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Littlejohn and Reinsch, 1997; Mitchell and Littlejohn, 2000) RE VIS TA DE LA ES CUE LA DE FÍ SI CA • Vol. VI, No. 1 • ISSN 2412-2564 Un sistema cuántico que gira sin momento angular y sin cambios en su forma A rotating quantum system without angular momentum and shape deformations E. SERRANO-ENSÁSTIGA1 Recibido: 24 abril de 2018 / Aceptado: 3 de junio de 2018 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 1Instituto de Ciencias Nucleares, Universidad classify the degrees of freedom of the system in 3 rotational and 3n − 6 Nacional Autónoma de México, Apartado shape coordinates. It is known that there exists an interaction between Postal 70-543, Ciudad de México 04510, the shape and orientation degrees of freedom. In particular, changes of México. , 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 Agradecimientos localized wavefunctions and study their evolution under a rigid rotor-like El autor le agradece a Daniel Flores por la Hamiltonian, concluding that this kind of wavefunctions may rotate by invitación a publicar en la Rev. Esc. Fis. their own. Acknowledgment La dinámica independiente de traslaciones de un sistema cuántico de The author thanks Daniel Flores for the n cuerpos tiene muchas aplicaciones, e.g., en sistemas moleculares y invitation to publish in the Rev. Esc. Fis. nucleares, donde es común clasificar los grados de libertad del sistema journal. en 3 rotacionales y 3n − 6 coordenadas de forma. Es sabido que existe una interacción entre los grados rotacionales y de forma. En particular, cambios en la forma pueden llegar a inducir un cambio en la orientación del sistema. En este trabajo, mostramos un sistema cuántico que gira pero sin deformar su densidad de probabilidades de forma y con momento angular cero. El cambio de la orientación es monitoreada con una función de onda de orientaciones localizada. Caracterizamos estas funciones de onda localizados y estudiamos su evolución bajo un Hamiltoniano tipo rotor rígido, concluyendo que esta clase de funciones de onda pueden rotar, independientes de su evolución de formas. KEYWORDS Geometric Methods, N Body Problem, Quantum Control,Rotational Motion PALABRAS CLAVES Métodos Geométricos, Problema de N Cuerpos, Control Cuántico, Movimiento Rotacional PACS 02.40.-k, 03.64.-w, 33.20.Wr, 36.20.Ng * Esta obra está bajo una licencia Creative Commons Reconocimiento - NoComercial 4.0 Internacional cbn * This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License. cbn 9 A ROTATING QUANTUM SYSTEM WITHOUT ANGULAR MOMENTUM AND SHAPE DEFORMATIONS I | INTRODUCTION A system of n isolating point particles has 3n coordinates separated in three coordinates of the center of mass position, three of the orientation of the system and 3n − 6 to describe the shape of the system. The last coordinates are related to the vibrational modes in small oscillations. The orientation and shape coordinates do not evolve independently in the most of the physical systems producing many phenomena, e.g., the falling cat problem (Kane and Scher, 1969; Montgomery, 1993), the re-orientation of a diver (Frolich, 1979), or the rovibrational excitations of molecules (Mitchell and Littlejohn, 1999; Ovchinnikov et al., 2001). The emergence of geometrical methods in the study of the n−body problem, pioneered by Guichardet (Guichardet, 1984), have provided extensions of the subject, in particular in the interplay between orientation and shape dynamics (Shapere and Wilczek, 1989; Tachibana and Iwai, 1986). The quantum version of the problem has also been benefited by these approaches, leading to new advances in the theory and its potential applications in molecular dynamics (Littlejohn et al., 1998a,9; Littlejohn and Reinsch, 1997; Mitchell and Littlejohn, 2000). Nowadays scientists have found and synthesized machine-like molecules (et al, 2015,1; Kottas et al., 2005), i.e., molecules managing a particular motion with operable degrees of freedom. The molecules are described in a semi-classical way because the considered constituent parts are too big to have a purely quantum behavior. These and incoming systems may present orientation-shape dynamics with new phenomena given by their quantum nature. For instance, the quantum version of the falling cat problem (Chryssomalakos et al., 2015) may represent the dynamics of the constituent atoms of a molecule. The aim of the paper is the presentation of quantum systems which, counter-intuitive to our (classical) experience, rotate with vanishing angular momentum expectation value and invariant (quantum) shape. The examples are made with the