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Journal of Physics Conference Series

Quantum self-controlling free-falling cats

Chryssomalis Chryssomalakos, H. Hernández-Coronado, Eduardo Serrano-Ensástiga

Published Jun 26, 2015• 0 citations
Publisher

Abstract

In the separation of rotations from internal motions in the n -body problem, there appear some gauge fields which physically represent Coriolis effects. These fields are also present in the "falling cat" problem: at the kinematical level they map changes in the cat's shape to changes in its orientation whereas at the dynamical level they show up as gauge potentials in the Hamiltonian. Classically, the vanishing angular momentum condition allows for the orientation degrees of freedom to decouple from the internal ones and the cat's re-orientation can be accounted for at the kinematical level, partially. In the quantum case the cat's reorientation description requires to be done on dynamical grounds. In this paper we explore the quantum version of the falling cat modelled as a three body problem.

DOI
10.1088/1742-6596/624/1/012007

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In the separation of rotations from internal motions in the n -body problem, there appear some gauge fields which physically represent Coriolis effects. These fields are also present in the "falling cat" problem: at the kinematical level they map changes in the cat's shape to changes in its orientation whereas at the dynamical level they show up as gauge potentials in the Hamiltonian. Classically, the vanishing angular momentum condition allows for the orientation degrees of freedom to decouple from the internal ones and the cat's re-orientation can be accounted for at the kinematical level, partially. In the quantum case the cat's reorientation description requires to be done on dynamical grounds. In this paper we explore the quantum version of the falling cat modelled as a three body problem.
The free translational dynamics of an n-body quantum system has many applications, e.g. in molecular and nuclei systems, where it is common to classify the degrees of freedom of the system in 3 rotational and 3n - 6 shape coordinates. It is known that there exists an interaction between the shape and orientation degrees of freedom. In particular, changes of the shape could induce an orientation change. In this work, it is shown a rotating quantum system which does not deform its shape probability density and with vanishing angular momentum. The orientation change is monitored with a localized orientation wavefunction. We characterize the localized wavefunctions and study their evolution under a rigid rotor-like Hamiltonian, concluding that this kind of wavefunctions may rotate by their own.
The quantum version of the free fall problem is a topic often skipped in undergraduate quantum mechanics courses, because its discussion usually requires wavepackets built on the Airy functions—a difficult computation. Here, on the contrary, we show that the problem can be nicely simplified both for a single particle and for general many-body systems by making use of a gauge transformation that corresponds to a change of reference frame from the laboratory frame to the one comoving with the falling system. Using this approach, the quantum mechanics problem of a particle in an external gravitational potential reduces to a much simpler one where there is no longer any gravitational potential in the Schrödinger equation. It is instructive to see that the same procedure can be used for many-body systems subjected to an external gravitational potential and a two-body interparticle potential that is a function of the distance between the particles. This topic provides a helpful and pedagogical example of a quantum many-body system whose dynamics can be analytically described in simple terms.
A several-particle system is called a molecule in the Born–Oppenheimer approximation. The nonrigidity of molecules involves difficulty in molecular dynamics. Guichardet [A. Guichardet, Ann. Inst. H. Poincaré 40, 329 (1984)] showed recently that the vibration motion cannot in general be separated from the rotation motion, by using the connection theory in differential geometry. The point of his theory is the observation that a center-of-mass system is made into a principal fiber bundle with rotation group as the structure group, and is equipped with a connection by the Eckart condition of rotationless constraint. The base manifold of this bundle is called the internal space. The fact that the connection has nonvanishing curvature gives rise to the nonseparability of vibration from rotation. This is a mathematical meaning of nonrigidity of molecules. As an application of the connection theory due to Guichardet, this paper establishes a gauge theory for nonrigid molecules on the basis of the observation that the vector bundle associated with the principal fiber bundle (the center-of-mass system) provides a setting for quantum mechanics of the ‘‘internal’’ molecular motion. The interest, however, centers on planar triatomic molecules in order to put forward the gauge theory in an explicit manner. The conclusion is this: The internal space of a planar triatomic molecule is diffeomorphic with R3−{0}, and endowed with Dirac’s monopole field which may be interpreted as a Coriolis field induced by the rotation. The angular momentum eigenvalues, which are twice the quantized monopole strengths, assign the complex line bundles over the internal space. The internal states of the molecule are described as the cross sections of the complex line bundle, on which the internal Hamiltonian operator acts in minimally coupling with the monopole field.
The separation of the dynamics of the quantum many-body systems into the overall rotation and internal motions leads to their quantization in rotating frames. Whereas the rotating body frame is unique for a rigid body, there is a great degree of freedom for choosing a body frame for a general rotating and vibrating N-particle system. This, in turn, implies the need to study many-body systems for a class of body frames.The concept of vibrational coordinates and of rotational coordinates is particularly clear for the case of a triatomic molecule: three particles form a single plane, in which the molecule vibrates, and at the same time the molecular plane rotates in the space-fixed frame of reference. The overall molecular rotation is made explicit by means of an orthogonal transformation, which brings the space-fixed into the body-fixed axes. The direction perpendicular to the molecular plane uniquely determines one of the three axes of the frame. Rotation about this direction will change the position of the axes in the molecular plane, in fact the axis system. It is already intuitively clear that this axis transformation will not affect the vibrational portion of the kinetic energy operator. The other parts of the kinetic energy, the rotational part and the rotation-vibration coupling part, will then need to adjust themselves to the new definition of the body frame. In this fashion, we may address the problem of the interaction of rotations and vibrations, which is one of the most important problems in molecular studies.The variation of the rovibrational kinetic energy operator following a change of orientation of the axis system is investigated in the present work from the physical and mathematical points of view. Various types of axis embedding are analyzed in a systematic fashion for triatomic molecules of arbitrary structure, including the bisector, principal axis, and Eckart frames. We also consider the bisector axis embedding for tetratomic molecules and the local axis embedding for pentatomic molecules. Different axis conventions lead sometimes to a complex entanglement of rotational and vibrational motions, which may pose serious obstacles to numerically exact approaches.
The problem of the reason of physical motion needs a review in the framework of quantum theory. The Aristotle's mistake, Galileo-Newton progress, Einstein physical geometry established the fundamental role of the spacetime geometry in the motion of fields and bodies. Quantum theory poses a new question about the motion of the quantum states and its reason in the quantum state space. The standard approach of quantum theory uses so-called method of the classical analogy where the action functional contains in the additive manner three terms: matter (free particles) + free fields + interaction term. Such approach leads to the quantum state space as some space of functions defined on the spacetime.I think if one tries to understand the peculiarity of the self-interacting quantum particles together with its ``field shell" then the classical scheme should be replaced. Then the role of the spacetime should be revised: the space of the unlocated pure quantum degrees of freedom and its geometry will play the fundamental role and the local dynamical spacetime arises as representation of the internal quantum motions (inverse representation).