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Molecular Physics

Rotation of a self-bound many-body system

John Lekner

Published Aug 1, 1974• 5 citations
Publisher

Abstract

We discuss the rotational excitations of highly quantum many-body systems (for example, polyatomic molecules or microdroplets of helium). For a general system, the states Fί, where and ί is a rotationally invariant ground or vibrational state, are shown to be eigenfunctions of L 2 and Lz , with eigenvalues L(L+1)ħ 2 and Lħ (for even L). These wavefunctions preserve the translational invariance and the permutation and inversion symmetries of the N-particle state ί. For harmonic pair interactions, the f = 1 wavefunctions are shown to be exact eigenstates of the N-body hamiltonian. For large N, the states Fί(f=1) represent surface oscillations of the type first proposed by Bohr. An inequality for the rotational excitation energy is obtained variationally; it depends on two, three, and four-particle correlations. Other translationally invariant angular momentum eigenfunctions are also discussed.

DOI
10.1080/00268977400102961

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•Jan 1, 1977•2 citations
We discuss the rotational excitations of highly quantum many-body systems (for example, polyatomic molecules or microdroplets of helium). For a general system, the states Fί, where and ί is a rotationally invariant ground or vibrational state, are shown to be eigenfunctions of L 2 and Lz , with eigenvalues L(L+1)ħ 2 and Lħ (for even L). These wavefunctions preserve the translational invariance and the permutation and inversion symmetries of the N-particle state ί. For harmonic pair interactions, the f = 1 wavefunctions are shown to be exact eigenstates of the N-body hamiltonian. For large N, the states Fί(f=1) represent surface oscillations of the type first proposed by Bohr. An inequality for the rotational excitation energy is obtained variationally; it depends on two, three, and four-particle correlations. Other translationally invariant angular momentum eigenfunctions are also discussed.
The quantum mechanical two-body problem is analyzed. Separating the center of mass from the relative motion Hamiltonian and focusing on “central potentials,” the stationary Schrödinger equation for the relative motion in spherical coordinates is split into radial and angular equations. The universal angular equation is identified as the eigenvalue equation of the angular momentum operator, whose proper solutions are the spherical harmonics. For fixed interparticle distance, the two-body system is mapped on a “rigid rotor” Hamiltonian, whose eigenstates coincide with the angular momentum eigenstates. In diatomic molecules, timescale separation between fast vibrations (radial motion) and slow rotations (angular motion) enables one to invoke a rigid rotor approximation for interpreting rotational absorption spectrum in the microwave regime. Deviations from the predictions of the rigid rotor model and their manifestation in experiments are analyzed by explicit solution of the stationary Schrödinger equation for two particles in the presence of vibration–rotation coupling.
This thesis is a collection of theoretical investigations into different aspects of the broad subject of quantum many-body theory. The results are grouped into three main parts, which in turn are divided into separate self-contained sections. Some of the work is presented in the form of published papers and papers that have been submitted for publication. The first section of Part A introduces some of the concepts involved in many-body problems, by developing methods to evaluate expectation values of the form . In the rest of Part A I consider collective excitations of finite quantum systems. The calculations are confined to nuclei because the results can then be compared with the extensive investigations that have been made into collective nuclear modes. In Section AII, wavefunctions are proposed for rotational excitations of even-even nuclei. Both isoscalar and isovector nuclear modes are discussed. In particular, the l2,m> isoscalar states are investigated for both spherical and deformed even-even nuclei, and the simplest isovector wavefunction is shown to give a good description of the giant dipole resonance. In section AIII wavefunctions are proposed for compressional vibrational states of spherical nuclei. Section AIV discusses sum rules for nuclear transitions of a given electric multipolarity. It is found that the 2+ and 1- states investigated in section AII and all but one of the vibrational states discussed in AIII each exhaust a large part of the appropriate sum rule. In Part B I consider the problem of how to describe flow in quantum fluids. In particular, we want to be able to identify the physical motion represented by any given many-body wavefunction. Section BI derives a guantum mechanical velocity field for a many-body system, paying special attention to the need for a quantum continuity equation. It is found that when the wavefunction has the usual time dependence e-iwt , that the quantum velocity formula averages over all oscillatory motion, so that much of the physical nature of the flow field is lost. In section BII a particular wavefunction is proposed to represent the quantum excitation corresponding to any given potential flow field. The results obtained by considering specific examples are very encouraging. In Part C I investigate the properties of surfaces. Section CI presents a theoretical description of the tension, energy and thickness of a classical liquid-vapour interface. In section CII the classical results are extended to describe the surface of a quantum system, namely superfluid helium four. Problems occur for the quantum system if the correlations arising from the zero-point-motion of the phonon modes are included in the ground state wavefunction. Finally, in section CIII discuss generalized virial theorems that give the change in the free energy of a system undergoing an infinitesimal deformation. For example, a particular deformation gives the expression used in CII, for the surface tension of a plane quantum surface.
