ACKNOWLEDGMENTS
This book would not have been possible without the help of a number of talented people who made different contributions at various stages of the book’s development. Among those deserving special thanks are Joel Allred, David Goldberg, Leonard Mlodinow, and Karen Pelaez.
a O. Lummer and E. Pringsheim, Transactions of the German Physical Society 2 (1900), p. 163
b H. Rubens and F. Kurlbaum, Proceedings of the Imperial Academy of Science, Berlin, October 25, 1900, p. 929.
c H. Beckmann, Inaugural dissertation, Tubingen 1898. See also H. Rubens, Weid. Ann. 69 (1899) p. 582.
d M. Planck, Ann. d. Phys. 1 (1900), p. 719.
e Compare with equation (8).
f M. Planck, loc. cit., pp. 730 ff.
g Moreover one should compare the critiques previously made of this theorem by W. Wien (Report of the Paris Congress 2, 1900, p. 40) and by O. Lummer (loc. cit., 1900, p. 92.).
h L. Boltzmann, Proceedings of the Imperial Academy of Science, Vienna, (II) 76 (1877), p. 428.
i Joh. v. Kries, The Principles of Probability Calculation (Freiburg, 1886), p. 36.
j W. Wien, Proceedings of the Imperial Academy of Science, Berlin, February 9, 1893, p. 55.
k M. Thiesen, Transactions of the German Physical Society 2 (1900), p. 66.
l Perhaps one should speak more appropriately of a “white” radiation, to generalize what one already understands by total white light.
m M. Planck, Ann. D. Phys. 1 (1900), p. 99.
n F. Kurlbaum, Wied. Ann. 65 (1898), p. 759.
o O. Lummer and Pringsheim, Transactions of the German Physical Society 2 (1900), p. 176.
p This assumption is equivalent to the supposition that the average kinetic energies of gas molecules and electrons are equal to each other at thermal equilibrium. It is well known that, with the help of this assumption, Herr Drude derived a theoretical expression for the ratio of thermal and electrical conductivities of metals.
q M. Planck, Ann. Phys. 1, 99 (1900).
r This problem can be formulated in the following manner. We expand the Z component of the electrical force (Z) at an arbitrary point during the time interval between t = 0 and t = T in a Fourier series in which Aν ≥ 0 and 0 ≤ αν ≤ 2π : the time T is taken to be very large relative to all the periods of oscillation that are present:
If one imagines making this expansion arbitrary often at a given point in space at randomly chosen instants of time, one will obtain various sets of values of Aν and αν . There then exist for the frequency of occurrence of different sets of values of Aν and αν (statistical) probabilities dW of the form:
The radiation is then as disordered as conceivable if
i.e., if the probability of a particular value of A or α is independent of other values of A or α. The more closely this condition is fulfilled (namely, that the individual pairs of values of Aν and αν are dependent upon the emission and absorption processes of specific groups of oscillators) the more closely will radiation in the case being considered approximate a perfectly random state.
s M. Planck, Ann. Phys. 4, 561 (1901).
t This assumption is an arbitrary one. One will naturally cling to this simplest assumption as long as it is not controverted experiment.
u If E is the energy of the system, one obtains:−d · ( E − TS) = pdv = TdS = RT · (n/ N) · (dv/v) ;
thereforepv = R · (n/N) · T.
v P. Lenard, Ann. Phys., 8, 169, 170 (1902).
w If one assumes that the individual electron is detached from a neutral molecule by light with the performance of a certain amount of work, nothing in the relation derived above need be changed; one can simply consider Pʹ as the sum of two terms.
x P. Lenard, Ann. Phys. 8, pp. 163, 185, and Table I, Fig. 2 (1902).
y P. Lenard, Ref. 9, p. 150 and p. 166–168.
z Should be ΠE (translator’s note).
aa P. Lenard, Ann. Phys., 12, 469 (1903).
ab J. Stark, Die Electrizitët in Gasen (Leipzig, 1902, p. 57)
ac In the interior of gases the ionization potential for negative ions is, however, five times greater.
