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arXiv:1410.1010

Posmon spectrosopy of quantum state on a circle

Q. H. Liu

Published Oct 4, 2014• 0 citations
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Abstract

Developing the analysis of the distribution of the particle's position-momentum dot product, the so-called \textit{posmom} $\mathbf{x}% \cdot \mathbf{p}$\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., $z$-axis component of the angular momentum. The posmom has two parity symmetries, specifically, invariant under two operations $m_{x}$ and $m_{y}$ representing mirror symmetry about $x$ and $y$ 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 $S^{1}$ in two-dimensional flat space $R^{2}$ is physically reasonable.

DOI
10.48550/arxiv.1410.1010

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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.
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.
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.
Covariant integral quantisation using coherent states for semi-direct product groups is implemented for the motion of a particle on the circle. In this case, the phase space is the cylinder, which is viewed as a left coset of the Euclidean group E(2). Coherent states issued from fiducial vectors are labeled by points in the cylinder and depend also on extra parameters. We carry out the corresponding quantisations of the basic classical observables, particularly the angular momentum and the 2π-periodic discontinuous angle function. We compute their corresponding lower symbols. The quantum localisation on the circle is examined through the properties of the angle operator yielded by our procedure, its spectrum and lower symbol, its commutator with the quantum angular momentum, and the resulting Heisenberg inequality. Comparison with other approaches to the long-standing question of the quantum angle is discussed.
A careful study of the physical properties of a family of coherent states on the circle, introduced some years ago by de Bièvre and González in [DG 92], is carried out. They were obtained from the Weyl-Heisenberg coherent states in L2(R)L^2(\\R)L2(R) by means of the Weil-Brezin-Zak transformation, they are labeled by the points of the cylinder S1timesRS^1 \\times \\RS1timesR, and they provide a realization of L2(S1)L^2(S^1)L2(S1) by entire functions (similar to the well-known Fock-Bargmann construction). In particular, we compute the expectation values of the position and momentum operators on the circle and we discuss the Heisenberg uncertainty relation.
The eigenstates of a free quantum particle propagating in a plane are derived in the context of phase space quantum mechanics. Two possibilities are analysed. First, the case of a particle with fixed energy and angular momentum is considered. A special choice of coordinates on the four-dimensional phase space, suitable for representing the eigenstates of the particle under consideration, is presented. A phase-space counterpart of the product of operators, known as the Moyal product, is derived in these coordinates. The eigenvalue equations are solved, and their physically acceptable solutions, called Wigner functions, are identified. Second, a particle with fixed components of the Cartesian momentum is considered. A relationship is found between the Wigner function of the particle with fixed Cartesian momentum components and some functions characterising the particle with fixed energy and angular momentum.