The Dreams That Stuff is Made of

by Stephen Hawking

  The laws of quantum mechanics are basically statistical. Although the parameters of an atomic system are determined in their entirety by an experiment, the result of a future observation of the system is not generally accurately predictable. But at any later point of time there are observations which yield accurately predictable results. For the other observations only the probability for a particular outcome of the experiment can be given. The degree of certainty which still attaches to the laws of quantum mechanics is, for example, responsible for the fact that the principles of conservation for energy and momentum still hold as strictly as ever. They can be checked with any desired accuracy and will then be valid according to the accuracy with which they are checked. The statistical character of the laws of quantum mechanics, however, becomes apparent in that an accurate study of the energetic conditions renders it impossible to pursue at the same time a particular event in space and time.

  For the clearest analysis of the conceptual principles of quantum mechanics we are indebted to Bohr who, in particular, applied the concept of complementarity to interpret the validity of the quantum-mechanical laws. The uncertainty relations alone afford an instance of how in quantum mechanics the exact knowledge of one variable can exclude the exact knowledge of another. This complementary relationship between different aspects of one and the same physical process is indeed characteristic for the whole structure of quantum mechanics. I had just mentioned that, for example, the determination of energetic relations excludes the detailed description of space-time processes. Similarly, the study of the chemical properties of a molecule is complementary to the study of the motions of the individual electrons in the molecule, or the observation of interference phenomena complementary to the observation of individual light quanta. Finally, the areas of validity of classical and quantum mechanics can be marked off one from the other as follows: Classical physics represents that striving to learn about Nature in which essentially we seek to draw conclusions about objective processes from observations and so ignore the consideration of the influences which every observation has on the object to be observed; classical physics, therefore, has its limits at the point from which the influence of the observation on the event can no longer be ignored. Conversely, quantum mechanics makes possible the treatment of atomic processes by partially foregoing their space-time description and objectification.

  So as not to dwell on assertions in excessively abstract terms about the interpretation of quantum mechanics, I would like briefly to explain with a well-known example how far it is possible through the atomic theory to achieve an understanding of the visual processes with which we are concerned in daily life. The interest of research workers has frequently been focused on the phenomenon of regularly shaped crystals suddenly forming from a liquid, e.g. a supersaturated salt solution. According to the atomic theory the forming force in this process is to a certain extent the symmetry characteristic of the solution to Schrödinger’s wave equation, and to that extent crystallization is explained by the atomic theory. Nevertheless this process retains a statistical and - one might almost say - historical element which cannot be further reduced: even when the state of the liquid is completely known before crystallization, the shape of the crystal is not determined by the laws of quantum mechanics. The formation of regular shapes is just far more probable than that of a shapeless lump. But the ultimate shape owes its genesis partly to an element of chance which in principle cannot be analysed further.

  Before closing this report on quantum mechanics, I may perhaps be allowed to discuss very briefly the hopes that may be attached to the further development of this branch of research. It would be superfluous to mention that the development must be continued, based equally on the studies of de Broglie, Schrödinger, Born, Jordan, and Dirac. Here the attention of the research workers is primarily directed to the problem of reconciling the claims of the special relativity theory with those of the quantum theory. The extraordinary advances made in this field by Dirac about which Mr. Dirac will speak here, meanwhile leave open the question whether it will be possible to satisfy the claims of the two theories without at the same time determining the Sommerfeld fine-structure constant. The attempts made hitherto to achieve a relativistic formulation of the quantum theory are all based on visual concepts so close to those of classical physics that it seems impossible to determine the fine-structure constant within this system of concepts. The expansion of the conceptual system under discussion here should, furthermore, be closely associated with the further development of the quantum theory of wave fields, and it appears to me as if this formalism, notwithstanding its thorough study by a number of workers (Dirac, Pauli, Jordan, Klein, Wigner, Fermi) has still not been completely exhausted. Important pointers for the further development of quantum mechanics also emerge from the experiments involving the structure of the atomic nuclei. From their analysis by means of the Gamow theory, it would appear that between the elementary particles of the atomic nucleus forces are at work which differ somewhat in type from the forces determining the structure of the atomic shell; Stem’s experiments seem, furthermore, to indicate that the behaviour of the heavy elementary particles cannot be represented by the formalism of Dirac’s theory of the electron. Future research will thus have to be prepared for surprises which may otherwise come both from the field of experience of nuclear physics as well as from that of cosmic radiation. But however the development proceeds in detail, the path so far traced by the quantum theory indicates that an understanding of those still unclarified features of atomic physics can only be acquired by foregoing visualization and objectification to an extent greater than that customary hitherto. We have probably no reason to regret this, because the thought of the great epistemological difficulties with which the visual atom concept of earlier physics had to contend gives us the hope that the abstracter atomic physics developing at present will one day fit more harmoniously into the great edifice of Science.

