An Introduction to the geometric quantization procedure which is the attempt at finding a procedure that maps classical observables to quantum observables.\n\nFrom this idea, we made a more mathematical approach to some very interesting physics.\nWe derived a mapping and while analysing it. It becomes clear that more conditions have to be imposed for this to produce physically correct answers. \nFrom geometric quantization, we produced the canonical quantization and the operator form of the Hamiltonian of the harmonic oscillator.\nAlong the way, we also found some physical consequences of the math.
We expound the generalized covariance condition in quantization. This condition is combined with the other standard conditions to determine the quantum-mechanical operator of a quadratic Hamiltonian
H2, represented by a quadratic function of momenta with a nonsingular coefficient matrix depending on coordinates, and the operator of an arbitrary quadratic quantity
F2 belonging to a system whose Hamiltonian is of the
H2 type. It is shown that the operator of
F2 depends also on
H2 and that H${\mathrm{^}}_{2}$ or F${\mathrm{^}}_{2}$ cannot contain a curvature term. We also present a modified ordering method with certain conditions under which the operator of a quadratic quantity can be found by replacing some linear functions of momenta with their operators. Finally, a path-integration formula is formed and shown to yield the expression of H${\mathrm{^}}_{2}$ that we derive by the canonical quantization method.
Abstract A fundamental problem regarding the Dirac quantization of a free particle on an ( ) curved hypersurface embedded in N flat space is the impossibility to give the same form of the curvature‐induced quantum potential, the geometric potential as commonly called, as that given by the Schrödinger equation method where the particle moves in a region confined by a thin‐layer sandwiching the surface. This problem is resolved by means of a previously proposed scheme that hypothesizes a simultaneous quantization of positions, momenta, and Hamiltonian, among which the operator‐ordering‐free section is identified and is then found sufficient to lead to the expected form of geometric potential.
The Schrodinger variational approach (1926) to quantization of the natural Hamilton mechanics in
2n-dimensional phase space is revised in the modern paradigm of quantum mechanics in application to the system the Hamilton function of which is a positive-definite quadratic form of the n momenta with the coefficients depending on coordinates in the generic configuration space
Vn. The quantum Hamilonian thus obtained has a paradoxical potential term depending on choice of coordinates in
Vn, which was discovered first by B. DeWitt in 1952 in the framework of canonical quantization of the system by a particular ordering of the operators of observables of momenta and coordinates. It is shown that the Schrodinger approach in the standard paradigm of quantum mechanics determines uniquely the ordering selected by DeWitt among a continuum of other possibilties to determine the quantum Hamiltonian. Two particular classes of observables of localization in
Vn are considered in detail. It is shown that, in general, the quantum-mechanical potential does not vanish even in the Euclidean configuration space
Vn except the case when Cartesian coordinates are taken as the observables of localization of the system. It is noted also that, in the quasi-classical approach to the quantization considered by DeWitt in 1957, the quantum Green function (propagator) is also non-unique and depends on the choice of a line in
Vn connecting these points. All three formalisms have the same local asymtotics of the quantum Hamiltonian if the normal Riemannian cordinates are used, at least implicitly, to localize the system in
Vn. Keywords: Hamilton function; Quantization; Quantum-Mechanical Potential; Observables of Localization; Quantum Anomaly of (non-relativistic) General Covariance.