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Q. H. Liu

Publications and source records attributed to Q. H. Liu.

49 records · Page 3Linked to original sources
arXiv2013

Can the canonical quantization be accomplished within the intrinsic geometry?

For particles constrained on a curved surface, how to perform quantization within Dirac's canonical quantization scheme is a long-standing problem. On one hand, Dirac stressed that the Cartesian coordinate system has fundamental importance in passing from the classical Hamiltonian to its quantum mechanical form while preserving the classical algebraic structure between positions, momenta and Hamiltonian to the extent possible. On the other, on the curved surface, we have no exact Cartesian coordinate system within intrinsic geometry. These two facts imply that the three-dimensional Euclidean space in which the curved surface is embedded must be invoked otherwise no proper canonical quantization is attainable. Since the minimum surfaces, catenoid and helicoid studied in this paper, have vanishing mean curvature, we explore whether the intrinsic geometry offers a proper framework in which the quantum theory can be established in a self-consistent way. Results show that it does for quantum motions on catenoid and it does not for that on helicoid, but neither is compatible with Schrödinger theory. In contrast, in three-dimensional Euclidean space, the geometric momentum and potential are then in agreement with those given by the Schrödinger theory.

quant-ph↗
arXiv2013

Geometric momentum for a particle constrained on a curved hypersurface

A strengthened canonical quantization scheme for the constrained motion on a curved hypersurface is proposed with introduction of the second category of fundamental commutation relations between Hamiltonian and positions/momenta, whereas those between positions and moments are categorized into the first. As an $N-1$ ($N\geq2$) dimensional hypersurface is embedded in an N dimensional Euclidean space, we obtain the proper momentum that depends on the mean curvature. For the surface is the spherical one, a long-standing problem on the form of the geometric potential is resolved in a lucid and unambiguous manner, which turns out to be identical to that given by the so-called confining potential technique. In addition, a new dynamical group SO(N,1) symmetry for the motion on the sphere is demonstrated.

quant-ph↗
arXiv2013

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

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.

quant-ph↗
arXiv2013

Quantum motion on a torus as a submanifold problem in a generalized Dirac's theory of second-class constraints

A generalization of the Dirac's canonical quantization theory for a system with second-class constraints is proposed as the fundamental commutation relations that are constituted by all commutators between positions, momenta and Hamiltonian so they are simultaneously quantized in a self-consistent manner, rather than by those between merely positions and momenta so the theory either contains redundant freedoms or conflicts with experiments. The application of the generalized theory to quantum motion on a torus leads to two remarkable results: i) The theory formulated purely on the torus, i.e., based on the so-called the purely intrinsic geometry, conflicts with itself. So it provides an explanation why an intrinsic examination of quantum motion on torus within the Schrodinger formalism is improper. ii) An extrinsic examination of the torus as a submanifold in three dimensional flat space turns out to be self-consistent and the resultant momenta and Hamiltonian are satisfactory all around.

math-ph↗
arXiv2012

On relation between geometric momentum and annihilation operators on a two-dimensional sphere

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.

quant-ph↗
arXiv2012

Geometric momentum in the Monge parametrization of two dimensional sphere

A two dimensional surface can be considered as three dimensional shell whose thickness is negligible in comparison with the dimension of the whole system. The quantum mechanics on surface can be first formulated in the bulk and the limit of vanishing thickness is then taken. The gradient operator and the Laplace operator originally defined in bulk converges to the geometric ones on the surface, and the so-called geometric momentum and geometric potential are obtained. On the surface of two dimensional sphere the geometric momentum in the Monge parametrization is explicitly explored. Dirac's theory on second-class constrained motion is resorted to for accounting for the commutator [x_{i},p_{j}]=i \hbar(δ_{ij}-x_{i}x_{j}/r^2) rather than [x_{i},p_{j}]=i\hbarδ_{ij} that does not hold true any more. This geometric momentum is geometric invariant under parameters transformation, and self-adjoint.

math-ph↗
arXiv2011

Geometric Momentum: the Proper Momentum for a Free Particle on a Two-dimensional Sphere

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_{θ} breaks one of the relations, while three components of the momentum expressed in the three-dimensional Cartesian system of axes as p_{i} (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 p_{i} 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.

quant-ph↗
arXiv2011

Geometric Momentum for a Particle on a Curved Surface

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}.

quant-ph↗
arXiv2009

Raising and lowering operators for angular momentum quantum numbers l in spherical harmonics

Two vector operators aimed at shifting angular momentum quantum number l in spherical harmonics |lm>, primarily proposed by Prof. X. L. Ka in 2001, are further studied. For a given magnetic quantum number m, specific states |lm> in spherical harmonics with the lowest angular momentum quantum numbers l are obtained and the state with minimum angular momentum quantum number in whole set of the spherical harmonics is |0,0>. How to use these states to generate whole set of spherical harmonics is illustrated.

quant-ph↗
arXiv2009

Constraint-induced mean curvature dependence of Cartesian momentum operators

The Hermitian Cartesian quantum momentum operator $\mathbf{p}$ for an embedded surface $M$ in $R^{3}$ is proved to be a constant factor $-i\hbar $ times the mean curvature vector field $H\mathbf{n}$ added to the usual differential term. With use of this form of momentum operators, the operator-ordering ambiguity exists in the construction of the correct kinetic energy operator and three different operator-orderings lead to the same result. PACS: 03.65.-w Quantum mechanics, 04.60.Ds Canonical quantization

math-ph↗
arXiv2000

A series representation of the nonlinear equation for axisymmetrical fluid membrane shape

Whatever the fluid lipid vesicle is modeled as the spontaneous-curvature, bilayer-coupling, or the area-difference elasticity, and no matter whether a pulling axial force applied at the vesicle poles or not, a universal shape equation presents when the shape has both axisymmetry and up-down symmetry. This equation is a second order nonlinear ordinary differential equation about the sine $sinψ(r)$ of the angle $ψ(r)$ between the tangent of the contour and the radial axis $r$. However, analytically there is not a generally applicable method to solve it, while numerically the angle $ψ(0)$ can not be obtained unless by tricky extrapolation for $r=0$ is a singular point of the equation. We report an infinite series representation of the equation, in which the known solutions are some special cases, and a new family of shapes related to the membrane microtubule formation, in which $sinψ(0)$ takes values from 0 to $π/2$, is given.

cond-mat.soft↗
arXiv2000

The Classical Limit of Quantum Mechanics and the Fejer Sum of the Fourier Series Expansion of a Classical Quantity

In quantum mechanics, the expectation value of a quantity on a quantum state, provided that the state itself gives in the classical limit a motion of a particle in a definite path, in classical limit goes over to Fourier series form of the classical quantity. In contrast to this widely accepted point of view, a rigorous calculation shows that the expectation value on such a state in classical limit exactly gives the Fejér's arithmetic mean of the partial sums of the Fourier series.

quant-ph↗