  QUANTISATION AS A PROBLEM OF PROPER VALUES; PARTS I-IV

  BY

  ERWIN SCHRODINGER

  This translation is part of a work originally published as a separate work, Four Lectures on Wave Mechanics, which was originally published at Glasgow in 1928.db

  QUANTISATION AS A PROBLEM OF PROPER VALUES (PART I)

  § 1. In this paper I wish to consider, first, the simple case of the hydrogen atom (non-relativistic and unperturbed), and show that the customary quantum conditions can be replaced by another postulate, in which the notion of “whole numbers”, merely as such, is not introduced. Rather when integralness does appear, it arises in the same natural way as it does in the case of the node-numbers of a vibrating string. The new conception is capable of generalisation, and strikes, I believe, very deeply at the true nature of the quantum rules.

  The usual form of the latter is connected with the Hamilton-Jacobi differential equation,

  (1)

  A solution of this equation is sought such as can be represented as the sum of functions, each being a function of one only of the independent variables q .

  Here we now put for S a new unknown ψ such that it will appear as a product of related functions of the single co-ordinates, i.e. we put

  (2)

  The constant K must be introduced from considerations of dimensions; it has those of action. Hence we get

  (1ʹ)

  Now we do not look for a solution of equation ( 1ʹ), but proceed as follows. If we neglect the relativistic variation of mass, equation ( 1’) can always be transformed so as to become a quadratic form (of ψ and its first derivatives) equated to zero. (For the one-electron problem is holds even when mass-variation is not neglected.) We now seek junction ψ , such that for any arbitrary variation of it the integral the said quadratic form, taken over the whole co-ordinate space,dc stationary, ψ being everywhere real, single-valued, finite, and continuously differentiable up to the second order. The quantum conditions are replaced by this variation problem.

  First, we will take
for H the Hamilton function for Keplerian motion, and show that ψ can be so chosen for all positive, but only for discrete set of negative values of E. That is, the above variation problem has a discrete and a continuous spectrum of proper values.

  The discrete spectrum corresponds to the Balmer terms and the ntinuous to the energies of the hyperbolic orbits. For numerical agreement K must have the value h2π .

  The choice of co-ordinates in the formation of the variational equations being arbitrary, let us take rectangular Cartesians. Then (1ʹ) becomes in our case

  (1”)

  e = charge, m = mass of an electron, r 2 = x 2 + y 2 + z2 . Our variation problem then reads

  (3)

  the integral being taken over all space. From this we find in the usual way

  (4)

  Therefore we must have, firstly,

  (5)

  And secondly,

  (6)

  df is an element of the infinite closed surface over which the integral is taken.

  (It will turn out later that this last condition requires us to supplement our problem by a postulate as to the behaviour of δψ at infinity, in order to ensure the existence of the above-mentioned continuous spectrum of proper values. See later.)

  The solution of (5) can be effected, for example, in polar coordinates, r, Θ, φ, if ψ be written as the product of three functions, each only of r, of Θ, or of φ. The method is sufficiently well known. The function of the angles turns out to be a surface harmonic, and if that of r be called χ , we get easily the differential equation,

  (7)

  The limitation of n to integral values is necessary so that the surface harmonic may be single-valued. We require solutions of (7) that will remain finite for all non-negative real values of r . Nowdd equation (7) has two singularities in the complex r-plane, at r = 0 and r = ∞, of which the second is an “indefinite point” (essential singularity) of all integrals, but the first on the contrary is not (for any integral). These two singularities form exactly the bounding points of our real interval. In such a case it is known now that the postulation of the finiteness of χ at the bounding points is equivalent to a boundary condition. The equation has in general no integral which remains finite at both end points; such an integral exists only for certain special values of the constants in the equation. It is now a question of defining these special values. This is the jumping-off point of the whole investigation.de

  Let us examine first the singularity at r = 0. The so-called indicial equation which defines the behaviour of the integral at this point, is

  (8)

  with roots

  (8ʹ)

  The two canonical integrals at this point have therefore the exponents n and −(n + 1). Since n is not negative, only the first of these is of use to us. Since it belongs to the greater exponent, it can be represented by an ordinary power series, which begins with r n . (The other integral, which does not interest us, can contain a logarithm, since the difference between the indices is an integer.) The next singularity is at infinity, so the above power series is always convergent and represents a transcendental integral function. We therefore have established that:

  The required solution is (except for a constant factor) a single-valued definite transcendental integral function, which at r = 0 belongs to the exponent n.

