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International Journal of Geometric Methods in Modern Physics

Distribution of x ⋅ p for quantum states on a circle

Quanhui Liu, L. Qin, X. L. Huang, Diyu Zhang, D. M. Xun

Published Apr 17, 2015• 0 citations
Publisher

Abstract

We first give the proper definition of the particle's position-momentum dot product, the so-called posmomx ⋅ p, to quantum states on a circular circle, in which the momentum turns out to be the geometric one that is recently intensively studied. Second, we carry out the posmom distributions for eigenstates of the free motion on the circle, i.e. [Formula: see text], (m = 0, ±1, ±2, …). The results are not only potentially experimentally testable, but also reflect a fact that the embedding of the circle S1 in two-dimensional flat space R2 is physically reasonable.

DOI
10.1142/s0219887815500784

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We first give the proper definition of the particle's position-momentum dot product, the so-called posmomx ⋅ p, to quantum states on a circular circle, in which the momentum turns out to be the geometric one that is recently intensively studied. Second, we carry out the posmom distributions for eigenstates of the free motion on the circle, i.e. [Formula: see text], (m = 0, ±1, ±2, …). The results are not only potentially experimentally testable, but also reflect a fact that the embedding of the circle S1 in two-dimensional flat space R2 is physically reasonable.
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.
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.
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.
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.