Quantum mechanics of a generalised rigid body (original) (raw)
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A covariant approach to the quantisation of a rigid body
This paper concerns the quantisation of a rigid body in the framework of ``covariant quantum mechanics'' on a curved spacetime with absolute time. The basic idea is to consider the multi-configuration space, i.e. the configuration space for nnn particles, as the nnn-fold product of the configuration space for one particle. Then we impose a rigid constraint on the multi-configuration space. The resulting space is then dealt with as a configuration space of a single abstract `particle'. The same idea is applied to all geometric and dynamical structures. We show that the above configuration space fits into the general framework of ``covariant quantum mechanics''. Hence, the methods of this theory can be applied to the rigid body. Accordingly, we find exactly two inequivalent choices of quantum structures for the rigid body. Then, we evaluate the quantum energy and momentum operators and the `rotational part' of their spectra. We provide a new mathematical interp...
A covariant approach to classical and quantum mechanics of a rigid body
1999
This paper concerns the quantisation of a rigid body in the framework of "covariant quantum mechanics" on a curved spacetime with absolute time. We consider the configuration space of n classical particles as the n-fold product of the configuration space of one particle. Then, we impose a rigid constraint and the resulting space is dealt with as a configuration space of a single abstract 'particle'. This classical framework turns out to be suitable for the formulation of covariant quantum mechanics according to this scheme. Thus, we quantise such a 'particle' accordingly. This scheme can model, e.g., the quantum dynamics of extremely cold molecules. We provide a new mathematical interpretation of two-valued wave functions on SO(3) in terms of single-valued sections of a new non-trivial quantum bundle. These results have clear analogies with spin.
Lie groups and quantum mechanics
Journal of Mathematical Analysis and Applications, 2013
Mathematical modeling should present a consistent description of physical phenomena. We illustrate an inconsistency with two Hamiltonians -the standard Hamiltonian and an example found in Goldstein -for the simple harmonic oscillator and its quantisation. Both descriptions are rich in Lie point symmetries and so one can calculate many Jacobi Last Multipliers and therefore Lagrangians. The Last Multiplier provides the route to the resolution of this problem and indicates that the great debate about the quantisation of dissipative systems should never have occurred.
Classical and quantum dynamics for an extended free rigid body
Differential Geometry and its Applications, 2010
In this paper, a free rigid body of dimension three is extended and analysed both in classical and quantum mechanics. The extension is performed by bringing the inverse inertia tensor, which is a positive-definite symmetric matrix for the ordinary rigid body, into an arbitrary real symmetric one. With an arbitrary real symmetric matrix chosen, associated is a Lie-Poisson structure on the Euclidean space of dimension three, through which the classical dynamics for an extended free rigid body is defined, and characterized by two first integrals. In parallel to this, the quantum dynamics is formulated as the problem of simultaneous spectral resolution of the two operators which are viewed as the quantization of the two classical first integrals. Intensive use is made of the unitary representation theory for Lie groups concerned. The explicit spectral resolution is obtained, in particular, when the extended free rigid body is an extended free symmetric top.
Geometric Aspects of the Quantization of a Rigid Body
Differential Equations - Geometry, Symmetries and Integrability, 2009
In this paper we review our results on the quantization of a rigid body. The fact that the configuration space is not simply connected yields two inequivalent quantizations. One of the quantizations allows us to recover classically double-valued wave functions as single valued sections of a non-trivial complex line bundle. This reopens the problem of a physical interpretation of these wave functions.
On the Classical-Quantum Relation of Constants of Motion
Frontiers in Physics, 2018
Groenewold-Van Hove theorem suggest that is not always possible to transform classical observables into quantum observables (a process known as quantization) in a way that, for all Hamiltonians, the constants of motion are preserved. The latter is a strong shortcoming for the ultimate goal of quantization, as one would expect that the notion of "constants of motion" is independent of the chosen physical scheme. It has been recently developed an approach to quantization that instead of mapping every classical observable into a quantum observable, it focuses on mapping the constants of motion themselves. In this article we will discuss the relations between classical and quantum theory under the light of this new form of quantization. In particular, we will examine the mapping of a class of operators that generalizes angular momentum where quantization satisfies the usual desirable properties.
In sections of the literature it is assumed, following early comments by Dirac, that a desirable quantization operation should preserve the Possion bracket (rather than merely agree in the limit h → 0). A celebrated mathematical theorem establishes that this is impossible; hence a consistent quantization is often considered to be unattainable. Here we question whether the premise of exact correspondence is sensible from a physical viewpoint. In particular, we give an exact quantum mechanical version of classical dynamics in which the commutator of any two quantized operators is proportional to the Poisson bracket of the corresponding functions (in a manner which subtly evades the aforementioned theorem). We then relate this novel dynamical system to the standard quantum theory, and identify the bracket structure for the case of symmetric ordering. Our conclusion is that a consistent quantization should not be expected to preserve the Poisson bracket.
On the quantization of mechanical systems
arXiv (Cornell University), 2017
We show what seems to be the key for quantization of classical systems. Given a manifold M , each riemannian metric (nondegenerate, of arbitrary signature) canonically determines a quantization rule or "Correspondence Principle", which assigns to each classical magnitude (function in T M , subject to certain conditions) a differential operator in C ∞ (M). The issue about the order in which the p' and q' are to be taken in quantization loses all meaning, once the general rule has been fixed. Specified the Correspondence Principle, each "classical state" of the system, understood as a vector field on M , determines a wave equation for each magnitude. The Schrödinger equation is a particular example of these wave equations.
Geometry from Dynamics, Classical and Quantum
2015
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