ad To those physicists who, in spite of all this, regard the hypothesis of elementary disorder as gratuitous or as incorrect, I wish to refer the simple fact that in every calculation of a coefficient of friction, of diffusion, or of heat conduction, from molecular considerations, the notion of elementary disorder is employed, whether tacitly or otherwise, and that it is therefore essentially more correct to stipulate this condition instead of ignoring or concealing it. But he who regards the hypothesis of elementary disorder as self-evident, should be reminded that, in accordance with a law of H. Poincaré, the precise investigation concerning the foundation of which would here lead us too far, the assumption of this hypothesis for all times is unwarranted for a closed space with absolutely smooth walls,—an important conclusion, against which can only be urged the fact that absolutely smooth walls do not exist in nature.
ae Communicated by the Author. A brief account of this paper was communicated to the Manchester Literary and Philosophical Society in February, 1911.
af Proc. Roy. Soc. lxxxii, p. 495 (1909)
ag roc. Roy. Soc. lxxxiii, p. 492 (1910)
ah Camb. Lit. & Phil Soc. xv pt. 5 (1910)
ai Crowther, Proc. Roy. Soc. lxxxiv. p. 226 (1910)
aj The deviation of a particle throughout a considerable angle from an encounter with a single atom will in this paper be called ‘single’ scattering. The deviation of a particle resulting from a multitude of small deviations will be termed ‘compound’ scattering.
ak A simple consideration shows that the deflexion is unaltered if the forces are attractive instead of repulsive.
al Manch. Lit. & Phil. Soc. 1910.
am The effect of change of velocity in an atomic encounter is neglected in this calculation.
an Phil. Mag. xviii. p. 909 (1909)
ao Annal. d. Phys. iv. 23. p. 671 (1907)
ap Nagaoka, Phil. Mag. vii. p. 445 (1904).
aq Communicated by Prof. E. Rutherford, F.R.S.
ar E. Rutherford, Phil. Mag. xxi. p. 669 (1911).
as See also Geiger and Marsden, Phil. Mag. April 1913.
at J.J. Thomson, Phil. Mag. vii. p. 237 (1904).
au See f. inst., ‘ThĀ©orie du ravonnement et les quanta.’ Rapports de la réunion à Bruxelles, Nov. 1911. Paris, 1912.
av See f. inst., M. Planck, Ann. d. Phys. xxxi. p. 758 (1910); xxxvii. p. 642 (1912); Verh. deutsch. Phys. Ges. 1911, p. 138.
aw A. Einstein, Ann. d. Phys. xvii. p. 132 (1905); xx. p. 199 (1906); xxii. p. 180 (1907).
ax A. E. Haas, Jahrb. d. Rad. u. El. vii. p. 261 (1910). See further, A. Schidlof, Ann. d. Phys. xxxv. p. 90 (1911); E. Wertheimer, Phys. Zeitschr. xii. p. 409 (1911), Verh. deutsch. Phys. Ges. 1912, p. 431; F.A. Lindemann, Verh. deutsch. Phys. Ges. 1911, pp. 482, 1107; F. Haber, Verh. deutsch. Phys. Ges. 1911, p. 1117.
ay J.W. Nicholson, Month. Not. Roy. Astr. Soc. lxxii. pp. 49,130, 677, 693, 729 (1912).
az See f. inst. N. Bohr, Phil. Mag. xxv. p. 24 (1913). The conclusion drawn in the paper cited is strongly supported by the fact that hydrogen, in the experiments on positive rays of Sir J. J. Thomson, is the only element which never occurs with a positive charge corresponding to the loss of more than one electron (comp. Phil. Mag. xxiv. p. 672 (1912)).
ba F. Paschen, Ann. d. Phys. xxvii. p. 565 (1908).
bb E. C. Pickering, Astrophys. J. iv p. 369 (1896); v. p. 92 (1897).
bc A. Fowler, Month. Not. Roy. Astr. Soc. lxxiii Dec. 1912.
bd W. Ritz, Phys. Zeitschr. ix p. 521 (1908).
be J. W. Nicholson loc. cit. p. 679.
bf A. Einstein, Ann. d. Phys. xvii. p. 146 (1905).
