The Dreams That Stuff is Made of

by Stephen Hawking

  dw W. Heisenberg, Ztschr. f. Phys. 33, p. 879, 1925; M. Born and P. Jordan, ibid. 34, p. 858, 1925; M. Born, W. Heisenberg, and P. Jordan, ibid. 35, p. 557 1926; P. Dirac, Proc. Roy. Soc., London, 109, p. 642, 1925.

  dx In relativity mechanics and taking a magnetic field into account the statement of the Hamilton-Jacobi equation becomes more complicated. In the case of a single electron, it asserts that the four-dimensional gradient of the action function, diminished by a given vector (the four-potential), has a constant value. The translation of this statement into the language of the wave theory presents a good many difficulties.

  dy Cf. Conrant-Hilbert, Methods of Mathematical Physics, i. (Berlin, Springer, 1924), v. § 9, p. 261, eqn. 43, and further ii. § 10, 4, p. 76.

  dz We are led thus to an equation in r, which may be treated by the method shown in the Kepler problem of Part I. Moreover, the one-dimensional oscillator leads to the same equation if q2 be taken as variable. I originally solved the problem directly in that way. For the hint that it was a question of Hermite polynomials, I have to thank Herr E. Fues. The polynomial appearing in the Kepler problem (eqn. 18 of Part I.) is the (2n + 1)th differential coefficient of the (n + l)th polynomial of Laguerre, as I subsequently found.

  ea Cf. A: Sommerfeld, Atombau und Spektrallinien, 4th edit., p. 833. We do not consider here the additional non-harmonic terms in the potential energy.

  eb Physik. Ztschr. 27, p. 95, 1926.

  ec Last two paragraphs of Part II.

  ed Courant-Hilbert, chap. vi. §§ 2, 4, p. 337.

  ee Courant-Hilbert, chap. v. § 5, 2, p. 241.

  ef P. S. Epstein, Ann. d. Phys. 50, p. 489, 1916.

  eg N. Bohr, Kopenhagener Akademie (8), IV., 1, 2, p. 69 et seq., 1918.

  eh A. Sommerfeld, Atombau, 4th ed., p. 772.

  ei H. A. Kramers, Kopenhagener Akademie (8), III., 3, p. 287, 1919.

  ej G. E. Uhlenbeck and S. Goudsmit, Physica, 1925: Die Naturwissenschaften, 1926; Nature, 20th Feb., 1926; cf. also L. H. Thomas, Nature, 10th April, 1926.

  ek J. C. Slater, Proc. Amer. Nat. Acad. 11, p. 732, 1925.

  el A. Sommerfeld and A. Unsöld, Ztschr. f. Phys. 36, p. 259, 1926.

  em Cf. last two pages of previous paper.

  en Cf. Courant-Hilbert, chap. v. § 5, 1, p. 238 et seq.

  eo Cf. Courant-Hilbert, chap. v. § 10, 2, p. 277.

  ep Cf. Courant-Hilbert, chap. v. § 5, 1, p. 240, and § 10, p. 279.

  eq Even the limitation “linear” is not absolutely necessary.

  er It follows from the general theory that the perturbed system of functions u*kl(x) must be orthogonal if the perturbation completely removes the degeneracy, and may be assumed orthogonal although that is not the case.

  es Courant-Hilbert, chap. i. § 3. 3, p. 14.

  et So far as the actual details of the analysis are concerned, the simplest way to get (32ʹ ), or, in general, to get the wave equation for any special co-ordinates, is to transform not the wave equation itself, but the corresponding variation problem (cf. Part I. p. 12), and thus to obtain the wave equation afresh as an Eulerian variation problem. We are thus spared the troublesome evaluation of the second derivatives. Cf. Courant-Hilbert, chap. iv. § 7, p. 193.

  eu W. Pauli, jun., Ztschr. f. Phys. 36, p. 336, 1926; N. Bohr, Die Naturw. 1, 1926.

  ev Preceding paper of this collection.

  ew W. Heisenberg, Ztschr. f. Phys. 33, p. 879, 1925; M. Born and P. Jordan, Ztschr. f. Phys. 34, pp. 867, 886, 1925.

  ex W. Pauli, jun., Ztschr. f. Physik, 36, p. 336, 1926.

  ey J. Stark, Ann. d. Phys. 48, p. 193, 1915.

  ez H. A. Kramers, Dänische Akademie (8), iii. 3, p. 333 et seq., 1919.

  fa J. Stark, Ann. d. Phys. 43, p. 1001 et seq., 1914.

  fb N. Bohr, Dänische Akademie (8), iv. 1. 1, p. 35, 1918.

