On The Geometric Momentum For A Particle In A Central Potential

Geometric momentum is a quantum physical quantity that has been proposed in recent years.The original definition is the momentum description in the(N+1)-dimensional space when a particle is constrained in an N-dimensional hypersurface,the momentum of the particle is a geometric invariant,which depends only on the bending property of the surface,and is independent of the parameter transformation of the surface.It is considered to give a proper description of the momentum of the moving particles on the surface.However,from the introduction of geometric momentum and its application as an extended canonical quantization,it is not enough to obtain the physical essence of geometric momentum.On the one hand,the paper takes geometric momentum as a component of three-dimensional momentum and explores the physical meaning of geometric momentum.On the other hand,based on the physical measurability of geometric momentum,radial momentum is used as a measurement scheme for non-self-adjoint operators.Finally,based on geometric momentum as an operator with eigenfunction and eigenvalue in three-dimensional space,the relationship between extended geometric momentum and angular momentum algebra of magnetic monopole-charge system is studied.In view of the above research ideas,the research methods and innovations of the paper have the following aspects:First,taking the particle motion in the central potential field as a special case,the relationship between the three-dimensional momentum component and the geometric momentum is studied.On the one hand,the eigenfunctions and eigenvalues of spherical geometric momentum are studied.On the other hand,the geometric momentum in the representation of the radial position determination value is studied to achieve the generalization of the geometric momentum from the true constraint system.Secondly,the generalized geometric momentum has the same expression as the geometric momentum bound on the spherical surface with the same radial position,which solves the theoretical problem of the physical and mathematical significance of the Dirac radial momentum operator.Geometric momentum itself also has a distinct physical meaning.Third,study the algebraic relationship between the angular momentum of the magnetic monopole-charge system and the geometric momentum of the determined radius sphere.Furthermore,the geometrical properties of geometric momentum as the infinitesimal parallel moving generator are studied,and the magnetic charge of magnetic monopole quantization,magnetic flux quantization and A-B phase shift are studied.Based on the above research methods and innovations,through detailed research and discussion,this paper has mainly achieved the following conclusions:First,it solves the radial momentum operator of the spherical coordinate system proposed by Dirac,and discusses that the operator has corresponding experimental measurable problems.This problem has been severely criticized and questioned by the mathematics and physics since its introduction.It is considered that the operator is nonself-adjointness,and it is impossible to have a complete set of eigenfunctions,so it cannot be measured.It is noted that the operator can have the expected value and its uncertainty.The paper proposes the self-adjoint operator decomposition method of the radial momentum operator,that is,the radial momentum operator defines the difference between the dimensional three-dimensional momentum operator and the geometric momentum operator.The space of these two different self-adjoint operators is different.It is found that the measurement of radial momentum is actually the difference between two different measurement results,so it is a measurable quantity in quantum mechanics.In order to clearly explain the rationality of this scheme,especially the ground state of hydrogen atom is taken as an example to give the distribution of possible values of radial momentum.Second,note that in the flat space,the momentum is the generator of the parallel movement of the quantum state.If the generalized geometric momentum has meaning,it should promote the non-trivial physical effect when the quantum state moves on the spherical surface.As a typical example,we apply geometric momentum to a magnetic monopole-charge system.For this system,we first define the proper orbital angular momentum-the magnetic monopole angular momentum,and then introduce the magnetic monopole geometric momentum.The six operators can form a closed so(3,1)algebra.Then,let the quantum state of the charge rotate for one week under the push of the magnetic monopole geometric momentum,and find that the quantum state exhibits two changes: the first is the well-known rotation;the other is the integrable phase factor due to the magnetic monopole.At the same time,we also found a new way of magnetic charge quantization.Thirdly,on the basis of studying the hydrogen atom in three-dimensional space,the measurement problem of n-dimensional radial momentum is generalized.The research shows that the probability of radial motion of electron low-moment region in the ground state of hydrogen atom is larger than that of three-dimensional.Fourthly,based on the extended regular quantization,the Ellen Fest theorem of moving particles on hypersurfaces is studied.It is found that the free moving particles moving on the hypersurface are driven by another curvature in addition to the centripetal force.