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Concept# Free particle

Summary

In physics, a free particle is a particle that, in some sense, is not bound by an external force, or equivalently not in a region where its potential energy varies. In classical physics, this means the particle is present in a "field-free" space. In quantum mechanics, it means the particle is in a region of uniform potential, usually set to zero in the region of interest since the potential can be arbitrarily set to zero at any point in space.
Classical free particle
The classical free particle is characterized by a fixed velocity v. The momentum is given by
\mathbf{p}=m\mathbf{v}
and the kinetic energy (equal to total energy) by
E=\frac{1}{2}mv^2=\frac{p^2}{2m}
where m is the mass of the particle and v is the vector velocity of the particle.
Quantum free particle
Mathematical description
Schrödinger equation and Matter wave
A free particle with mass m in non-relativistic quantum me

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Related lectures (59)

Simon Nessim Henein, Ilan Vardi

Eye movements consist of spherical rotations, with orientation generally constrained by Listing’s law. Our main result is a complete explicit formulation of ballistic eye movement under the Listing constraint. We present a conceptual framework for eye movement bas-ing the dynamics of Listing motion on the equator of the sphere of unit quaternions, which we call the Listing sphere. Analytical dynamics shows that ballistic Listing motion corre-sponds to free particle motion on the Listing sphere. Thus, ballistic Listing movement is greatly simplified by transposition to the Listing sphere, where it consists of shortest dis-tance trajectories. This proves that ballistic eye motion consists of constant speed rotation along circles passing through the occipital point. The relevance of the occipital point in eye movement was already noted by Helmholtz, which we explain by the fact that it cor-responds to the equator of the Listing sphere. We designed a physical mechanism produc-ing the correspondence between eye movement and particle motion on the Listing sphere. Our straightforward description of ballistic eye motion under the pure Listing kinematic constraint serves as a useful idealized benchmark in the study of actual physiological eye movements, whose orientations are known to deviate slightly from the Listing constraint.

2015Jan Hugo Dil, Mauro Fanciulli, Stefan Peter Muff

After photon absorption, electrons from a dispersive band of a solid require a finite time in the photoemission process before being photoemitted as free particles, in line with recent attosecond-resolved photoemission experiments. According to the Eisenbud-Wigner-Smith model, the time delay is due to a phase shift of different transitions that occur in the process. Such a phase shift is also at the origin of the angular dependent spin polarization of the photoelectron beam, observable in spin degenerate systems without angular momentum transfer by the incident photon. We propose a semiquantitative model which permits us to relate spin and time scales in photoemission from condensed matter targets and to better understand spin-and angle-resolved photoemission spectroscopy (SARPES) experiments on spin degenerate systems. We also present the first experimental determination by SARPES of this time delay in a dispersive band, which is found to be greater than 26 as for electrons emitted from the sp-bulk band of the model system Cu(111).

We study the large deviations of the power injected by the active force for an active Ornstein-Uhlenbeck particle (AOUP), free or in a confining potential. For the free-particle case, we compute the rate function analytically in d-dimensions from a saddle-point expansion, and numerically in two dimensions by (a) direct sampling of the active work in numerical solutions of the AOUP equations and (b) Legendre-Fenchel transform of the scaled cumulant generating function obtained via a cloning algorithm. The rate function presents asymptotically linear branches on both sides and it is independent of the system's dimensionality, apart from a multiplicative factor. For the confining potential case, we focus on two-dimensional systems and obtain the rate function numerically using both methods (a) and (b). We find a different scenario for harmonic and anharmonic potentials: in the former case, the phenomenology of fluctuations is analogous to that of a free particle, but the rate function might be non-analytic; in the latter case the rate functions are analytic, but fluctuations are realised by entirely different means, which rely strongly on the particle-potential interaction. Finally, we check the validity of a fluctuation relation for the active work distribution. In the free-particle case, the relation is satisfied with a slope proportional to the bath temperature. The same slope is found for the harmonic potential, regardless of activity, and for an anharmonic potential with low activity. In the anharmonic case with high activity, instead, we find a different slope which is equal to an effective temperature obtained from the fluctuation-dissipation theorem.