How Important is the Proper Treatment of Translational Invariance in the Analysis of Electron Scattering from Nuclei ?

  • K. W. Schmid
Chapter
Part of the NATO ASI Series book series (NSSB, volume 334)

Abstract

We shall consider the nucleus as a non-relativistic, finite quantum many-body system. Obviously its physics should be the same all over the world and even in a space-lab orbiting around it. This is what we call translational and Galilean invariance. It has the consequence that the corresponding Hamiltonian, however complicated it may be, cannot depend on the center of mass coordinate of the constituents and only in a trivial way on the total linear momentum:
$$ \hat H = {{\hat H}_{int}}({{\vec r}_i} - \vec R,{{\hat p}_i} - {1 \over A}\hat P,{\tau _i},{\sigma _i}) + {{{{\hat P}^2}} \over {2MA}} $$
(1)
Here i runs from 1 to the number of nucleons A, M is the nucleon mass, P = ∑ i=1 A P i the operator of the total linear momentum, \( \vec R \equiv \sum {_{i = 1}^A} {{\vec r}_i}/A \) the center of mass coordinate, and τ i , σ i denote the isospin and spin quantum numbers of the individual constituents. The eigenfunctions of (1)
$$ \left| {{\rm{\psi }}\rangle {\rm{ = }}\left| {{{\rm{\psi }}_{int}}} \right.} \right.\rangle \left| {\vec P\rangle } \right. $$
(2)
can thus be factorized into an internal part and a trivial plane wave describing the motion of the system as a whole. Obviously we are only interested in the former and hence we have to solve the Schrödinger equation
$$ {{\hat H}_{int}}\left| {{\rm{\psi }}\rangle {\rm{ = }}{{\rm{E}}_{int}}\left| {{{\rm{\psi }}_{int}}} \right.} \right.\rangle $$
(3)
for the bound (or the corresponding coupled channel equations for the scattering) states of the system. This is precisely what is done for few nucleon problems: one writes the Hamiltonian in Jacobi coordinates and solves the corresponding Schrödinger (or Fadeev) equations. However, nucleons are Fermions and thus the internal wave function has to be antisymmetric. Since the Jacobi coordinates are functions of all the particle coordinates, the antisymmetrisation has to be done explicitly. Because the number of terms needed is equal to the factorial of the number of identical particles involved, in many-body physics such an explicit antisymmetrisation is impossible.

Keywords

Convection 208Pb 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    J. P. Elliot and T. H. R. Skyrme, Proc. Phys. Soc. A (London) 232, 561 (1955)ADSCrossRefGoogle Scholar
  2. [2]
    R. E. Peierls and J. Yoccoz, Proc. Phys. Soc. A (London) 70, 381 (1957)MathSciNetADSMATHCrossRefGoogle Scholar
  3. [3]
    J. Yoccoz, Varenna Lectures 36, 474 (1966)Google Scholar
  4. [4]
    K. W. Schmid and F. Grümmer, Z. Phys. A 336, 5 (1990)ADSGoogle Scholar
  5. [5]
    K. W. Schmid and F. Grümmer, Z. Phys. A 337, 267 (1990)ADSGoogle Scholar
  6. [6]
    K. W. Schmid and P.-G. Reinhard, Nucl.Phys. A 530, 283 (1991)ADSCrossRefGoogle Scholar
  7. [7]
    L. J. Tassie and C. F. Barker, Phys.Rev. 111, 940 (1958)ADSCrossRefGoogle Scholar
  8. [8]
    G. Orlandini and M. Traini, Rep. Prog. Phys. 54, 257 (1991)ADSCrossRefGoogle Scholar
  9. [9]
    C. Ciofidegli Atti, Nucl.Phys. A 463, 127c (1987); G. Salmé and E. Pace, unpublishedGoogle Scholar
  10. [10]
    K. F. von Reden et al., Phys.Rev. C 41, 41 (1990)ADSGoogle Scholar
  11. [11]
    R. Schiavilla et al., Nucl.Phys. A 473, 267 (1987)ADSMATHCrossRefGoogle Scholar
  12. [12]
    J. Carlson and R. Schiavilla, Phys. Rev. Lett. 68, 3682 (1992)ADSCrossRefGoogle Scholar
  13. [13]
    P. Mulders, Phys. Rep. 185, 83 (1990)ADSCrossRefGoogle Scholar
  14. [14]
    K. W. Schmid and G. Schmidt, submitted for publ.Google Scholar
  15. [15]
    K. W. Schmid, L. Egido and G. Schmidt, in preparationGoogle Scholar
  16. [16]
    Z. Papp, Phys. Rev. A 46, 4437 (1992)ADSCrossRefGoogle Scholar
  17. [a]
    Z. Papp, ibid 38, 2457 (1988)ADSGoogle Scholar
  18. [b]
    Z. Papp, Comp. Phys. Comm. 71, 426 and 435 (1992)ADSCrossRefGoogle Scholar
  19. [17]
    K. W. Schmid and G. Schmidt, in preparationGoogle Scholar
  20. [18]
    E. W. Schmid and G. Spitz, Z Phys. A 321, 581 (1985)ADSCrossRefGoogle Scholar
  21. [19]
    see, e.g., G. DoDang et al., Phys. Rev. C 35, 1637 (1987)ADSCrossRefGoogle Scholar
  22. [20]
    H. Eikemeier and H. H. Hackenbroich, Nucl.Phys. A 169, 407 (1971)ADSCrossRefGoogle Scholar
  23. [21]
    D. M. Brink and E. Boeker, Nucl.Phys. 91, 1 (1967)CrossRefGoogle Scholar
  24. [22]
    R. Machleidt, Adv. Nucl. Phys. 19, 189 (1989)CrossRefGoogle Scholar
  25. [23]
    H. Müther and P. U. Sauer, preprint 1992, to be published in Computational nuclear physics. Vol II ,edited by K. Langanke, J. A. Maruhn and S. E. Koonin (Springer, Berlin)Google Scholar

Copyright information

© Springer Science+Business Media New York 1994

Authors and Affiliations

  • K. W. Schmid
    • 1
  1. 1.Institut für Theoretische PhysikUniversität TübingenGermany

Personalised recommendations