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On velocity-dependent potentials in quantum mechanics

1970
19 pages
Published in:
  • Nucl.Phys.B 16 (1970) 431-449
  • Published: 1970
DOI:
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Abstract: (Elsevier)
Velocity-dependent potentials are investigated in both the Lagrangian and the Hamiltonian formalism in quantum mechanics. In order to achieve a consistent method, a canonical transformation is introduced for the type of Lagrangian 1 2 q i g ij (q) q j − V(q) where g ij and V are functions of position operators q i only. It is found that the proper Hamiltonian satisfying the canonical equation of motion should be H = p i q i − L − Z where Z is a function of q i and is expressed in terms of g ij ( q ). This formulation has been examined by some examples. The Euler-Lagrange equation that is consistent with the canonical equation of motion is derived and it turns out to be in an apparently dissipated form. However, the physical system could be a non-dissipative one. Finally, the Schroedinger equation is investigated according to the above argument.
  • quantum mechanics
  • potential