Authors: Steven Kenneth Kauffmann
The commutator of the Dirac free-particle's velocity operator with its Hamiltonian operator is nonzero and independent of Planck's constant, which violates the quantum correspondence-principle requirement that commutators of observables must vanish when Planck's constant vanishes, as well as violating the absence of spontaneous acceleration of relativistic free particles. The consequent physically pathological "zitterbewegung" is of course completely absent when the natural relativistic square-root free-particle Hamiltonian operator is used; nevertheless the energy spectrum of that pathology-free natural relativistic square-root free-particle Hamiltonian is exactly matched by the positive-energy sector of the Dirac free-particle Hamiltonian's energy spectrum. Contrariwise, however, Foldy-Wouthuysen unitary transformation of the positive-energy sector of any hydrogen-type Dirac 4 x 4 Hamiltonian to 2 x 2 form reveals a "zitterbewegung"-induced "Darwin-term" smearing of the proton charge density which is completely absent in the straightforward relativistic extension of the corresponding hydrogen-type nonrelativistic Pauli 2 x 2 Hamiltonian. Compensating for an atomic proton's physically absent "electron zitterbewegung"-induced charge smearing would result in a misleadingly contracted impression of its charge radius.
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