Abstract

As is well known, the eigen-vibrations of a cubical cavity, with edge l and volume V=l3 whose walls are perfect reflectors, have the following properties: The components k i of the wave vector whose magnitudes amount to 2π divided by the wave-length, have the eigenvalues
$$k_t = 2\pi \frac{{s_t }} {{2l}},\,\,\,\,s_1 ,s_2 ,s_3 = 1,2,3,...$$
where the integers s i are to be restricted to positive numbers, since we are dealing with standing waves. The number dN of eigen-vibrations with wave vector components k i lying between k i and k i + dk i is then given by
$$dN = V \cdot 2 \cdot 8 \cdot \frac{1} {{\left( {2\pi } \right)^3 }}\,\,dk_1 \,dk_2 \,dk_3 ,$$
where the factor 2 takes care of the two polarisation directions of the wave, while the factor 8 arises because for standing waves we have to restrict ourselves to the positive octant of the k-space.

Keywords

Cond 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

  1. 2.
    Cf. C.G. Darwin, Proc. Roy. Soc. London (A), 136, 36 (1932).ADSCrossRefGoogle Scholar
  2. 3.
    These relations can be found in a somewhat different form (four-dimension) in P. Jordan and W. Pauli, Z. Phys. 47, 151 (1927) and in the form given here in W. Heisenberg and W. Pauli, ibid., 56, 1 (1927).ADSCrossRefGoogle Scholar
  3. 4.
    Cf. L. Landau and R. Peierls, Z. Phys. 62, 188 (1930).ADSCrossRefGoogle Scholar
  4. 5.
    A discussion on the introduction of the quantities F and F * for avoiding the zero-point energy can be found in L. Rosenfeld and J. Solomon, Journ. de Phys. Series 7, 2, 139 (1931) and also in J. Solomon, Doctoral Thesis, Paris (1931). The C.R. between i F and F* given in these references arc, incorrect, since they are not compatible with the conditions, div \(\vec F = 0\) and div \(\vec F* = 0\).Google Scholar
  5. 6.
    Cf. L. Landau and R. Peierls, Z. Phys. 69, 56 (1931) in particular, Secs. 3 and 4; W. Heisenberg, The Physical Principles of Quantum Theory, loc. cit., Chapter 3, Sec. 2.ADSCrossRefGoogle Scholar
  6. 9.
    L. Landau and R. Peierls, Z. Phys., 62, 188 (1930)ADSCrossRefGoogle Scholar
  7. cf. also J.R. Oppenheimer, Phys. Rev., 38, 725 (1931).ADSCrossRefGoogle Scholar
  8. 2.
    Reviews: L. Rosenfeld, Mém. de l’Inst. Henri Poincaré, 2, 24 (1932)Google Scholar
  9. E. Fermi, Rev. Mod. Phys. 1, 87 (1932).ADSCrossRefGoogle Scholar
  10. Original Papers; G. Mie, Ann. d. Phys. (4) 85, 711 (1928)ADSCrossRefGoogle Scholar
  11. W. Heisenberg and W. Pauli, I, Z. Phys. 56, 1 (1929); II, ibid, 59, 168 (1929). (Remarks on these: L. Rosenfeld, ibid., 58, 540 (1929) and 63, 574 (1930)ADSCrossRefGoogle Scholar
  12. E. Fermi, Lincei Rend. (6), 9, 881 (1929), 12, 431 (1930)Google Scholar
  13. L. Rosenfeld, Ann. d. Phys., 5, 113 (1930).)ADSCrossRefGoogle Scholar
  14. 4.
    P.A.M. Dirac, Proc. Roy. Soc. London, 114, 243, 710 (1927). Dirac employed standing waves and not progressive waves for quantising the radiation. Secondly, at that time his theory of the electron had not appeared and, therefore, he used in the Hamiltonian for each particle the non-relativistic operator \(\left( {1/2m} \right)\sum {\pi _j^2 }\) instead of \(c\left( {\sum {\alpha _j \pi _j + mc\beta } } \right)\). This is approximately correct for small velocities of particles. The fact that the equations (26.23) and (26.24) follow from quantum electrodynamics was shown byADSCrossRefGoogle Scholar
  15. J.R. Oppenheimer, Phys. Rev., 35, 461 (1930) andADSCrossRefGoogle Scholar
  16. E. Fermi, Lincei Rend., 12, 431 (1930).Google Scholar
  17. 5.
    L. Landau and R. Peierls, Z. Phys. 62, 188 (1930).ADSCrossRefGoogle Scholar
  18. 6.
    W. Heisenberg, Ann. d. Phys. (5), 9, 338 (1931). In contrast to Heisenberg’s approach, we do not use here the method of quantisation of matter waves. The interaction between the electrons of the atom can then be arbitrary.ADSCrossRefGoogle Scholar
  19. 1.
    I thank Prof. R. Peierls for these remarks. For the case of the self-energy of an oscillator, cf., also L. Rosenfeld, Z. Phys., 70, 454 (1931).ADSCrossRefGoogle Scholar
  20. 2.
    I. Waller, Z. Phys., 62, 673 (1930).ADSCrossRefGoogle Scholar
  21. 3.
    J.R. Oppenheimer, Phys. Rev., 35, 461 (1930).ADSCrossRefGoogle Scholar
  22. 4.
    Cf. M. Born and G. Rumer, Z. Phys., 69, 141 (1931).ADSCrossRefGoogle Scholar
  23. 5.
    Approximate ansatz for these have been presented in G. Breit, Phys. Rev., 34, 553 (1929); 36, 383 (1930) (magnetic interaction, but without retardation). See H. Bethe and E. Salpeter, Quantum Mechanics of One-and Two-electron problem, loc. cit. andADSMathSciNetCrossRefGoogle Scholar
  24. C. Moller, Z. Phys., 70. 786 (1931) (Collisions with weak interaction, treated as retarded), also Ann. d. Phys., 14, 531 (1932). See further G. Wentzel, Wave Mechanics of Collision and Radiation Processes, loc. cit.ADSCrossRefGoogle Scholar
  25. Cf. also A.D. Fokker, Physica, 12, 145 (1932) and Z. Phys., 58, 386 (1929), where a two-body problem with partly retarded and partly advanced potentials is treated. For this problem, no radiation is emitted classically.Google Scholar
  26. 6.
    L. Rosenfeld, Z. Phys., 65, 589 (1930).ADSCrossRefGoogle Scholar
  27. J. Solomon, Z. Phys., 71, 162 (1931).ADSCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1980

Authors and Affiliations

  • Wolfgang Pauli

There are no affiliations available

Personalised recommendations