This thesis is a collection of theoretical investigations into different aspects of the broad subject of quantum many-body theory. The results are grouped into three main parts, which in turn are divided into separate self-contained sections. Some of the work is presented in the form of published papers and papers that have been submitted for publication. The first section of Part A introduces some of the concepts involved in many-body problems, by developing methods to evaluate expectation values of the form . In the rest of Part A I consider collective excitations of finite quantum systems. The calculations are confined to nuclei because the results can then be compared with the extensive investigations that have been made into collective nuclear modes. In Section AII, wavefunctions are proposed for rotational excitations of even-even nuclei. Both isoscalar and isovector nuclear modes are discussed. In particular, the l2,m> isoscalar states are investigated for both spherical and deformed even-even nuclei, and the simplest isovector wavefunction is shown to give a good description of the giant dipole resonance. In section AIII wavefunctions are proposed for compressional vibrational states of spherical nuclei. Section AIV discusses sum rules for nuclear transitions of a given electric multipolarity. It is found that the 2+ and 1- states investigated in section AII and all but one of the vibrational states discussed in AIII each exhaust a large part of the appropriate sum rule. In Part B I consider the problem of how to describe flow in quantum fluids. In particular, we want to be able to identify the physical motion represented by any given many-body wavefunction. Section BI derives a guantum mechanical velocity field for a many-body system, paying special attention to the need for a quantum continuity equation. It is found that when the wavefunction has the usual time dependence e-iwt , that the quantum velocity formula averages over all oscillatory motion, so that much of the physical nature of the flow field is lost. In section BII a particular wavefunction is proposed to represent the quantum excitation corresponding to any given potential flow field. The results obtained by considering specific examples are very encouraging. In Part C I investigate the properties of surfaces. Section CI presents a theoretical description of the tension, energy and thickness of a classical liquid-vapour interface. In section CII the classical results are extended to describe the surface of a quantum system, namely superfluid helium four. Problems occur for the quantum system if the correlations arising from the zero-point-motion of the phonon modes are included in the ground state wavefunction. Finally, in section CIII discuss generalized virial theorems that give the change in the free energy of a system undergoing an infinitesimal deformation. For example, a particular deformation gives the expression used in CII, for the surface tension of a plane quantum surface.
This thesis is a collection of theoretical investigations into different aspects of the broad subject of quantum many-body theory. The results are grouped into three main parts, which in turn are divided into separate self-contained sections. Some of the work is presented in the form of published papers and papers that have been submitted for publication. The first section of Part A introduces some of the concepts involved in many-body problems, by developing methods to evaluate expectation values of the form . In the rest of Part A I consider collective excitations of finite quantum systems. The calculations are confined to nuclei because the results can then be compared with the extensive investigations that have been made into collective nuclear modes. In Section AII, wavefunctions are proposed for rotational excitations of even-even nuclei. Both isoscalar and isovector nuclear modes are discussed. In particular, the l2,m> isoscalar states are investigated for both spherical and deformed even-even nuclei, and the simplest isovector wavefunction is shown to give a good description of the giant dipole resonance. In section AIII wavefunctions are proposed for compressional vibrational states of spherical nuclei. Section AIV discusses sum rules for nuclear transitions of a given electric multipolarity. It is found that the 2+ and 1- states investigated in section AII and all but one of the vibrational states discussed in AIII each exhaust a large part of the appropriate sum rule. In Part B I consider the problem of how to describe flow in quantum fluids. In particular, we want to be able to identify the physical motion represented by any given many-body wavefunction. Section BI derives a guantum mechanical velocity field for a many-body system, paying special attention to the need for a quantum continuity equation. It is found that when the wavefunction has the usual time dependence e-iwt , that the quantum velocity formula averages over all oscillatory motion, so that much of the physical nature of the flow field is lost. In section BII a particular wavefunction is proposed to represent the quantum excitation corresponding to any given potential flow field. The results obtained by considering specific examples are very encouraging. In Part C I investigate the properties of surfaces. Section CI presents a theoretical description of the tension, energy and thickness of a classical liquid-vapour interface. In section CII the classical results are extended to describe the surface of a quantum system, namely superfluid helium four. Problems occur for the quantum system if the correlations arising from the zero-point-motion of the phonon modes are included in the ground state wavefunction. Finally, in section CIII discuss generalized virial theorems that give the change in the free energy of a system undergoing an infinitesimal deformation. For example, a particular deformation gives the expression used in CII, for the surface tension of a plane quantum surface.