On Relation Between Geometric Momentum and Annihilation Operators on a Two-Dimensional Sphere

Abstract

With a recently introduced geometric momentum that depends on the extrinsic curvature and offers a proper description of momentum on two-dimensional sphere, we show that the annihilation operators whose eigenstates are coherent states on the sphere take the expected form αx + iβp, where α and β are two operators that depend on the angular momentum and x and p are the position and the geometric momentum, respectively. Since the geometric momentum is manifestly a consequence of embedding the two-dimensional sphere in the three-dimensional flat space, the coherent states reflects some aspects beyond the intrinsic geometry of the surfaces. PACS numbers: 03.65.-w Quantum mechanics; 02.40.-k Differential geometry; 92.60.hc Mesosphere; 73.20.Fz Quantum localization on surfaces and interfacesThe coherent states on two-dimensional sphere, generated by the annihilation operators, were discovered around the turn of present century, independently by Hall [1][2][3][4][5][6] in the Bargmann representation, and by in the position representation, respectively. Once each group of them got to know the work of the other, both soon realized that their coherent states are essentially the same [6,9], and the equivalence was also noted by other group [11]. However, it is puzzling that in the annihilation operators they introduced, there is a fundamental quantity that is represented by a non-hermitian operator and has the same dimension of linear momentum, but it does not bear a transparent physical nor geometric meaning. This article points out that the physical and geometric interpretation of the fundamental quantity is easily available, based on the geometric momentum that is recently introduced to offer a proper description of momentum on sphere [12], whose general form for an arbitrary twodimensional curved surface with M denoting the mean curvature, is given by [12][13][14], *

Geometric momentum and angular momentum for charge-monopole system

“…In consequence, we have e r · P ⊥ + P ⊥ · e r = 0. The components in Cartesian coordinates give the standard projections [1][2][3][4][5][6][7][8][9][10],…”

On relationship between canonical momentum and geometric momentum

“…where, Π r , Π θ and Π ϕ are so-called the geometric momenta [9][10][11][12][13][14][15][16][17][18][19][20] for the corresponding surfaces, though the last one M ϕ = 0 is trivial,…”

“…[1][2][3][4][5][6][7][8][9][10][11][12][13] In this paper, we show that canonical momenta P ξ associated with its conjugate canonical positions, or coordinates, ξ, are closely related to mean curvatures of the surface ξ = const. So, the geometric momenta [10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26] that are under extensive studies and applications are closely related to natural decompositions of the momentum operator in gaussian normal, curvilinear in general, coordinates.…”

Distribution of xp in some molecular rotational states

“…where the momenta p i (i = 1, 2, 3) are the so-called geometric momentum p = −i (∇ s 2 + M n) [15][16][17][18][19][20][21] on S 2 where M is the mean curvature −1/r and n is the normal vector, which is proposed for a proper description of momentum in quantum mechanics for a particle constrained on S 2 , and explicitly we have, [15][16][17][18][19][20][21]…”