bg R. W. Wood, Physical Optics p. 513 (1911).
bh Compare J.J. Thomson, Phil. Mag, xxiii. p. 456 (1912).
bi E. Rutherford, Phil. Mag. xxiv. pp. 453 & 893 (1912).
bj Loc. cit.
bk Loc. cit.
bl In the considerations leading to this hypothesis we have assumed that the velocity of the electrons is small compared with the velocity of light. The limits of validity of this assumption will be discussed in Part II.
bm For the result of the continued work of Coster and Hevesy with the new element, for which they have proposed the name hafnium, the reader may be referred to their letters in Nature of January 20, February 10 and 24, and April 7.
bn TRANSLATORS’ NOTE.—In the English edition, Professor Heisenberg’s lectures on the mathematical part of the theory have been reproduced in more detail. This seemed advisable since a treatment of the general transformation theory and the quantum theory of wave fields was not available in English at the time the manuscript was prepared. The former has since been treated in several texts (E. U. Condon and P. M. Morse, Quantum Mechanics; A. E. Ruark and H. C. Urey, Atoms, Molecules and Quanta; both published by McGraw-Hill).
The English text also deviates in several other points from the German, but these are felt to be unessential changes.
bo W. Heisenberg, Zeitschrift für Physik, 43, 172, 1927.
bp Proceedings of the Royal Society, A, 85, 285, 1911; see also Jahrbuch der Radioaktivitat, 10, 34, 1913.
bq Physical Review, 30, 705, 1927; Proceedings of the National Academy, 14, 317, 1928.
br Proceedings of the Royal Society, A, 117, 600, 1928; A, 119, 651, 1928.
bs Annalen der Physik, 85, 981, 1928.
bt Japanese Journal of Physics, 5, 83, 1928.
†Annalen der Physik, 17, 145, 1905.
* Physical Review, 25, 306, 1925.
bu Verhandlungen der Deutschen Physikalische Gesellschaft, 15, 613, 1913.
bv Nature, 121, 580, 1928; Naturwissenschaften, 16, 245, 1928.
bw N. Bohr, Nature, 121, 580, 1928.
bx The following considerations apply equally to any of the three space co-ordinates of the electron, therefore only one is treated explicitly.
by In this connection one should particularly remember that the human language permits the construction of sentences which do not involve any consequences and which therefore have no content at all—in spite of the fact that these sentences produce some kind of picture in our imagination; e.g., the statement that besides our world there exists another world, with which any connection is impossible in principle, does not lead to any experimental consequence, but does produce a kind of picture in the mind. Obviously such a statement can neither be proved nor disproved. One should be especially careful in using the words “reality,” “actually,” etc., since these words very often lead to statements of the type just mentioned.
bz Kennard, Zeitschrift für Physik, 44, 326, 1927.
ca Loc. cit.
cb N. Bohr, loc. cit.
cc N. Bohr, loc. cit.
cd Ibid.
ce The translators believe that the literal rendering of the German phrase (“pure case”) does not at all convey the concept involved.
cf Kennard, loc, cit.; C. G. Darwin, Proceedings of the Royal Society, A, 117, 258, 1927.
cg P. Ehrenfest, Zeitschrift f ür Physik, 45, 455, 1927.
ch For a single photon the configuration space has only three dimensions; the Schrödinger equation of a photon can thus be regarded as formally identical with the Maxwell equations.
ci Loc. cit.
cj H. Weyl, Zeitschrift für Physik, 46, 1, 1927.
ck Nature, 121, 580, 1928.
cl Ibid.
cm It need scarcely be remarked that the term “observation” as here used does not refer to the observation of lines on photographic plates, etc., but rather to the observation of “the electrons in a single atom,” etc. Cf. p. 1.
cn N. Bohr, loc. cit.
co Kennard, Zeitschrift für Physik, 44, 326, 1927.
cp M. Born, Zeitschrift für Physik, 38, 803, 1926.
cq A. Einstein, Berliner Berichte, p. 334, 1926; A. Rupp, ibid., p. 341, 1926.
cr Naturwissenschaften, 11, 873, 1923.
