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American Journal of Physics

Cranking model calculation of the moment of inertia using the method of Dalgarno and Lewis

Samuel M. Harris

Published Apr 1, 1977• 4 citations
Publisher

Abstract

We present a calculation of the moment of inertia of a system of noninteracting particles moving in a common deformed, harmonic-oscillator potential well. Our calculation employs the cranking model formula, but instead of performing the usual sum over intermediate states, we use the method of Dalgarno and Lewis to replace that sum by an expectation value of appropriately defined operators. It is suggested that this calculation may serve as a useful example in a quantum mechanics or nuclear theory course.

DOI
10.1119/1.10847

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Canadian Journal of Physics•Apr 1, 1978•22 citations
We present a calculation of the moment of inertia of a system of noninteracting particles moving in a common deformed, harmonic-oscillator potential well. Our calculation employs the cranking model formula, but instead of performing the usual sum over intermediate states, we use the method of Dalgarno and Lewis to replace that sum by an expectation value of appropriately defined operators. It is suggested that this calculation may serve as a useful example in a quantum mechanics or nuclear theory course.
The effects of possible superfluidity on the cranking moment of a large many-fermion system moving under periodic boundary conditions are investigated within the framework of the theory of superconductivity recently formulated by Bogolyubov. The Hamiltonian is initially subjected to Bogolyubov's general unitary transformation. The collective excitations of the fermions are then considered in the usual pair approximation; the appropriate cranking terms are linear in the boson pair operators. On performing a unitary transformation which transforms away these linear terms, one obtains an expression for the moment of inertia of the system which includes both the effects of possible superfluidity and collective excitation. This expression, by virtue of its being stationary with respect to arbitrary variations in the amplitude associated with the latter unitary transformation, is then utilized as a variational principle for the moment of inertia. For the normal state, the result previously obtained by the author, that the moment of inertia has the rigid value, is rederived in more compact form. For the superfluid state, one finds that collective excitations effect a marked increase in the superfluid moment at intermediate coupling strengths although the resulting moment is still quite small compared to the rigid value. In contrast to the normal state case, where particle-hole pairs play a major role, this increase is almost entirely due to excitations consisting of particle-pairs or hole-pairs. The precise magnitude of the apparent resonance in the moment produced by the ddd-wave part of the cranking interaction is dependent to some extent on the features of the particle-particle potential which leads to the superfluid state. Variational expressions for the moment are exhibited for both Yukawa and delta-function shell potentials. These results are identical in charged and neutral Fermi systems. A calculation of the cranking moment at finite temperatures is presented in an Appendix along with an interpretation of it in terms of Bardeen's two-fluid model of superconductivity.
It is shown that the moment of inertia of a noninteracting many-body fermion system moving under periodic boundary conditions has the classical or rigid value when calculated on the "cranking" model of Inglis. By investigating the analogous "pushing" case for the inertial mass we show that the rigid moment can be associated with rigid rotation in spite of apparent surface currents. The effect of particle-particle forces is investigated in the lowest order of perturbation theory. The terms corresponding to a level shift or effective mass are just compensated by other terms and there is no change in the moment of inertia. The possible general validity of these results and their consequence is discussed.
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.
It is shown that in the random-phase approximation the effect of particle-particle interaction on the moment of inertia of a many-fermion system moving under periodic boundary conditions vanishes when calculated on the "cranking" model of Inglis. This result is obtained by performing a unitary transformation on the equivalent Hamiltonian which reproduces the sequence of diagrams given by the random-phase approximation. The close resemblance of the general problem to the soluble model problem of a rotating dense electron gas is exploited in the calculation. This work extends that of Amado and Brueckner who have demonstrated only that interaction effects cancel in lowest order. The nature of their approximations and the connection to the problem of collective excitations are discussed.
The Hamiltonian describing a system of particles in a rotating coordinate system is derived and it is shown that the simple classical solution of rigid flow is forbidden in quantum mechanics, even at very low angular velocities. This effect is closely parallel to the Aharonov–Bohm effect, which likewise has its origin in the single-valuedness requirement of the wave function. An analytical approach to perturbation theory is used to include the effects of the Coriolis and centrifugal forces and to derive the current flows for some independent-particle systems, that is, for the Inglis cranking model. It is shown, by explicit construction, that the currents are not rigid even when the moment of inertia assumes the rigid-flow value, as it does for the harmonic oscillator single-particle potential under conditions of self-consistency. Furthermore, it is shown that, for a more general potential, even the moment of inertia is not rigid.