  We must now investigate the behaviour of this function at infinity on the positive real axis. To that end we simplify equation (7) by the substitution

  (9)

  where α is so chosen that the term with 1/r 2 drops out. It is easy to verify that then α must have one of the two values n, −(n + 1). Equation (7) then takes the form,

  (7ʹ)

  Its integrals belong at r = 0 to the exponents 0 and −2α −. For the α-value, α = n, the first of these integrals, and for the second α value, α = −(n + 1), the second of these integrals is an integral function and leads, according to (9), to the desired solution, which is single-valued. We therefore lose nothing if we confine ourselves to one of the two α-values. Take, then,

  (10)

  Our solution U then, at r = 0, belongs to the exponent 0. Equation (7ʹ) is called Laplace’s equation. The general type is

  (7”)

  Here the constants have the values

  (11)

  This type of equation is comparatively simple to handle for this reason: The so-called Laplace’s transformation, which in general leads again to an equation of the second order, here gives one of the first. This allows the solutions of (7”) to be represented by complex integrals. The resultdf only is given here. The integral

  (12)

  is a solution of (7”) for a path of integration L, for which

  (13)

  The constants c1, c2, α1, α2 have the following values. c1 and c2 are the roots of the quadratic equation

  (14)

  and

  (14ʹ)

  In the case of equation (7ʹ) these become, using (11) and (10),

  (14”)

  The representation by the integral (12) allows us, not only to survey the asymptotic behaviour of the totality of solutions when r tends to infinity in a definite way, but also to give an account of this behaviour for one definite solution, which is always a much more difficult task.

  We shall at first exclude the case where α1 and α2 are real integers. When this occurs, it occurs for both quantities simultaneously, and when, and only when,

  (15)

  Therefore we assume that (15) is not fulfilled.

  The behaviour of the totality of solutions when r tends to infinity in a definite manner—we think always of r becoming infinite through real positive values—is characteriseddg by the behaviour of the two linearly independent solutions, which we will call U1 and U2, and which are obtained by the following specialisations of the path of integration L. In each case let z come from infinity and return there along the same path, in such a direction that

  (16)

  i.e. the real part of zr is to become negative and infinite. In this way condition (13) is satisfied. In the one case let z make a circuit once round the point c1 (solution U1), and in the other, round c2 (solution U2).

  Now for very large real positive values of r , these two solutions are represented asymptotically (in the sense used by Poincaré) by

  (17)

  in which we are content to take the first term of the asymptotic series of integral negative powers of r.

  We have now to distinguish between the two cases.

  (1) E > 0.This guarantees the non-fulfilment of (15), as it makes the left hand a pure imaginary. Further, by (14”), c 1 and c 2 also become pure imaginaries. The exponential functions in (17), since r is real, are therefore periodic functions which remain finite. The values of α1 and α2 from (14”) show that both U1 and U2 tend to zero like r −n−1 . This must therefore be valid for our transcendental integral solution U, whose behaviour we are investigating, however it may be linearly compounded from U1 and U2. Further, (9) and (10) show that the function χ, i.e. the transcendental integral solution of the original equation (7), always tends to zero like 1/r , as it arises from U through multiplication by rn . We can thus state:The Eulerian differential equation (5) of our variation problem has, for every positive E , solutions, which are everywhere single-valued, finite, and continuous; and which tend to zero with 1/r at infinity, under continual oscillations. The surface condition (6) has yet to be discussed.

  (2) E < 0. In this case the possibility (15) is not eo ipso excluded, yet we will maintain that exclusion provisionally. Then by (14”) and (17), for r → ∞, U1 grows beyond all limits, but U2 vanishes exponentially. Our integral function U (and the same is true for χ ) will then remain finite if, and only if, U is identical with U2, save perhaps for a numerical factor. This, however, can never be, as is proved thus: If a closed circuit round both points c1 and c2 be chosen for the path L, thereby satisfying condition (13) since the circuit is really closed on the Riemann surface of the integrand, on account of
α1 + α2 being an integer, then it is easy to show that the integral (12) represents our integral function U. (12) can be developed in a series of positive powers of r , which converges, at all events, for r sufficiently small, and since it satisfies equation (7ʹ), it must coincide with the series for U. Therefore U is represented by (12) if L be a closed circuit round both points c1 and c2. This closed circuit can be so distorted, however, as to make it appear additively combined from the two paths, considered above, which belonged to U1 and U2; and the factors are non-vanishing, 1 and e 2π iα1 . Therefore U cannot coincide with U2 , but must contain also U1 . Q.E.D.