Collective motion in quantum mechanics

1957 ◽  
Vol 239 (1218) ◽  
pp. 399-412 ◽  
Keyword(s):  
Collective Motion ◽  
Electron Gas ◽  
Quantum Mechanical ◽  
Particle Model ◽  
Coupling Theory ◽  
Rotating System ◽  
Plasma Oscillations ◽  
Nucleon Recoil ◽  
The Moment ◽  
General Method

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.

scholarly journals The temperature dependence of free electron susceptibility

1935 ◽  
Vol 152 (877) ◽  
pp. 672-692 ◽  
Keyword(s):  
Magnetic Field ◽  
Electron Spin ◽  
Free Electron ◽  
Low Temperatures ◽  
Electron Gas ◽  
Spin Effect ◽  
Degenerate State ◽  
Diamagnetic Contribution ◽  
The Moment ◽  
Diamagnetic Effect

One of the earliest applications of the Fermic-Dirac statistics was that of pauli to the treatment of the paramagnetism, due to the electron spin, of an electron gas. The result he obtained, for low temperatures, may be put in the form M p = 3/2 Nμ 2 H/ε 0 , where M p is the total magnetic moment due to the spin effect, N the number of electrons, μ the Bohr magneton, and ε 0 the maximum electron energy in the completely degenerate state. It was later shown by Landau that electrons, apart from the spin effect, gave a diamagnetic contribution to the susceptibility. The diamagnetic effect (which is zero on a classical basis) arises from the discreteness of the energy states of an electron in a magnetic field. For low temperatures the result obtained is M D = -½ Nμ 2 H/ε 0 , where M D is the diamagnetic contribution to the moment. The spin effect was further considered by Bloch, who gave, as a higher approximation at low temperatures, M P = 3/2 Nμ 2 H/ε 0 {1-π 2 /12( k T/ε 0 ) 2 }.

Spin Effects on the Plasma Oscillations of an Electron Gas in a Magnetic Field

1964 ◽  
Vol 133 (1A) ◽  
pp. A104-A106 ◽  
Author(s):  
S. Gartenhaus ◽  
G. Stranahan
Keyword(s):  
Magnetic Field ◽  
Electron Gas ◽  
Plasma Oscillations ◽  
Spin Effects

scholarly journals Optimality, reduction and collective motion

2015 ◽  
Vol 471 (2177) ◽  
pp. 20140606 ◽  
Author(s):  
Eric W. Justh ◽  
P. S. Krishnaprasad
Keyword(s):  
Collective Motion ◽  
Optimal Control Problems ◽  
Coupling Parameter ◽  
Rigid Motion ◽  
Particle Model ◽  
Explicit Integration ◽  
Hidden Symmetry ◽  
Necessary Conditions For Optimality ◽  
Special Solutions ◽  
Optimality Principles

The planar self-steering particle model of agents in a collective gives rise to dynamics on the N -fold direct product of SE (2), the rigid motion group in the plane. Assuming a connected, undirected graph of interaction between agents, we pose a family of symmetric optimal control problems with a coupling parameter capturing the strength of interactions. The Hamiltonian system associated with the necessary conditions for optimality is reducible to a Lie–Poisson dynamical system possessing interesting structure. In particular, the strong coupling limit reveals additional (hidden) symmetry, beyond the manifest one used in reduction: this enables explicit integration of the dynamics, and demonstrates the presence of a ‘master clock’ that governs all agents to steer identically. For finite coupling strength, we show that special solutions exist with steering controls proportional across the collective. These results suggest that optimality principles may provide a framework for understanding imitative behaviours observed in certain animal aggregations.

SUM RULE FOR MULTISCALE REPRESENTATIONS OF KINEMATICALLY DESCRIBED SYSTEMS

2002 ◽  
Vol 05 (04) ◽  
pp. 409-431 ◽  
Author(s):  
YANEER BAR-YAM
Keyword(s):  
Kinetic Energy ◽  
Degrees Of Freedom ◽  
Collective Motion ◽  
Space Time ◽  
Moment Of Inertia ◽  
Sum Rule ◽  
Effective Moment ◽  
External Frame ◽  
The Moment ◽  
Effective Moment Of Inertia

We derive a sum rule that constrains the scale based decomposition of the trajectories of finite systems of particles. The sum rule reflects a tradeoff between the finer and larger scale collective degrees of freedom. For short duration trajectories, where acceleration is irrelevant, the sum rule can be related to the moment of inertia and the kinetic energy (times a characteristic time squared). Thus, two nonequilibrium systems that have the same kinetic energy and moment of inertia can, when compared to each other, have different scales of behavior, but if one of them has larger scales of behavior than the other, it must compensate by also having smaller scales of behavior. In the context of coherence or correlation, the larger scale of behavior corresponds to the collective motion, while the smaller scales of behavior correspond to the relative motion of correlated particles. For longer duration trajectories, the sum rule includes the full effective moment of inertia of the system in space-time with respect to an external frame of reference, providing the possibility of relating the class of systems that can exist in the same space-time domain.

scholarly journals Negative Effective Mass in Plasmonic Systems

2020 ◽  
Author(s):  
Edward Bormashenko ◽  
Irina Legchenkova
Keyword(s):  
Effective Mass ◽  
Plasma Frequency ◽  
Electron Gas ◽  
Metallic Particle ◽  
Plasma Oscillations ◽  
Mechanical Coupling ◽  
Negative Effective Mass ◽  
External Frequency ◽  
Elastic Spring ◽  
Ionic Lattice

We report the negative effective mass metamaterials based on the electro-mechanical coupling exploiting plasma oscillations of a free electron gas. The negative mass appears as a result of vibration of a metallic particle with a frequency of ω which is close the frequency of the plasma oscillations of the electron gas m_2 relatively to the ionic lattice m_1. The plasma oscillations are represented with the elastic spring k_2=ω_p^2 m_2, where ω_p is the plasma frequency. Thus, the metallic particle vibrated with the external frequency ω is described by the effective mass m_eff=m_1+(m_2 ω_p^2)/(ω_p^2-ω^2 ) , which is negative when the frequency ω approaches ω_p from above. The idea is exemplified with two conducting metals, namely Au and Li.

scholarly journals Mössbauer experiments in a rotating system and physical interpretation of their results

2021 ◽  
pp. 34-43
Author(s):  
Alexander L. Kholmetskii ◽  
Tolga Yarman ◽  
Ozan Yarman ◽  
Metin Arik
Keyword(s):  
Energy Shift ◽  
Special Theory ◽  
Relativistic Energy ◽  
Theory Of Relativity ◽  
Rotating System ◽  
Rotating Systems ◽  
Extra Energy ◽  
Quantum Mechanical Description ◽  
Dilation Effect ◽  
The Moment

We discuss the results of modern Mössbauer experiments in a rotating system, which show the presence of an extra energy shift between the emitted and absorbed resonant radiation in addition to the relativistic energy shift of the resonant lines due to the time dilation effect in the co-rotating source and absorber with different radial coordinates. We analyse the available attempts to explain the origin of the extra energy shift, which include some extensions of special theory of relativity with hypothesis about the existence of limited acceleration in nature, with hypothesis about a so-called «time-dependent Doppler effect», as well as in the framework of the general theory of relativity under re-analysis of the metric effects in the rotating system, which is focused to the problem of correct synchronisation of clocks in a rotating system with a laboratory clock. We show that all such attempts remain unsuccessful until the moment, and we indicate possible ways of solving this problem, which should combine metric effects in rotating systems with quantum mechanical description of resonant nuclei confined in crystal cells.

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