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Progress of Theoretical Physics

Elementary Theory of Quantum-Mechanical Collective Motion of Particles, I

Sin‐itiro Tomonaga

Published May 1, 1955• 107 citations
Publisher

Abstract

An elementary theory of quantum-mechnaical collective motion of many-particle systems is developed. The fundamental idea is illustrated by taking an example of surface oscillation of an incompressible system of particles. The essential point of the theory is, roughly speaking, the neglect of 1/√N against unity, N being the number of particles of the system. The method is a natural generalization of separation of translational motion from the internal relative motion by the use of centre of mass coordinates. A description in terms of field concept is briefly discussed.

DOI
10.1143/ptp.13.467

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Progress of Theoretical Physics•Jan 1, 1959•15 citations
An elementary theory of quantum-mechnaical collective motion of many-particle systems is developed. The fundamental idea is illustrated by taking an example of surface oscillation of an incompressible system of particles. The essential point of the theory is, roughly speaking, the neglect of 1/√N against unity, N being the number of particles of the system. The method is a natural generalization of separation of translational motion from the internal relative motion by the use of centre of mass coordinates. A description in terms of field concept is briefly discussed.
An elementary theory is developed of a classical treatment with the ‘Collective Motion’ of many-particle systems. The fundamental concept of collective motion can be reduced into a certain transformation property of Hamiltonian. Regarded as the dynamical variables, the continuous parameters of such a transformation describe the collective motion of the system, while the individual motions are partly suppressed by the same numbers of subsidiary conditions, or constraints, logically derived in, as the numbers of transformation parameters. Then the general procedure of formulation is illustrated; as an example, the incompressible motion with three modes can be systematically formulated. As another example, the unified treatment is performed also with the vortex motion. The digression in the approximating procedure is also presumed.
Abstract A general method of discussing quantum-mechanical problems involving collective motion is proposed, in which the emphasis is placed on consideration of sets of states rather than single states, and in which the additional collective co-ordinates are not redundant but used to describe the sets. The method is applied to a number of relatively simple examples: plasma oscillations of an electron gas; some problems of nuclear structure including the α-particle model and the collective model; and to two problems in meson field theory, concerning nucleon isobars in strong-coupling theory and concerning nucleon recoil. One of the main aims is to determine as well as possible the parameters of the collective motion; in particular, a formula is given for the moment of inertia of a rotating system.
A theory of collective submanifold, recently, developed by the present authors (M. Y. and A. K.) is applied to a classical many-body system, the Hamiltonian of which depends on the relative coordinates. The solution enables the separation of the total degrees of freedom into the collective and the intrinsic ones, which may be called the generalized center of mass and the relative motions. The results can be applied to the large amplitude collective motion and its couplings with the intrinsic degrees of freedom, which cannot be given in the HF plus RPA method.
The concept of an ideal collective coordinate is introduced by means of the following example: Consider a one-dimensional vibration of a many-body system in the sense that a large subset of states ∣n〉|n〉∣n〉 of the system exhibits an energy spectrum and relative transition probabilities following the laws of the (in general anharmonic) oscillator described by H\mathcal{H}H (${p}_{\ensuremath{\alpha}}, \ensuremath{\alpha}$) $(\ensuremath{\alpha}|n)={\ensuremath{\omega}}_{n}(\ensuremath{\alpha}|n)$. We suppose the set of many-body states ∣n〉|n〉∣n〉 to extend indefinitely, and we take the transform $|\ensuremath{\alpha}〉=\ensuremath{\Sigma}|n〉$ ($n|\ensuremath{\alpha}$) to define a many-body generating state of the band which is precisely localized in $\ensuremath{\alpha}$ space. The basic assumption of collectivity, that changing the state of at most a few particles cannot much alter the value of $\ensuremath{\alpha}$, is shown to be sufficient to derive a phenomenological theory from the many-body starting point. The phenomenological aspects of a recent theory of rotations due to Villars is seen to be contained in the above formulation as a special case. A brief review is given of the generator coordinate and similar projection methods in order to exhibit their relationship with the present method.