  fc J. Stark, Ann. d. Phys. 43, p. 1004, 1914.

  fd Equation (38) at end of previous paper of this collection. The fourth allows for the fact that for the radiation it is a question of the square of the acceleration and not of the electric moment itself. In this equation (38) occurs explicitly another factor (Ek − Em)h. This is occasioned by the appearance of ∂/∂t in statement (36). Addition at proofcorrection: Now I recognise this ∂/∂t to be incorrect, though I hoped it would make the later relativistic generalisation easier. Statement (36), loc. cit., is to be replaced by ψ . The above doubts about the fourth power are therefore dissolved.

  fe Cf. M. Born and P. Jordan, Ztschr. f. Phys. 34, p. 887, 1925.

  ff R. Ladenburg, Ann. d. Phys. (4), 38, p. 249, 1912.

  fg R. Ladenburg and F. Reiche, Die Naturwissenschaften, 1923, p. 584.

  fh Cf. Ladenburg-Reiche, loc. cit., the first formula in the second column, p. 584. The factor ν0 in the above expression comes from the fact that the “transition probability” aki is still to be multiplied by the “energy quantum” to give the intensity of the radiation.

  fi I lately gave the proper functions (70) (see Part I.), but without noticing their connection with the Laguerre polynomials. For the proof of the above representation, see the Mathematical Appendix, section 1.

  fj It is to be noticed that the density function, generally denoted by ρ(x), reads as η sin θ in equation (67), because the equation must be multiplied by η2 sin θ, in order to acquire self-adjoint form.

  fk Courant-Hilbert, chap. ii. $ 11, 5, p. 78, equation (72).

  fl Courant-Hilbert, chap. iii. § 4, 2, p. 115.

  fm Courant-Hilbert, chap. v. § 5, 1, p. 240.

  fn Courant-Hilbert, chap. ii. § 11, 5, p. 78, equation (68).

  fo Cf. Ann. d. Phys. 79, pp. 361, 489; 80, p. 437, 1926 (Parts I., II., III.); further, on the connection with Heisenberg’s theory, ibid. 79, p. 734 (p. 4 5).

  fp E .g., for a vibrating plate, ∇2∇2 u + = 0. Cf. Courant-Hilbert, chap. v. , p. 256.

  fq Cf. Part III. § and 2, text beside equations (8) and (24).

  fr Cf. what follows, and § 7.

  fs Cf. end of paper on Quantum Mechanics of Heisenberg, etc., and also the Calculation of Intensities in the Stark Effect in Part III. At the first quoted place, the real part of ψψ was proposed instead of ψψ. This was a mistake, which was corrected in Part III.

  ft We assume as previously, for the sake of simplicity, the proper functions un (x ) to be real, but notice that it may sometimes be much more convenient or even imperative to work with complex aggregates of the real proper functions, e.g. in the proper functions of the Kepler problem to work with instead of mφ.

  fu H. A. Kramers, Nature, May 10, 1924; ibid. August 30, 1924; Kramers and W. Heisenberg. Ztschr. f. Phys. 31, p. 681, 1925. The description given in the latter paper of the polarisation of the scattered light (equation 27) from correspondence principles, is almost identical formally with ours.

  fv It is hardly necessary to say that the two directions which, for simplicity, we have designated as “z-direction” and “y-direction” do not require to be exactly perpendicular to one another. The one is the direction of polarisation of the incident wave; the other is that polarisation component of the secondary wave, in which we are specially interested.

  fw Born, Heisenberg, and Jordan, Ztschr. f. Phys. 35, p. 572, 1926.

  fx Cf. especially the concluding words of Heisenberg’s latest exposition of his theory, Math. Ann. 95, p. 683, 1926, in connection with this difficulty of comprehending the course of an event in time.

  fy Further discussed in § 5.

  fz H. Weyl, Math. Ann. 68, p. 220, 1910; Gött. Nachr. 1910. Cf. also E. Hilb, Sitz.-Ber. d. Physik. Mediz. Soc. Erlangen, 43, p. 68, 1911; Math. Ann. 71, p. 76, 1911. I have to thank Herr Weyl not only for these references but also for very valuable oral instruction in these not very simple matters.

  ga I have to thank Herr Fues for this exposition.

  gb As Herr E.Fues informs me, we can very often omit the limiting process in practice and write u(ξ, E) fo rth einne rintegral,viz.viz. always, when ∫ρ(ξ)f(ξ)u(ξ,E) dξ exists.

