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Physica Scripta

The Path Integral Approach for the Nuclear Collective Motion

D. Galetti, Salomon Sylvain Mizrahi, B. M. Pimentel

Published Feb 1, 1983• 4 citations
Publisher

Abstract

A procedure is developed for treating a nuclear collective motion via the path integral formalism by defining a collective propagator in the "phase space" of the collective subspace. This collective subspace is projected out of the full many-body Hilbert space by an approach of the Generator Coordinate Method. Under the approximation of quadratic "momentum" dependence we write the propagator in the coordinate representation from which we derive the collective Schrödinger equation in terms of a canonical pair and .

DOI
10.1088/0031-8949/27/2/003

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Physical Review•Jul 20, 1968•21 citations
A procedure is developed for treating a nuclear collective motion via the path integral formalism by defining a collective propagator in the "phase space" of the collective subspace. This collective subspace is projected out of the full many-body Hilbert space by an approach of the Generator Coordinate Method. Under the approximation of quadratic "momentum" dependence we write the propagator in the coordinate representation from which we derive the collective Schrödinger equation in terms of a canonical pair and .
A numerical procedure is introduced which allows us to extract a collective Hamiltonian, expressed in terms of a collective canonical pair $\stackrel{^}{p}$ and $\stackrel{^}{q}$. The starting point is a microscopic many-body approach, namely the generator coordinate method, and the Weyl formalism permits the description of the collective dynamics in the $p\ensuremath{-}q$ space. As an illustration we compare numerical calculations with exact results, obtained analytically, for the quadratic energy kernel with the Gaussian overlap approximation case and the Goldhaber-Teller dipole model applied to the 4He^{4}\mathrm{He}4He nucleus.NUCLEAR STRUCTURE Numerical derivation of collective Hamiltonians. Generator coordinate method, Weyl procedure. Comparison with analytical results.
This is the third in a series of papers which intends to develop a new microscopic theory capable by itself to determine the “maximaldecoupled” collective subspace in the many-particle Hilbert space as well as the collective Hamiltonian specifying the subspace. The fundamental principle to formulate the theory is called the invariance principle of the Schrödinger equation. The main purpose of this paper is to show that, when we positively utilize the invariance principle, we can directly obtain a quantum theory of collective motion going beyond the random phase approximation.
A Quantum mechanical formalism of collective motion is proposed by using the idea of path integral method. Time evolution of many· body system is described by the propagator which joins the many-body state vectors parametrized by complex collective parameters. A general form of path integral representation for the propagator is derived in the complex parameter space. In particular the case of the coherent state representation is discussed in detail. It is pointed out that the time-dependent variation principle and especially the time dependent Hartree-Fock are naturally obtained as a classical limit.
We present a general method for constructing path intergrals for the nuclear many-body problem. This method uses continuous and overcomplete sets of vectors in the Hilbert space. The state labels play the role of classical coordinates which are quantized as bosons. The equations of motion for the classical coordinates are obtained by calculating the functional integral in the saddle point approximation. In the particular case where the over-complete set considered is the set of all Slater determinants, the classical equations of motion are the time-dependent Hartree-Fock equations. The functional integral provides a way of requantizing these classical equations. This quantization involves boson degrees of freedom and is in some cases very similar to the method of boson expansion. It is shown that the functional integral formalism provides a unifying framework to describe various approaches to the nuclear many-body problem.NUCLEAR STRUCTURE Functional integrals on continuous over-complete sets. Time-dependent Hartree and Hartree-Fock theories. Boson representations for fermion systems.
The path integral formulation of quantum mechanics constructs the propagator by evaluating the action S for all classical paths in coordinate space. A corresponding momentum path integral may also be defined through Fourier transforms in the endpoints. Although these momentum path integrals are especially simple for several special cases, no one has, to my knowledge, ever formally constructed them from all classical paths in momentum space. I show that this is possible because there exists another classical mechanics based on an alternate classical action R. Hamilton's Canonical equations result from a variational principle in both S and R. S uses fixed beginning and ending spatial points while R uses fixed beginning and ending momentum points. This alternative action's classical mechanics also includes a Hamilton-Jacobi equation. I also present some important points concerning the beginning and ending conditions on the action necessary to apply a Canonical transformation. These properties explain the failure of the Canonical transformation in the phase space path integral. It follows that a path integral may be constructed from classical position paths using S in the coordinate representation or from classical momentum paths using R in the momentum representation. Several example calculations are presented that illustrate the simplifications and practical advantages made possible by this broader view of the path integral. In particular, the normalized amplitude for a free particle is found without using the Schrodinger equation, the internal spin degree of freedom is simply and naturally derived, and the simple harmonic oscillator is calculated.
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.