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arXiv:1209.2209

Momentum spectrometry of spherical harmonics and a probe of geometric embedding effect

Q. H. Liu

Published Sep 11, 2012• 1 citation
arXivHTMLPDF

Abstract

As a submanifold is embedded into higher dimensional flat space, quantum mechanics gives various embedding quantities, e.g., the geometric momentum and geometric potential, etc. For a particle moving on a two-dimensional sphere or a free rotation of a spherical top, the projections of the geometric momentum p and the angular momentum L onto a certain Cartesian axis form a complete set of commuting observables as [p_{i},L_{i}]=0 (i=1,2,3). We have therefore a (p_{i},L_{i}) representation for the states on the two-dimensional spherical surface. The geometric momentum distribution of the ground states for a freely rotating rigid rotor seems within the resolution power of present momentum spectrometer and can be measured to probe the embedding effect.

DOI
10.48550/arxiv.1209.2209

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As a submanifold is embedded into higher dimensional flat space, quantum mechanics gives various embedding quantities, e.g., the geometric momentum and geometric potential, etc. For a particle moving on a two-dimensional sphere or a free rotation of a spherical top, the projections of the geometric momentum p and the angular momentum L onto a certain Cartesian axis form a complete set of commuting observables as [p_{i},L_{i}]=0 (i=1,2,3). We have therefore a (p_{i},L_{i}) representation for the states on the two-dimensional spherical surface. The geometric momentum distribution of the ground states for a freely rotating rigid rotor seems within the resolution power of present momentum spectrometer and can be measured to probe the embedding effect.
As a manifold is embedded into a higher dimensional Euclidean space, quantum mechanics gives various embedding quantities. In the present study, two embedding quantities for a two-dimensional curved surface in the three-dimensional flat space, the geometric momentum and the geometric potential, are derived in a unified manner. For a particle moving on a two-dimensional sphere or a free rotation of a spherical top, the projections of both the geometric momentum p\mathbf{p}p and the angular momentum L\mathbf{L}L onto a certain Cartesian axis form a complete set of commuting observables as [pi,Li]=0[p_{i},L_{i}]=0[pi​,Li​]=0 (i=1,2,3i=1,2,3i=1,2,3), thus constituting a dynamical (pi,Li)(p_{i},L_{i})(pi​,Li​) representation for the states on the two-dimensional spherical surface. The geometric momentum distribution of the states represented by spherical harmonics is successfully obtained, and this distribution for a homonuclear diatomic molecule seems within the resolution power of present momentum spectrometers and can be measured to probe the embedding effect.
In Dirac's canonical quantization theory on systems with second-class constraints, the commutators between the position, momentum, and Hamiltonian form a set of algebraic relations that are fundamental in construction of both the quantum momentum and the Hamiltonian. For a free particle on a two-dimensional sphere or a spherical top, results show that the well-known canonical momentum ${p}_{\ensuremath{\theta}}$ breaks one of the relations, while three components of the momentum expressed in the three-dimensional Cartesian system of axes as pi{p}_{i}pi​ (i=1,2,3)(i=1,2,3)(i=1,2,3) are satisfactory all around. This momentum is not only geometrically invariant but also self-adjoint, and we call it geometric momentum. The nontrivial commutators between pi{p}_{i}pi​ generate three components of the orbital angular momentum; thus the geometric momentum is fundamental to the angular one. We note that there are five different forms of the geometric momentum proposed in the current literature, but only one of them turns out to be meaningful.
The geometric momentum was originally introduced for defining the momentum of particle constrained on a hypersurface, but it is in fact not necessarily defined on a curved surface only. If a coordinate system contains a family of hypersurfaces and a normal vector on hypersurface used as a unit vector, the geometric momentum can be defined on the family of hypersurfaces and can be used to determine a complete set of commuting observables. For instance, the spherical polar coordinate system is such a kind of coordinate, in which for a given value of radial position, the spherical surface is a hypersurface. It is well-known that any vector in the space can be decomposed into components along each axis of the spherical polar coordinates, but the geometric momentum has a different decomposition, for it requires a projection of the momentum on the hypersurface, and then needs to decompose the projection into the Cartesian coordinates of the original space where the whole spherical coordinates are defined. Explicitly, with a relation-iħ▽= p Σ + p n where-iħ▽ can be usual momentum operator in Cartesian coordinates, and p Σ is the momentum component on the hypersurface which turns out to be the geometric momentum, and p n is the momentum component along the radial direction, we have a nontrivial definition of radial momentum as p n ≡-iħ▽- p Σ. Once-iħ▽ and p Σ are measurable, p n is then indirectly measurable. The three-dimensional isotropic harmonic oscillator can be described in both the Cartesian and the spherical polar coordinates, whose quantum states thus can be examined in terms of both momentum and geometric momentum distributions. The distributions of the radial momentum are explicitly given for some states. The radial momentum operator that was introduced by Dirac has clear physical significance, in contrast to widely spreading belief that it is not measurable due to its non-self-adjoint.
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.
Developing the analysis of the distribution of the particle's position-momentum dot product, the so-called \textit{posmom} x\mathbf{x}% \cdot \mathbf{p}x\textbf{,} to quantum states on a circular circle on two-dimensional Cartesian coordinates, we give its posmometry (introduced recently by Y. A. Bernard and P. M. W. Gill, Posmom: The Unobserved Observable, J. Phys. Chem. Lett. 1\textbf{(}2010\textbf{)}1254) for eigenstates of the free motion on the circle, i.e., zzz-axis component of the angular momentum. The posmom has two parity symmetries, specifically, invariant under two operations mxm_{x}mx​ and mym_{y}my​ representing mirror symmetry about xxx and yyy axis respectively. The complete eigenfunction set of the posmom is then four-valued and consists of four basic parts each of them is defined within a distinct quadrant of the circle. The results are not only potentially experimentally testable, but also reflect a fact that the embedding of the circle S1S^{1}S1 in two-dimensional flat space R2R^{2}R2 is physically reasonable.
Abstract This chapter treats rotations in three dimensions and the angular momentum quantum mechanically. The angular momentum is a 3-vector that plays the same role in the kinetic energy of rotating particles as the linear momentum 3-vector plays in the kinetic energy for translational motion. The quantum operator for the three components do not commute, i.e. the full vector cannot be known. Each component commutes with the operator for the square-length of the angular momentum vector, and therefore one opts for calculating the square-length and one of the components, z, referred to as the projection. The angular momentum is quantized. The (azimuthal) quantum number ℓ = 0,1,2,3… quantifies the square-length as ℓ(ℓ+1), and the projection is quantified by the (magnbetic) quantum number mℓ = -ℓ…ℓ in integer steps. The eigenfunctions are called spherical harmonics. A ‘rigid rotor’ model is set up to treat the rotational spectrum of a diatomic molecule.