How to Construct a Coordinate Representation of a Hamiltonian Operator on a Torus

Abstract

The dynamical system of a point particle constrained on a torus is quantizedà la Dirac with two kinds of coordinate systems respectively; the Cartesian and toric coordinate systems. In the Cartesian coordinate system, it is difficult to express momentum operators in coordinate representation owing to the complication in structure of the commutation relations between canonical variables. In the toric coordinate system, the commutation relations have a simple form and their solutions in coordinate representation are easily obtained with, furthermore, two quantum Hamiltonians turning up. A problem comes out when the coordinate system is transformed, after quantization, from the Cartesian to the toric coordinate system. *

Geometric momentum: The proper momentum for a free particle on a two-dimensional sphere

“…The following is a list of various forms of the GM in the current literature. Because (6)- (8) give the most general GM of p (α,β)i satisfying the fundamental commutators (20 ) (v) The fifth choice is made by many groups based on different theoretical grounds and it should be α = 1 and β = 0 13,17,[19][20][21][22][23]30 .…”

“…Previous works find that various momentum including the usual canonical one and GM as well are all definable 14,19 . For a two-dimensional sphere [13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30] , the well-known canonical momenta p θ = −i (∂ θ + cot θ/2) and p ϕ = −i ∂ ϕ and the momenta ( 6)-( 8) all seem to be permissible. In fact, neither of these momenta is all equally physical, nor are they all completely compatible with Dirac's theory.…”

Coherent states for the quantum mechanics on a torus

“…where ϕ 1 is the azimuthal angle and ϕ 2 polar angle; R and r are the outer and inner radius of the torus, respectively. In such a case the scalar product is given by [11] f |g = 1 4π is of the form…”

Quantum motion on a torus as a submanifold problem in a generalized Dirac’s theory of second-class constraints

“…Notice that the classification theorem for compact surfaces states that [16], every compact orientable surface is homeomorphic either to a sphere or to a connected sum of tori, implying that if there is any difficulty associated with quantum mechanics for a particle constrained on a sphere or a torus, enormous theoretical problems would arise from dealing with an arbitrary two-dimensional curved surface in quantum mechanics. It forms one of the reasons that the sphere [6] and the torus [17][18][19][20] are used to test various theories. The main purpose of the present study is to take the torus to show that Dirac formalism is complementary to the Schrödinger one.…”