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

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


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:
$$ \hat H = {{\hat H}_{int}}({{\vec r}_i} - \vec R,{{\hat p}_i} - {1 \over A}\hat P,{\tau _i},{\sigma _i}) + {{{{\hat P}^2}} \over {2MA}} $$
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)
$$ \left| {{\rm{\psi }}\rangle {\rm{ = }}\left| {{{\rm{\psi }}_{int}}} \right.} \right.\rangle \left| {\vec P\rangle } \right. $$
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
$$ {{\hat H}_{int}}\left| {{\rm{\psi }}\rangle {\rm{ = }}{{\rm{E}}_{int}}\left| {{{\rm{\psi }}_{int}}} \right.} \right.\rangle $$
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.


Form Factor Rest Frame Proper Treatment Translational Invariance Ground State Wave Function 
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Copyright information

© Springer Science+Business Media New York 1994

Authors and Affiliations

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

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