three-body system. The outline of this paper is as follows: Sec. II reviews the classic n-body problem, following closely (Littlejohn and Reinsch, 1997). In Sec. III is presented the three-body model and an example of a system which rotates under a cyclic shape deformation and with vanishing total angular momentum. Sec. IV reviews the quantum version of the n-body problem and the approximations used in this work. The three-body quantum system and our counter-intuitive examples are presented in Sec. V. Conclusions are summarized in Sec. VI. II | CLASSICALCASE The first step is to consider a body consisting of n point-like particles with an interacting potential V and no other external force. The Lagrangian of the system is n 1 2 L = ∑ jr˙saj +V(rsa); (1) 2 a=1 where rsa is the a-th particle’s position vector with respect to a fixed inertial frame called the space frame. The absence of external forces suggests the factorization of the center of mass dynamics writing the Lagrangian with relative vectors, e.g., the mass-weighted Jacobi vectors, p N rsa = µa ∑ Tabrsb ; a = 1;:::;n − 1; (2) b=1 where µa are the reduced masses and T is a numerical matrix (see (Littlejohn and Reinsch, 1997) for more details) such that the vectors rsa are invariant under translations. Using the previous coordinate REF-UNAH / Vol. 6 - No. 1 / 10 - 24 A ROTATING QUANTUM SYSTEM WITHOUT ANGULAR MOMENTUM AND SHAPE DEFORMATIONS transformation, the Lagrangian takes the form n 1 ˙ 2 2 L = MjRsj + ∑ jr˙ saj +V(rsa): (3) 2 a=1 The configuration space of the whole system is R3n = R3 × C, where C = R3n−3 is the translation- reduced configuration space. For the rest of the paper, Rs = R˙ s = 0 and C is referred simply as the configuration space. A point in C specifies the shape and orientation of the body. Three degrees of freedom define the orientation system (for instance, the Euler angles) and there are 3n − 6 remaining coordinates qµ which specify the body’s shape, where qµ are independent functions on C invariant under proper rotations, µ µ q (rs;1;:::;rs;n−1) = q (Rrs;1;:::;Rrs;n−1); (4) for all R 2 SO(3). To define the body’s orientation operationally, on the other hand, we require (i) an orthonormal frame called the body frame, fixed to the body for each shape and given by the relations µ ra = ra(q ), a = 1;:::;n − 1, (the quantities referred to the body frame are written without the sub- index s), and (ii) the rotation R 2 SO(3) that maps the body frame to the space frame parametrized, for instance, by Euler angles R = R(qi). Fixing a body frame for each shape is a choice of gauge, that defines as reference orientation the one where the body and space frames coincide. The factorization of (non-collinear) configurations in “shape times orientation” leads to a description of C as a SO(3) principal fiber bundle (Littlejohn and Reinsch, 1997; Nakahara, 1990) and its mathematical construction is as follows. We let p 2 C be some specific configuration, and we consider the set of configurations given by the orbit of p under the action of SO(3) Op = fRpjR 2 SO(3)g. The configurations of Op have the same shape as p, and they form a surface in C. The non-collinear configurations of the system have an orbit of dimension 3, each of which is a copy of SO(3). The orbit of collinear configurations are a copy of the two-sphere S2, and when all the particles coincide in a single point p (an n-body collision), the orbit is just the point p itself; in this case the orbit is zero-dimensional. For systems with number of particles n ≥ 3, the configurations without orbits diffeomorphic to SO(3) is a set of measure zero. The shape space of C is the quotient space S = C=SO(3), the space in which a single point represents a whole orbit, i.e., the equivalence class of configurations of the same shape. If we exclude the configurations without orbit diffeomorphic to SO(3), then the remaining configuration space qualifies properly speaking as a principal fiber bundle, in which the shape space is the base space, the structure group is SO(3), and the fibers are the orbits. The whole configuration space C will be considered as a principal fiber bundle, just taking into account that there are points in the base space where the fiber shrinks and silly situations may occur. The total space C has a surjective projection map p : C ! S mapping each point p of C to its equivalence class in S. In the fiber-bundle picture, a choice of gauge is a locally smooth specification of a unique point in each fiber, i.e., the graph of a continuous map s : S ! C such that p(s(x)) = x for all x 2 S.
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