&n
bsp; gc K.F. Herzfeld and K. L. Wolf, Ann. d. Phys. 76, p. 71, 567, 1925; H. Kollmann and H. Mark, Die Nw. 14, p. 648, 1926.

  gd A very interesting and successful attempt to compare the action of flying charged particles with the action of light waves, through a Fourier decomposition of their field, is to be found in a paper by E. Fermi, Ztschr. f. Phys. 29, p. 315, 1924.

  ge End of Part II. (p. 39); paper on Heisenberg’s quantum mechanics (p. 60).

  gf Cf. paper on Heisenberg’s theory, equation (31). The quantity there denoted by ΔD − is our “density function” ρ (x) (e.g. r 2 sin in spherical polars). T is the kinetic energy as function of the position co-ordinates and momenta, the suffix at T denoting differentiation with respect to a momentum. In equations (31) and (32), loc. cit., unfortunately by error the suffix k is used twice, once for the summation and then also as a representative suffix in the argument of the functions.

  gg Pauli, ‘Z. f. Physik,’ vol. 43, p. 601 (1927).

  gh Darwin,’ Roy. Soc. Proc.,’ A, vol. 116, p. 227 (1927).

  gi Gordon, ‘Z. f. Physik,’ vol. 40, p. 117 (1926).

  gj Klein, ‘Z. f. Physik,’ vol. 41, p. 407 (1927).

  gk Jordan, ‘Z. f. Physik,’ vol. 40, p. 809 (1927); Dirac, ‘Roy. Soc. Proc.,’ A, vol. 113, p. 621 (1927).

  gl Pauli, loc. cit.

  gm We say that a anticommutes with b when ab = −ba.

  gn See ‘Roy. Soc. Proc.,’ A, vol. 111, p. 281 (1926).

  go See B. L. v. d. Waerden, Die gruppentheoretische Methode in der Quantentheorie (Berlin, 1932).

  gp In the spinor calculus this is a spinor with 2j undotted and 2k dotted indices.

  gq see M. Fierz, Helv. Phys. acta 12, 3 (1939); also L. de Broglie, Comptes rendus 208, 1697 (1939); 209, 265 (1939).

  gr By “gauge-transformation of the first kind” we understand a transformation U→Ueiα U* →U*e−iα with an arbitrary space and time function α. By “gauge-transformation of the second kind” we understand a transformation of the type

  as for those of the electromagnetic potentials.

  gs The general proof for this has been given by M. Fierz, Helv. Phys. Acta 13, 45 (1940).

  gt See for instance W. Pauli in the article “Wellen-mechanik” in the Handbuch der Physik, Vol. 24/2, p. 260.

  gu But we exclude operation like (k2 + k2 )½, which operate at finite distances in the coordinate space.

  gv M. Fierz, Helv. Phys. Acta 12, 3 (1939).

  gw The author therefore considers as not conclusive the original argument of Dirac. according to which the field equation must be of the first order.

  gx On account of the existence of such conditions the canonical formalism is not applicable for spin > 1 and therefore the discussion about the connection between spin and statistics by J. S. de Wet, Phys. Rev. 57, 646 (1940), which is based on that formalism is not general enough.

  gy The consistent development of this method leads to the “many-time formalism” of Dirac, which has been given by P. A. M. Dirac, Quantum Mechanics (Oxford, second edition, 1935).

  gz See P. A. M. Dirac, Proc. Camb. Phil. Soc. 30, 150 (1934).

  ha For the canonical quantization formalism this postulate is satisfied implicitly. But this postulate is much more general than the canonical formalism.

  hb See W. Pauli, Ann. de 1’Inst. H. Poincare 6, 137 (1936), esp. § 3.

  hc This contradiction may be seen also by resolving U(r) into eigenvibrations according to

  The equation (21) leads then, among others, to the relation

  a relation, which is not possible for brackets with the + sign unless U± (k) and (k) vanish.

  hd M. Fierz and W. Pauli, Proc. Roy. Soc. A173, 211 (1939).

  he R. Ladenburg, Z. Physik, 4 (1921) 451; R. Ladenburg and F. Reiche, Naturwiss., 11 (1923) 584.

  hf H. A. Kramers, Nature, 113 (1924) 673.

  hg H. A. Kramers and W. Heisenberg, Z. Physik, 31 (1925) 681.

  hh M. Born, Z. Physik, 26 (1924) 379; M. Born and P. Jordan, Z. Physik, 33 (1925) 479.

  hi W. Heisenberg, Z. Physik, 33 (1925) 879.

  hj M. Born and P. Jordan, Z. Physik, 34 (1925) 858.

  hk M. Born, W. Heisenberg, and P. Jordan, Z. Physik, 35 (1926) 557.

  hl P. A. M. Dirac, Proc. Roy. Soc. (London), A 109 (1925) 642.

  hm W. Pauli, Z. Physik, 36 (1926) 336.

