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曲线上自旋为零粒子的哈密顿量、几何动量与力算符：物理方法

Hamiltonian, Geometric Momentum and Force Operators for a Spin Zero Particle on a Curve: Physical Approach

M. S. Shikakhwa, N. Chair

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AI总结：

本文采用物理挤压方法构建三维空间曲线上自旋零粒子的厄米哈密顿量、几何动量与力算符，并以圆柱螺旋线为例验证。

摘要：

本文通过在假定存在强法向力的情况下，将张成垂直于曲线的管道的坐标挤压到曲线上，构建了被限制在嵌入三维空间的曲线中的自旋为零粒子的哈密顿量。我们沿用此前用于将粒子限制在曲面上的新方法，即从三维动量算符的一个表达式出发，在该表达式中，沿曲线方向和垂直于曲线方向的分量分别是厄米的。在这种情况下，用以法向方向上的动量算符表示的动能算符随后成为一个厄米算符。当舍弃该算符并将围绕曲线的管道厚度设为零时，人们会自动得到包含如预期的几何势项的厄米曲线哈密顿量。研究表明，该势的起源在于原始三维动量算符的排序或对称化，以使其成为厄米算符。本文还构建了粒子被限制在曲线上时的厄米动量算符，并发现它类似于被限制在曲面上的粒子的所谓几何动量，因为它具有一个沿曲线法向且与曲率成正比的项。本文还推导了曲线上粒子的力算符，并表明对于具有恒定曲率和挠率的曲线，该力算符会简化为一个——表面上看——垂直于曲线的单分量，它是经典表达式的对称化再加上一个量子项。随后，本文针对被限制在嵌入三维空间的圆柱螺旋线上的粒子这一特定情形，推导了上述所有量。

英文摘要：

The Hamiltonian for a spin zero particle that is confined to a curve embedded in the 3D space is constructed by squeezing the coordinates spanning a tube normal to the curve onto the curve assuming strong normal forces. 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.

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