of the relation λ = h/µνg which enters into the proof. Equation (14), on the other hand, applies to any pair of canonic conjugates p and q. This greater generality of (14) is rather specious, however. As Bohrca has emphasized, if a measurement of its co-ordinate is to be possible at all, the electron must be practically free.

  § 2. ILLUSTRATIONS OF THE UNCERTAINTY RELATIONS

  The uncertainty principle refers to the degree of indeterminateness in the possible present knowledge of the simultaneous values of various quantities with which the quantum theory deals; it does not restrict, for example, the exactness of a position measurement alone or a velocity measurement alone. Thus suppose that the velocity of a free electron is precisely known, while the position is completely unknown. Then the principle states that every subsequent observation of the position will alter the momentum by an unknown and undeterminable amount such that after carrying out the experiment our knowledge of the electronic motion is restricted by the uncertainty relation. This may be expressed in concise and general terms by saying that every experiment destroys some of the knowledge of the system wh
ich was obtained by previous experiments. This formulation makes it clear that the uncertainty relation does not refer to the past; if the velocity of the electron is at first known and the position then exactly measured, the position for times previous to the measurement may be calculated. Then for these past times ΔpΔq is smaller than the usual limiting value, but this knowledge of the past is of a purely speculative character, since it can never (because of the unknown change in momentum caused by the position measurement) be used as an initial condition in any calculation of the future progress of the electron and thus cannot be subjected to experimental verification. It is a matter of personal belief whether such a calculation concerning the past history of the electron can be ascribed any physical reality or not.