cs Nature, 121, 501; 122, 12, 1928.
ct Proceedings of the Royal Society, A, 114, 243, 710, 1927.
cu Ibid., 117, 610, 1928.
cv Dirac (loc. cit) uses the original Schrödinger form in place of the Hamiltonian function (73). With the use of (73) the calculation is somewhat simpler, since the quadratic terms in φi drop out of the interaction energy. The results are the same as those of Dirac.
cw G. Breit, Journal of the Optical Society of America, 14, 324, 1927.
cx J. W. Gibbs, Elementary Principles in Statistical Mechanics, pp. 70–72, 1902.
cy Zeitschrift für Physik, 26, 178, 1924.
cz P. A. M. Dirac, Proceedings of the Royal Society, A, 117, 610, 1928.
da Zeitschrift für Physik, 53, 157, 1929.
db Reprinted courtesy of the American Mathematical Society.
dc I am aware this formulation is not entirely unambiguous.
dd For guidance in the treatment of (7) I owe thanks to Hermann Weyl.
de For unproved propositions in what follows, see L. Schlesinger’s Differential Equations (Collection Schubert, No. 13, Göschen, 1900, especially chapters 3 and 5).
df Cf. Schlesinger. The theory is due to H. Poincaré eA and J. Horn.
dg If (15) is satisfied, at least one of the two paths of integration described in the text cannot be used, as it yields a vanishing result.
dh L. de Broglie, Ann. de Physique (10) 3, p. 22, 1925. (Thèses, Paris, 1924.)
di Physik. Ztschr. 27, p. 95, 1926.
dj This procedure will not be pursued further in the present paper. It was only intended to give a provisional, quick survey of the external connection between the wave equation and the Hamilton-Jacobi equation. ψ is not actually the action function of a definite motion in the relation stated in (2) of Part I. On the other hand the connection between the wave equation and the variation problem is of course very real; the integrand of the stationary integral is the Lagrange function for the wave process.
dk Cf. e.g. E. T. Whittaker’s Anal. Dynamics, chap. xi.
dl Felix Klein has since 1891 repeatedly developed the theory of Jacobi from quasi-optical considerations in non-Euclidean higher space in his lectures on mechanics. Cf. F. Klein, Jahresber. d. Deutsch. Math. Ver. 1, 1891, and Zeits. f. Math. u. Phys. 46, 1901 (Ges.-Abh. ii. pp. 601 and 603). In the second note, Klein remarks reproachfully that his discourse at Halle ten years previously, in which he had discussed this correspondence and emphasized the great significance of Hamilton’s optical works, had “not obtained the general attention, which he had expected”. For this allusion to F. Klein, I am indebted to a friendly communication from Prof. Sommerfeld. See also Atombau, 4th ed., p. 803.
dm See especially A. Einstein, Verh. d. D. Physik. Ges. 19, pp. 77, 82, 1917. The framing of the quantum conditions here is the most akin, out of all the older attempts, to the present one. De Broglie has returned to it.
dn Cf. for the optical case, A. Sommerfeld and Iris Runge, Ann. d. Phys. 35, p. 290, 1911. There (in the working out of an oral remark of P. Debye), it is shown, how the equation of first order and second degree for the phase (“Hamiltonian equation”) may be accurately derived from the equation of the second order and first degree for the wave function (“wave equation”), in the limiting case of vanishing wave length.
do Cf. A. Einstein, Berl. Ber. p. 9 et seq., 1925.
dp In Part I. this appeared merely as an approximate equation, derived from a pure speculation.
dq L. de Broglie, Ann. de Physique (10) 3, p. 22, 1925. (Thèses, Paris, 1924.)
dr P. Debye, Ann. d. Phys. 30, p. 755, 1909.
ds M. v. Laue, idem 44, p. 1197 (§ 2), 1914.
dt Loc. cit.
du Cf. especially the papers of Heisenberg, Born, Jordan, and Dirac quoted later, and further N. Bohr, Die Naturwissenschaften, January 1926.
dv The introduction of f (qk) means that not only the “density” but also the “elasticity” varies with the position.
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