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Journal of Mathematical Physics

A geometric setting for the quantum planar n-body system, and a U(n−1) basis for the internal states

Toshihiro Iwai

Published Jun 1, 1988• 10 citations
Publisher

Abstract

In a previous paper [J. Math. Phys. 28, 964 (1987)], the author showed that the internal motion of a molecule, a many-body system in the Born–Oppenheimer approximation, can be well described in terms of the gauge theory or of the connection theory in differential geometry. However, the scope of that paper centers on the planar triatomic molecule in order to put forward the gauge theory in an explicit manner. This paper is a continuation of the previous one and gives the generalization to the planar multiatomic molecule. The internal space of the n-atomic molecule proves to be diffeomorphic to R+×CPn−2, the product of the positive real numbers and the complex projective space. The internal states of the molecule are described as cross sections in complex line bundles over the internal space. Introduction of the complex line bundles is a geometric consequence of the angular momentum conservation law, because cross sections in each complex line bundle are in one-to-one correspondence with eigenstates that have a fixed total angular momentum eigenvalue. The internal Hamiltonian operator is obtained, which acts on the cross sections in the complex line bundle. Further, boson calculus is performed to obtain a complete basis of internal states of the molecule, using the harmonic oscillator annihilation and creation operators. As a result, carrier spaces of unitary irreducible representations of the unitary group U(n−1), which are characterized by two integers, are realized as finite-dimensional subspaces of the space of the square integrable cross sections in the complex line bundle.

DOI
10.1063/1.527925

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In a previous paper [J. Math. Phys. 28, 964 (1987)], the author showed that the internal motion of a molecule, a many-body system in the Born–Oppenheimer approximation, can be well described in terms of the gauge theory or of the connection theory in differential geometry. However, the scope of that paper centers on the planar triatomic molecule in order to put forward the gauge theory in an explicit manner. This paper is a continuation of the previous one and gives the generalization to the planar multiatomic molecule. The internal space of the n-atomic molecule proves to be diffeomorphic to R+×CPn−2, the product of the positive real numbers and the complex projective space. The internal states of the molecule are described as cross sections in complex line bundles over the internal space. Introduction of the complex line bundles is a geometric consequence of the angular momentum conservation law, because cross sections in each complex line bundle are in one-to-one correspondence with eigenstates that have a fixed total angular momentum eigenvalue. The internal Hamiltonian operator is obtained, which acts on the cross sections in the complex line bundle. Further, boson calculus is performed to obtain a complete basis of internal states of the molecule, using the harmonic oscillator annihilation and creation operators. As a result, carrier spaces of unitary irreducible representations of the unitary group U(n−1), which are characterized by two integers, are realized as finite-dimensional subspaces of the space of the square integrable cross sections in the complex line bundle.
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.
Quantum mechanics for internal motions of the three-body system is set up on the basis of the complex vector bundle theory. The three-body system is called a triatomic molecule in the Born–Oppenheimer approximation. The internal states of the molecule are described as cross sections in the complex vector bundle assigned by an eigenvalue of the square of the total angular momentum operator. This bundle is equipped with a linear connection, which is a natural consequence of a geometric interpretation of the so-called Eckart condition. The coupling of the internal motion with the rotation is understood naturally in terms of this connection. The internal Hamiltonian operator is obtained which includes the internal motion–rotation coupling and a centrifugal potential. The complex vector bundle for the triatomic molecule proves to be a trivial bundle, though the geometric setting for the internal motion is independent of whether the bundle is trivial or not.
We study the quantum mechanical harmonic oscillator in two and three dimensions, with particular attention to the solutions as basis states for representing their respective symmetry groups — O(2), O(1,1), O(3), and O(2,1). The goal of this study is to establish a correspondence between Hilbert space descriptions found by solving the Schrodinger equation in polar coordinates, and Fock space descriptions constructed by expressing the symmetry operators in terms of creation/annihilation operators. We obtain wavefunctions characterized by a principal quantum number, the group Casimir eigenvalue, and one group generator whose eigenvalue is m + s , for integer m and real constant parameter s . For the three groups that contain O(2), the solutions split into two inequivalent representations, one associated with s = 0, from which we recover the familiar description of the oscillator as a product of one-dimensional solutions, and the other with s > 0 (in three dimensions, solutions are found for s = 0 and s = 1/2) whose solutions are non-separable in Cartesian coordinates, and are hence overlooked by the standard Fock space approach. The O(1,1) solutions are singlet states, restricted to zero eigenvalue of the symmetry operator, which represents the boost, not angular momentum. For O(2), a single set of creation and annihilation operators forms a ladder representation for the allowed oscillator states for any s, and the degeneracy of energy states is always finite. However, in three dimensions, the integer and half-integer eigenstates are qualitatively different: the former can be expressed as finite dimensional irreducible tensors under O(3) or O(2,1) while the latter exhibit infinite degeneracy. Creation operators that produce the allowed integer states by acting on the non-degenerate ground state are constructed as irreducible tensor products of the fundamental vector representation. However, the half-integer eigenstates are infinite-dimensional, as expected for the non-compact O(2,1), and no Fock space description is found for this case. For all s ≠ 0 solutions, the SU(N) symmetry of the harmonic oscillator Hamiltonian recently discovered by Bars is spontaneously broken by the ground state. The connection of this symmetry breaking to the non-separability into one-dimensional Cartesian solutions is demonstrated. We conclude with a discussion of the O(2,1) solutions, and the problem of timelike ghost excitations.
Quantum systems whose states are tightly distributed among several invariant subspaces (variable spin systems) can be described in terms of distributions in a four-dimensional phase-space T∗S2 in the limit of large average angular momentum. The cotangent bundle T∗S2 is also the classical manifold for systems with E(3) symmetry group with appropriately fixed Casimir operators. This allows us to employ the asymptotic form of the star-product proper for variable (integer) spin systems to develop a deformation quantization scheme for a particle moving on the two-dimensional sphere, whose observables are elements of e(3) algebra and the corresponding phase-space is T∗S2. We show that the standard commutation relations of the e(3) algebra are recovered from the corresponding classical Poisson brackets and the explicit expressions for the eigenvalues and eigenfunctions of some quantized classical observables (such as the angular momentum operators and their squares) are obtained.
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.