On Geometric Momentum For Isotropic Oscillators

The introduction of geometric momentum gives an appropriate definition of the momentum of moving particle constrained on a supersurface.If the curvilinear coordinate system consists of a superfamily group and a normal vector above it,the geometric momentum can be a mechanical quantity in the coordinate system.In many cases,the physical quantity that occurs with geometric momentum is geometric potential,which are new geometric effects that are produced when particles are constrained to a surface.These geometric effects have also been confirmed by experiments,giving a very reasonable explanation for many phenomena in condensed matter physics.Apart from the above two physical quantities,Dirac proposed the radial momentum operator in the spherical coordinate system in?The Principles of Quantum Mechanics?and considered that it has real eigenvalues.However,related studies have confirmed that the radial momentum operator is not self-adjoint,it is impossible to measure without complete eigenfunction.However,there are few studies on radial momentum measurement schemes,most of the discussion focuses on the self-adjointness.In this paper,a possible measurement scheme is given,that is,the Dirac momentum operator p_r is defined as the difference between the momentum operator p and the geometric momentum?by using Gaussian normal coordinates.the measurement is performed in different state spaces,this avoids the difficulty that the radial momentum is not self-adjoint and can't be directly measured,thus giving the theoretical measurement of equivalent radial momentum.In this paper,the three-dimensional and four-dimensional isotropic harmonic oscillator models are used to analyze the momentum and geometric momentum probability distributions of the the ground state and first excited states,the equivalent radial momentum distribution is derived.The results show that the distribution trend of the momentum and geometric momentum is the same,and it's the same concept under the classical limit,the difference in quantitative is equivalent radial momentum,this shows the physical meaning of Dirac's introduction of radial momentum.In addition,the four-dimensional isotropic harmonic oscillator model can be used to popularize the conclusions to the N dimensional space,further supporting what Dirac calls pr?real,and r true conjugate momentum?.