Quantisation of the Classical Model of Dirac Electron in the Schrödinger Picture

M. F. Araujo de Resende, Thales Machado F

In this paper, we present a detailed review/analysis of the Dirac quantisation of Hamiltonian systems with constraints. To this end, we use, as a guide, the physical example provided by the dynamics of a solid ball rolling, without slipping, down an inclined plane under the action of gravity. After all, however simple this physical system may be, it provides a rich framework for this analysis since, in addition to allowing us to discuss scenarios involving holonomic and non-holonomic constraints, it is also a gauge system. Indeed, due to this latter fact, we have carefully detailed how the transition, from classical to quantum mechanics, must be guided by the Dirac-Bergmann algorithm and by the consequent replacement of Dirac brackets with commutators. As a central result, we demonstrate that the restriction of the Hamiltonian operator of this system with constraints to the physical Hilbert subspace (which is identified with the quantisation of these constraints) reproduces the same Schrödinger equation that can be originally obtained in intrinsic terms, a fact that only reinforces the consistency of the Dirac quantisation method.

R. Gerritsma, Gerhard Kirchmair, F. Zähringer +3 more

Nima Moshayedi

Dževad Belkić

In this chapter, the authors show that various pictures of quantum mechanics are interrelated by the unitary transformations. Calling upon the correspondence principle, the quantum-mechanical scattering problem would be quite analogous to the mentioned classical model. The two analysed pictures of quantum mechanics are not the only possible ones. Quite the opposite, all the unitary transformations carried out on the state vectors and the observables of the Schrödinger picture, in fact, define a new formalism of quantum mechanics. There are infinitely many such formalisms but they are all rigorously equivalent. Among all of the possible, alternative pictures of quantum mechanics, of particular importance for scattering theory is the so-called interaction picture, which was first introduced by Dirac. Hence, the alternative name: the Dirac picture.

Heinz Otto Cordes

Hisaiti Yamasaki

Progress of Theoretical Physics Vol. 34 No. 3 (1965) pp. 462–472 Explicit Representation of the Transformation Property of Physical Quantity of Dirac Electron with the Hyperbolic Metric of Velocity Hisaiti Yamasaki Progress of Theoretical Physics Vol. 36 No. 1 (1966) pp. 72–85 A New Derivation of Classical Models of the Spinning Electron from the WKB Solutions to the Pauli and Dirac Equations Hisaiti Yamasaki Progress of Theoretical Physics Vol. 39 No. 2 (1968) pp. 372–393 The Frenkel-Kramers Model and the Zitterbewegung in the Extended Heisenberg Picture of the Dirac Electron Hisaiti Yamasaki