  hn E. Schrödinger, Ann. Physik, [4] 79 (1926) 361, 489, 734; 80 (1926) 437; 81 (1926) 109.

  ho L. de Broglie, Thesis Paris, 1924; Ann. Phys. (Paris), [10] 3 (1925) 22.

  hp W. Elasser, Naturwiss., 13 (1925) 711.

  hq C. J. Davisson and L. H. Germer, Phys. Rev., 30 (1927) 707.

  hr G. P. Thomson and A. Reid, Nature, 119 (1927) 890; G. P. Thomson, Proc. Roy. Soc. (London), A 117 (1928) 600.

  hs E. Schrödinger, Brit. J. Phil. Sci., 3 (1952) 109, 233.

  ht M. Born and N. Wiener, Z. Physik, 36 (1926) 174.

  hu M. Born, Z. Physik, 37 (1926) 863; 38 (1926) 803; Göttinger Nachr. Math. Phys. Kl., (1926) 146.

  hv G. Wentzel, Z. Physik, 40 (1926) 590.

  hw W. Heisenberg, Z. Physik, 43 (1927) 172.

  hx H. Faxén and J. Holtsmark, Z. Physik, 45 (1927) 307.

  hy H. Bethe, Ann. Physik, 5 (1930) 325.

  hz N. F. Mott, Proc. Roy. Soc. (London), A 124 (1929) 422, 425; Proc. Cambridge Phil. Soc., 25 (1929) 304.

  ia M. Born, Z. Physik, 40 (1926) 167; M. Born and V. Fock, Z. Physik, 51 (1928) 165.

  ib P. A. M. Dirac, Proc. Roy. Soc. (London), A 109 (1925) 642; 110 (1926) 561; 111 (1926) 281; 112 (26) 674.

  ic N. Bohr, Naturwiss., 16 (1928) 245; 17 (1929) 483; 21 (1933) 13. “Kausalität und Komplementarität” (Causality and Complementarity), Die Erkenntnis, 6 (1936) 293.

  id M. Born, Phil. Quart., 3 (1953) 134; Physik. Bl., 10 (1954) 49.

  ie A. Einstein, B. Podolsky and N. Rosen, Phys. Rev. 47, 777 (1935).

  if Cf. N. Bohr, Atomic Theory and Description of Nature, I (Cambridge, 1934).

  ig The deductions contained in the article cited may in this respect be considered as an immediate consequence of the transformation theorems of quantum mechanics, which perhaps more than any other feature of the formalism contribute to secure its mathematical completeness and its rational correspondence with classical mechanics. In fact, it is always possible in the description of a mechanical system, consisting of two partial systems (1) and (2), interacting or not, to replace any two pairs of canonically conjugate variables (q1 p1), (q2 p2) pertaining to systems (1) and (2), respectively, and satisfying the usual commutation rules[q1 p1] = [q2 p2] = ih/2π,

  [q1 q2 ] = [p1 p2] = [q1 p2] = [q2 p1]= 0,

  by two pairs of new conjugate variables (Q1 P1), (Q2 P2) related to the first variables by a simple orthogonal transformation, corresponding to a rotation of angle θ in the planes (q1 q2), (p1 p2)q1 = Q1 cos θ − Q2 sin θ p1 = P1 cos θ − P2 sin θ

  q2 = Q1 sin θ + Q2 cos θ p2 = P1 sin θ + P2 cos θ.

  Since these variables will satisfy analogous commutation rules, in particular[Q1 P1] = ih/2π, [Q1 P2] = 0,

  it follows that in the description of the state of the combined system definite numerical values may not be assigned to both Q1 and P1, but that we may clearly assign such values to both Q1 and P2 . In that case it further results from the expressions of these variables in terms of (q1 p1) and (q2 p2), namelyQ1 = q1 cos θ + q2 sin θ, P2 = − p1 sin θ + p2 cos θ,

  that a subsequent measurement of either q2 or p2 will allow us to predict the value of q1 or p1 respectively.

  ih The obvious impossibility of actually carrying out, with the experimental technique at our disposal, such measuring procedures as are discussed here and in the following does clearly not affect the theoretical argument, since the procedures in question are essentially equivalent with atomic processes, like the Compton effect, where a corresponding application of the conservation theorem of momentum is well established.

  ii As will be seen, this description, a part from a trivial normalizing factor, corresponds
exactly to the transformation of variables described in the preceding footnote if (q1 p1), (q2 p2) represent the positional coordinates and components of momenta of the two particles and if θ = − π /4. It may also be remarked that the wave function given by formula (9) of the article cited corresponds to the special choice of P2 = 0 and the limiting case of two infinitely narrow slits.