Geometric momentum and angular momentum for charge-monopole system

Abstract

For a charge-monopole pair, though the definition of the orbital angular momentum is different from the usual one, and the transverse part of the momentum that includes the vector potential as an additive term turns out to be the so-called geometric momentum that is under intensive study recently. For the charge is constrained on the spherical surface with monopole at the origin, the commutation relations between all components of geometric momentum and the orbital angular momentum satisfy the so(3, 1) algebra. With construction of the geometrically infinitesimal displacement operator based on the geometric momentum, the so(3, 1) algebra implies the Aharonov-Bohm phase shift. The related problems such as charge and flux quantization are also addressed. PACS numbers: 14.80.Hv Magnetic monopoles, 03.65.-w Quantum mechanics.For a particle in the centrally symmetrical potential field, for instance, an electron interacts with proton that is placed at the origin of the coordinates, the orbital angular momentum is a conservative quantity, which is defined by,where the position of the particle is defined by its radius vector r, and P is the canonical momentum. Once splitting the momentum P into a radial P and a transverse part P ⊥ in the following, with r ≡ |r| and e r ≡ r/r, P ≡ e r (e r · P) , and P ⊥ ≡ P − e r (e r · P) ,we immediately find that in the angular momentum only the transverse part P ⊥ remains and its radial part P has zero effect in it for we have r × P = 0 and an equivalent definition of the orbital angular momentum,In quantum mechanics, P ≡ −i ∇. The splitting of P means the gradient operator ∇ for the centrally symmetrical system is also splittable in the similar fashion. In fact, the transverse momentum with a given radius r is the geometric one which is recently under extensive investigations [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15], and its explicit form is given shortly, c.f., (11a)-(11c).In section II, we show how to split the gradient operator ∇ in spherical polar coordinates. In section III, we apply this splitting to the charge-monopole system. In final section IV, a brief conclusion is given. In whole of present paper, only the nonrelativistic motion of the charge is considered and the spin is ignored. I. RADIAL AND TRANSVERSE DECOMPOSITION OF THE GRADIENT OPERATOR IN SPHERICAL POLAR COORDINATESFor a spherically symmetrical potential field, the most useful coordinates are the spherical polar ones, and the transformation relation between the Cartesian coordinates (x, y, z) and the spherical polar coordinates (r, θ, ϕ) is,where θ is the polar angle from the positive z-axis with 0 ≤ θ < π, and ϕ is the azimuthal angle in the xy-plane from the x-axis with 0 ≤ ϕ < 2π. The gradient operator in the Cartesian coordinates, ∇ cart ≡ e x ∂ x + e y ∂ y + e z ∂ z , can be expressed in (r, θ, ϕ) by either of the following two ways, ∇ sp = e r ∂ ∂r + e θ 1 r ∂ ∂θ + e ϕ 1 r sin θ ∂ ∂ϕ , and (5a) ∇ sp = ∂ ∂r e r + 1 r ∂ ∂θ e θ + 1 r sin θ ∂ ∂ϕ e ϕ . (5b) * Electronic address: quanhuiliu@gmail.com