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arXiv:2401.13664 · The European Physical Journal Plus

Hamiltonian, geometric momentum and force operators for a spin zero particle on a curve: physical approach

M. S. Shikakhwa, Noureddine Chair

Published Jun 27, 2024• 2 citations
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Abstract

Abstract The Hamiltonian for a spin zero particle that is confined to a 1D curve embedded in the 3D space is constructed. Confinement is achieved by starting with the particle living in a small tube surrounding the curve, and assuming an infinitely strong normal force that squeezes the thickness of the tube to zero, eventually pinning the particle to the curve. We follow the new approach that we applied to confine a particle to a surface, in that we start with an expression for the 3D momentum operators whose components along and normal to the curve directions are separately Hermitian. The kinetic energy operator expressed in terms of the momentum operator in the normal direction is then a Hermitian operator in this case. When this operator is dropped and the thickness of the tube surrounding the curve is set to zero, one automatically gets the Hermitian curve Hamiltonian that contains the geometric potential term as expected. It is demonstrated that the origin of this potential lies in the ordering or symmetrization of the original 3D momentum operators in order to render them Hermitian. The Hermitian momentum operator for the particle as it is confined to the curve is also constructed and is seen to be similar to what is known as the geometric momentum of a particle confined to a surface in that it has a term proportional to the curvature that is along the normal to the curve. The force operator of the particle on the curve is also derived, and is shown to reduce, for a curve with a constant curvature and torsion, to a -apparently- single component normal to the curve that is a symmetrization of the classical expression plus a quantum term. All the above quantities are then derived for the specific case of a particle confined to a cylindrical helix embedded in 3D space.

DOI
10.1140/epjp/s13360-024-05342-5

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Abstract The Hamiltonian for a spin zero particle that is confined to a 1D curve embedded in the 3D space is constructed. Confinement is achieved by starting with the particle living in a small tube surrounding the curve, and assuming an infinitely strong normal force that squeezes the thickness of the tube to zero, eventually pinning the particle to the curve. We follow the new approach that we applied to confine a particle to a surface, in that we start with an expression for the 3D momentum operators whose components along and normal to the curve directions are separately Hermitian. The kinetic energy operator expressed in terms of the momentum operator in the normal direction is then a Hermitian operator in this case. When this operator is dropped and the thickness of the tube surrounding the curve is set to zero, one automatically gets the Hermitian curve Hamiltonian that contains the geometric potential term as expected. It is demonstrated that the origin of this potential lies in the ordering or symmetrization of the original 3D momentum operators in order to render them Hermitian. The Hermitian momentum operator for the particle as it is confined to the curve is also constructed and is seen to be similar to what is known as the geometric momentum of a particle confined to a surface in that it has a term proportional to the curvature that is along the normal to the curve. The force operator of the particle on the curve is also derived, and is shown to reduce, for a curve with a constant curvature and torsion, to a -apparently- single component normal to the curve that is a symmetrization of the classical expression plus a quantum term. All the above quantities are then derived for the specific case of a particle confined to a cylindrical helix embedded in 3D space.
Investigating a particle, a charged particle in an electromagnetic field and a particle with spin-orbit interaction confined to a space curve in square and circular cases, respectively, we first find that their effective Hamiltonian can be defined in terms of the ground states of their confined dimensions. And we obtain interesting results: the torsion-induced geometric potential appears in the square case, while in the circular case it vanishes; the geometrically induced gauge potential and Zeeman coupling appear in the circular case; and the geometric gauge potential, curvature and torsion simultaneously emerge in the spin-orbit interaction in the circular case. In the example, a helix nanowire, the previous results are all demonstrated. These results provide a way to generate an artificial gauge field and to manipulate spin transport by designing the geometry of curved nanodevice.
Abstract The Hermitian surface momentum operator for a particle confined to a 2D curved surface spanned by orthogonal coordinates and embedded in 3D space is expressed as a symmetric expression in derivatives with respect to the surface coordinates and so is manifestly along the surface. This is an alternative form to the one reported in the literature and usually named geometric momentum, which has a term proportional to the mean curvature along the direction normal to the surface, and so “apparently” not along the surface. The symmetric form of the momentum is the sum of two symmetric Hermitian operators along the two orthogonal directions defined by the surface coordinates. The centripetal force operator for a particle on the surface of a cylinder and a sphere is calculated by taking the time derivative of the momentum and is seen to be a symmetrization of the well-known classical expressions.
Abstract We construct the Hermitian Schrödinger Hamiltonian of spin-less particles and the gauge-covariant Pauli Hamiltonian of spin one-half particles in a magnetic field, which are confined to cylindrical and spherical surfaces. The approach does not require the use of involved differential-geometrical methods and is intuitive and physical, relying on the general requirements of Hermicity and gauge-covariance. The surfaces are embedded in the full three-dimensional space and confinement to the surfaces is achieved by strong radial potentials. We identify the Hermitian and gauge-covariant (in the presence of a magnetic field) physical radial momentum in each case and set it to zero upon confinement to the surfaces. The resulting surface Hamiltonians are seen to be automatically Hermitian and gauge-covariant. The well-known geometrical kinetic energy also emerges naturally.
When a two-dimensional curved surface is conceived as a limiting case of a curved shell of equal thickness d, where the limit d\rightarrow0 is then taken, the well-known geometric potential is induced by the kinetic energy operator, in fact by the second order partial derivatives. Applying this confining procedure to the momentum operator, in fact to the first order partial derivatives, we find the so-called geometric momentum instead. This momentum is compatible with the Dirac's canonical quantization theory on system with second-class constraints. The distribution amplitudes of the geometric momentum on the spherical harmonics are analytically determined, and they are experimentally testable for rotational states of spherical molecules such as C_{60}.
In the spirit of the thin-layer quantization approach, we give the formula of the geometric influences of a particle confined to a curved surface embedded in three-dimensional Euclidean space. The geometric contributions can result from the reduced commutation relation between the acted function depending on normal variable and the normal derivative. According to the formula, we obtain the geometric potential, geometric momentum, geometric orbital angular momentum, geometric linear Rashba and cubic Dresselhaus spin-orbit couplings. As an example, a truncated cone surface is considered. We find that the geometric orbital angular momentum can provide an azimuthal polarization for spin, and the sign of the geometric Dresselhaus spin-orbit coupling can be flipped through the inclination angle of generatrix.
Investigating the geometric effects resulting from the detailed behaviors of the confining potential, we consider square and circular confinements to constrain a particle to a space curve. We find a torsion-induced geometric potential and a curvature-induced geometric momentum just in the square case, while a geometric gauge potential solely in the circular case. In the presence of electromagnetic field, a geometrically induced magnetic moment couples with magnetic field as an induced Zeeman coupling only for the circular confinement also. As spin-orbit interaction is considered, we find some additional terms for the spin-orbit coupling, which are induced not only by torsion, but also curvature. Moreover, in the circular case, the spin also couples with an intrinsic angular momentum, which describes the azimuthal motions mapped on the space curve. As an important conclusion for the thin-layer quantization approach, some substantial geometric effects result from the confinement boundaries. Finally, these results are proved on a helical wire.