Generalized momentum equation of quantum mechanics

Simon Davis

The momentum operator may be generalized for the correspondence of the limit of quantum commutators of observables with the classical commutators. The substitution of the momentum operator into the nonrelativistic equation for the conservation of energy yields a generalization of the stationary wave equation. Replacing also the energy operator, a generalization of the time-dependent Schrödinger equations, there exists a special class of wavefunctions that satisfies the nonrelativistic wave equation in quantum theory.

E. U. Condon

V Hnizdo

In nonrelativistic quantum mechanics, the total (i.e. orbital plus spin) angular momentum of a charged particle with spin that moves in a Coulomb plus spin–orbit-coupling potential is conserved. In a classical nonrelativistic treatment of this problem, in which the Lagrange equations determine the orbital motion and the Thomas equation yields the rate of change of the spin, the particle's total angular momentum in which the orbital angular momentum is defined in terms of the kinetic momentum is generally not conserved. However, a generalized total angular momentum, in which the orbital part is defined in terms of the canonical momentum, is conserved. This illustrates the fact that the quantum-mechanical operator of momentum corresponds to the canonical momentum of classical mechanics.

Bogdan Mielnik

Mayeul Arminjon

According to Schroedinger's ideas, classical dynamics of point particles should correspond to the " geometrical optics " limit of a linear wave equation, in the same way as ray optics is the limit of wave optics. It is shown that, using notions of modern wave theory, the " geometrical optics " analogy leads to the correspondence between a classical Hamiltonian H and a " quantum " wave equation in a natural and general way. In particular, the correspondence is unambiguous also in the case where H contains mixed terms involving momentum and position. In the line of Schroedinger's ideas, it is also attempted to justify the occurrence, in QM, of eigenvalues problems, not merely for energy, but also for momentum. It is shown that the wave functions of pure momentum states can be defined in a physically more satisfying way than by assuming plane waves. In the case of a spatially uniform force field, such momentum states have a singularity and move undeformed according to Newton's second law. The mentioned unambiguous correspondence allows to uniquely extend the Klein-Gordon relativistic wave equation to the case where a constant gravitational field is present. It is argued that Schroedinger's wave mechanics can be extended to the case with a variable gravitational field only if one accepts that the wave equation is a preferred-frame one. From this viewpoint, generally-covariant extensions of the wave equations of QM seem rather formal. Finally, it is conjectured that there is no need for a quantum gravity.

Bogdan Mielnik